Category O Finiteness Duality and Blocks

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

The classical BGG category O

Definition

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Let ${α_{i}}_{i \in I}$ be the simple roots of the chosen positive system and write $Q^{+} = \sum_{i \in I} Z_{\geq 0} α_{i}$ and $μ \leq λ$ when $λ - μ \in Q^{+}$ . Let $W$ be the Weyl group of Root reflections and the Weyl group action and let $ρ$ be the Weyl vector of The Weyl vector rho for a chosen positive system; for $w \in W$ set $w \cdot λ = w (λ + ρ) - ρ$ .

The BGG category $O$ is the full subcategory of left $U (g)$ -modules $M$ satisfying all three conditions: $M$ is finitely generated; $M = ⨁_{μ \in h^{*}} M_{μ}$ where $M_{μ} = {v : h v = μ (h) v for all h \in h}$ ; and $U (n^{+}) v$ is finite dimensional for each $v \in M$ . Its morphisms are all $g$ -linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of PBW gives an ordered monomial basis for the enveloping algebra and Triangular decomposition from a chosen positive root system.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Noetherianity of the enveloping algebra

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

For every finite-dimensional complex Lie algebra $g$ (semisimplicity is unnecessary here), $U (g)$ is left and right Noetherian.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Let $x_{1}, \dots, x_{r}$ be an ordered basis of a finite-dimensional complex Lie algebra $g$ . Then the monomials $x_{1}^{a_{1}} x_{2}^{a_{2}} \dots x_{r}^{a_{r}} (a_{i} \in N_{0})$ form a basis of $U (g)$ . In particular, multiplication identifies $gr U (g)$ with the symmetric algebra $S (g)$ on the symbols of the $x_{i}$ . (PBW gives an ordered monomial basis for the enveloping algebra)

[F2]

Let $R$ be a Noetherian commutative ring. Then the polynomial ring $R [x]$ is a Noetherian commutative ring. No hypothesis beyond Noetherianity is placed on $R$ : it may have zero divisors, and it may be the zero ring. (Hilbert basis theorem: if $R$ is Noetherian then $R [x]$ is Noetherian)

Proof

1.1

Put $A = U (g)$ with its nonnegative PBW filtration. Then $gr A ≅ C [X_{1}, \dots, X_{d}]$ . A field is Noetherian since its only ideals are $0$ and itself; iterating the Hilbert basis theorem makes this polynomial ring Noetherian, also when $d = 0$ .

F1F2

2.1

For a left ideal $I$ , $gr I$ is a homogeneous ideal of $gr A$ . Choose finite homogeneous generators $σ (a_{j})$ and lift them to $a_{j} \in I$ of the same degrees. Homogeneous generators may be chosen by replacing generators of a homogeneous ideal by their homogeneous components. For $I = 0$ use the empty list.

constructstep 1.1

3.1

If $a \in I$ has degree $n$ , write its symbol as $\sum_{j} b_{j} σ (a_{j})$ with homogeneous $b_{j}$ of degree $n - \deg a_{j}$ , omitting negative degrees. Lift $b_{j}$ to $c_{j} \in A$ . Then $a - \sum_{j} c_{j} a_{j}$ has degree less than $n$ . Repetition terminates below degree zero, proving $I = \sum_{j} A a_{j}$ . This includes degree-zero elements.

algebrastep 2.1

4.1

For a right ideal use $gr I = \sum_{j} σ (a_{j}) b_{j}$ and subtract $\sum_{j} a_{j} c_{j}$ instead. Thus every left and every right ideal is finitely generated. An ascending chain stabilizes because its union is an ideal and its finitely many generators already belong to one member.

algebrastep 3.1∎

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Finite Borel-stable generators and weight flags

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Every $M \in O$ has a finite-dimensional, $b$ -stable, $h$ -semisimple generating subspace $E$ . There is a flag $0 = E_{0} \subset E_{1} \subset \dots \subset E_{r} = E$ of $b$ -submodules whose quotients are one dimensional and annihilated by $n^{+}$ . For $M = 0$ take $E = 0$ and the empty flag.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The BGG category $O$ is the full subcategory of left $U (g)$ -modules $M$ satisfying all three conditions: $M$ is finitely generated; $M = ⨁_{μ \in h^{*}} M_{μ}$ where $M_{μ} = {v : h v = μ (h) v for all h \in h}$ ; and $U (n^{+}) v$ is finite dimensional for each $v \in M$ . Its morphisms are all $g$ -linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

Proof

1.1

Choose finitely many generators and replace them by their finitely many weight components $v_{1}, \dots, v_{s}$ . They still generate $M$ . Set $E = \sum_{i} U (n^{+}) v_{i}$ . Local finiteness makes $E$ finite dimensional. For a weight vector $v_{i}$ , root monomials applied to $v_{i}$ are weight vectors; hence $E$ is $h$ -stable and $h$ -semisimple as well as $n^{+}$ -stable. For $M = 0$ choose no generators.

F1construct

2.1

If $E \neq 0$ , its finite weight set has a maximal element for the positive-root order. A nonzero vector at that weight spans a $b$ -stable line: every positive-root operator raises the weight and therefore kills it. The quotient by that line remains finite dimensional and a weight module, since weight components descend.

choosestep 1.1

3.1

Repeat the preceding construction in the quotient and take inverse images of the resulting flag. Dimension drops by one each time, so the process ends at zero and gives exactly $r = \dim E$ one-dimensional quotients. When $E = 0$ there is no step to perform.

algebrastep 2.1∎

PropositionStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

The support description of category O with finite generation

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Let $M$ be a finitely generated $h$ -semisimple $g$ -module. Then $M \in O$ if and only if $supp M \subset ⋃_{i = 1}^{r} (λ_{i} - Q^{+})$ for some finite list of weights. In either case every $M_{μ}$ is finite dimensional. The list may be empty for $M = 0$ ; finite generation is an independent hypothesis.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The BGG category $O$ is the full subcategory of left $U (g)$ -modules $M$ satisfying all three conditions: $M$ is finitely generated; $M = ⨁_{μ \in h^{*}} M_{μ}$ where $M_{μ} = {v : h v = μ (h) v for all h \in h}$ ; and $U (n^{+}) v$ is finite dimensional for each $v \in M$ . Its morphisms are all $g$ -linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every $M \in O$ has a finite-dimensional, $b$ -stable, $h$ -semisimple generating subspace $E$ . There is a flag $0 = E_{0} \subset E_{1} \subset \dots \subset E_{r} = E$ of $b$ -submodules whose quotients are one dimensional and annihilated by $n^{+}$ . For $M = 0$ take $E = 0$ and the empty flag. (Finite Borel-stable generators and weight flags)

[F3]

Let $x_{1}, \dots, x_{r}$ be an ordered basis of a finite-dimensional complex Lie algebra $g$ . Then the monomials $x_{1}^{a_{1}} x_{2}^{a_{2}} \dots x_{r}^{a_{r}} (a_{i} \in N_{0})$ form a basis of $U (g)$ . In particular, multiplication identifies $gr U (g)$ with the symmetric algebra $S (g)$ on the symbols of the $x_{i}$ . (PBW gives an ordered monomial basis for the enveloping algebra)

Proof

1.1

Assume $M \in O$ and choose the finite $b$ -stable weight space sum $E$ supplied by F2 (a finite-dimensional weight subspace, not necessarily a sum of full weight spaces). Ordering negative-root vectors before a basis of $b$ in PBW gives $M = U (n^{-}) E$ . Its support is contained in the union of $ν - Q^{+}$ over the finitely many weights $ν$ of $E$ .

F1F2F3

2.1

For a fixed $β \in Q^{+}$ , negative-root PBW monomials of weight $- β$ have exponent sum bounded by the height of $β$ , since every positive root has positive integral height. There are finitely many such monomials. The surjection $U (n^{-}) \otimes E \to M$ therefore has finite-dimensional weight spaces in its source, giving $\dim M_{μ} < \infty$ .

F3algebrastep 1.1

3.1

Conversely assume the finite-cone support condition. For a weight vector $v \in M_{μ}$ , a positive-root PBW monomial of weight $γ \in Q^{+}$ can act nontrivially only if $μ + γ = λ_{i} - β$ for some $i$ and $β \in Q^{+}$ . Thus $γ + β = λ_{i} - μ \in Q^{+}$ . For each eligible $i$ , the height of $γ$ is bounded by that fixed height. Only finitely many PBW monomials meet these bounds, so $U (n^{+}) v$ is finite dimensional. This does not presume finite-dimensional weight spaces.

F3algebrastep 2.1

4.1

Every vector is a finite sum of weight vectors, so the same local finiteness follows for it by summing their finite-dimensional orbit spaces. The two standing hypotheses now give $M \in O$ , and the preceding forward proof gives finite weight spaces. Empty support means $M = 0$ and all conclusions hold.

F1algebrastep 3.1∎

PropositionStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Verma and finite-dimensional weight modules belong to O

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Every Verma module $M (λ)$ , every quotient of it, and every finite-dimensional $h$ -semisimple $g$ -module belongs to $O$ .

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The BGG category $O$ is the full subcategory of left $U (g)$ -modules $M$ satisfying all three conditions: $M$ is finitely generated; $M = ⨁_{μ \in h^{*}} M_{μ}$ where $M_{μ} = {v : h v = μ (h) v for all h \in h}$ ; and $U (n^{+}) v$ is finite dimensional for each $v \in M$ . Its morphisms are all $g$ -linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Let $M$ be a finitely generated $h$ -semisimple $g$ -module. Then $M \in O$ if and only if $supp M \subset ⋃_{i = 1}^{r} (λ_{i} - Q^{+})$ for some finite list of weights. In either case every $M_{μ}$ is finite dimensional. The list may be empty for $M = 0$ ; finite generation is an independent hypothesis. (The support description of category O with finite generation)

[F3]

The weights of $M (λ)$ are exactly $λ - β$ for $β \in Q^{+}$ ; every weight space is finite dimensional, and $M (λ)_{λ} = C v_{λ}$ . (Weights of a Verma module lie below lambda)

Proof

1.1

The Verma weight formula gives a weight decomposition supported in $λ - Q^{+}$ ; it is cyclic on its highest vector. Hence the support criterion puts it in $O$ . A quotient is still cyclic, is still a weight module, and has support in the same cone: on the finite set of components of any vector, polynomial interpolation in Cartan elements isolates each component of a stable subspace. Thus subspaces and quotients inherit weight decompositions.

F2F3

2.1

A finite-dimensional weight module is generated by any vector-space basis, and every $U (n^{+})$ orbit is contained in this finite-dimensional space. All three axioms hold. The zero quotient and the zero finite-dimensional module use the empty generating set.

F1step 1.1∎

TheoremStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Category O is abelian and extension closed among weight modules

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

The category $O$ is closed under submodules, quotients and finite direct sums and is an abelian category. If $0 \to A \to E \to B \to 0$ is exact, $A, B \in O$ , and $E$ is $h$ -semisimple, then $E \in O$ . The middle-term weight hypothesis is essential.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The BGG category $O$ is the full subcategory of left $U (g)$ -modules $M$ satisfying all three conditions: $M$ is finitely generated; $M = ⨁_{μ \in h^{*}} M_{μ}$ where $M_{μ} = {v : h v = μ (h) v for all h \in h}$ ; and $U (n^{+}) v$ is finite dimensional for each $v \in M$ . Its morphisms are all $g$ -linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Let $M$ be a finitely generated $h$ -semisimple $g$ -module. Then $M \in O$ if and only if $supp M \subset ⋃_{i = 1}^{r} (λ_{i} - Q^{+})$ for some finite list of weights. In either case every $M_{μ}$ is finite dimensional. The list may be empty for $M = 0$ ; finite generation is an independent hypothesis. (The support description of category O with finite generation)

[F3]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . For every finite-dimensional complex Lie algebra $g$ (semisimplicity is unnecessary here), $U (g)$ is left and right Noetherian. (Noetherianity of the enveloping algebra)

[F4]

For every ring $R$ , the category $R - M o d$ of left $R$ -modules is an abelian category. (Modules over a ring form an abelian category)

[F5]

For any ordered basis of a finite-dimensional complex Lie algebra, the corresponding ordered monomials form a basis of its enveloping algebra. (PBW gives an ordered monomial basis for the enveloping algebra)

[F6]

For $λ \in h^{*}$ , the Verma module is $M (λ) = U (g) \otimes_{U (b)} C_{λ}$ . (Verma modules)

Proof

1.1

Write $R = U (g)$ . Every submodule of $R^{r}$ is finitely generated: for $r = 1$ this is left Noetherianity; for $r > 1$ project to the last coordinate, lift finite generators of the image ideal, and adjoin generators of the kernel, a submodule of $R^{r - 1}$ . The case $r = 0$ is zero. Taking inverse images under $R^{r} ↠ M$ proves that every submodule of a finitely generated module is finitely generated.

F3algebra

2.1

A submodule of a weight module is a sum of weight spaces: for the finitely many components of a vector choose $h \in h$ separating their distinct weights and apply the interpolation polynomials in $h$ . Quotients therefore inherit weight decompositions, and exact sequences of weight modules are exact at each weight. Submodules and quotients inherit local $n^{+}$ -finiteness, and quotients inherit finite generation. Finite direct sums inherit all three axioms, including the empty sum.

F1algebrastep 1.1

3.1

Kernels and cokernels of $O$ -maps are thus the ambient module kernels and cokernels. The full subcategory contains zero and finite biproducts; the ambient image-coimage isomorphism remains inside it. The module-category abelian axioms therefore hold in $O$ .

F4step 2.1

4.1

For the asserted extension, choose generators of $A$ and lifts of finitely many generators of $B$ . They generate $E$ , since subtracting their $R$ -linear combinations leaves an element of $A$ . Weightwise exactness gives $supp E = supp A \cup supp B$ . Each is in finitely many downward cones; the assumed weight decomposition and finite generation of $E$ allow F2 to be applied. Thus $E \in O$ , also when either end is zero.

F2algebrastep 3.1

5.1

The weight hypothesis cannot be omitted. For $g = {s l}_{2}$ , let $V = C v \oplus C w$ be a $b$ -module with $e v = e w = 0$ , $h v = λ v$ , and $h w = λ w + v$ . Then $0 \to C_{λ} \to V \to C_{λ} \to 0$ is exact. Ordering a PBW basis with $f$ before a basis of $b$ shows that $U (g)$ is free as a right $U (b)$ -module, so induction is exact and gives $0 \to M (λ) \to E \to M (λ) \to 0$ for $E = U (g) \otimes_{U (b)} V$ . Each end lies in $O$ : its PBW basis $f^{k} v_{λ}$ consists of weight vectors, it is cyclic, and $e$ acts locally nilpotently. But PBW also gives $1 \otimes v \neq 0$ in $E$ , while $(h - λ) (1 \otimes w) = 1 \otimes v$ and $(h - λ)^{2} (1 \otimes w) = 0$ . Thus $h$ is not semisimple on $E$ , so $E \notin O$ .

F1F5F6algebra∎

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

A nonzero O-object has a highest-weight vector

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Every nonzero $M \in O$ contains a nonzero weight vector killed by $n^{+}$ .

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Let $M$ be a finitely generated $h$ -semisimple $g$ -module. Then $M \in O$ if and only if $supp M \subset ⋃_{i = 1}^{r} (λ_{i} - Q^{+})$ for some finite list of weights. In either case every $M_{μ}$ is finite dimensional. The list may be empty for $M = 0$ ; finite generation is an independent hypothesis. (The support description of category O with finite generation)

Proof

1.1

Choose a weight $μ$ in the nonempty support. Above $μ$ , each containing cone $λ_{i} - Q^{+}$ has only finitely many possibilities: $μ \leq ν \leq λ_{i}$ implies $ν - μ, λ_{i} - ν \in Q^{+}$ and their sum is fixed. Bounding simple-root coefficients makes this set finite. Thus the support above $μ$ is a nonempty finite poset.

F1choose

2.1

Choose a maximal element $ν$ of that poset and $0 \neq v \in M_{ν}$ . Every positive-root operator sends $v$ to weight $ν + α$ , which would still be above $μ$ but is absent by maximality. All such operators kill $v$ , hence $n^{+} v = 0$ .

choosestep 1.1∎

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

The simple objects of O

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

The simple objects of $O$ are exactly the modules $L (λ)$ , $λ \in h^{*}$ , and $L (λ) ≅ L (μ)$ if and only if $λ = μ$ . Simplicity here excludes zero.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The category $O$ is closed under submodules, quotients and finite direct sums and is an abelian category. If $0 \to A \to E \to B \to 0$ is exact, $A, B \in O$ , and $E$ is $h$ -semisimple, then $E \in O$ . The middle-term weight hypothesis is essential. (Category O is abelian and extension closed among weight modules)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every nonzero $M \in O$ contains a nonzero weight vector killed by $n^{+}$ . (A nonzero O-object has a highest-weight vector)

[F3]

For a $g$ -module $V$ , sending a homomorphism $T : M (λ) \to V$ to $T (v_{λ})$ is a bijection onto the vectors $v \in V$ of weight $λ$ annihilated by $n^{+}$ . Here $M (λ)$ is def-verma-module. The nonzero vectors in this target are precisely the highest-weight vectors of weight $λ$ from def-highest-weight-vector-and-cyclic-highest-weight-module; the zero vector corresponds to the zero homomorphism. (The universal property of Verma modules)

[F4]

The proper submodule $J (λ)$ which is the sum of all proper submodules is the unique maximal submodule of $M (λ)$ . The quotient $L (λ) : = M (λ) / J (λ)$ is simple and is its unique simple quotient. (A Verma module has a unique simple quotient)

[F5]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every Verma module $M (λ)$ , every quotient of it, and every finite-dimensional $h$ -semisimple $g$ -module belongs to $O$ . (Verma and finite-dimensional weight modules belong to O)

Proof

1.1

For a nonzero simple object $S$ , closure under submodules means simplicity in $O$ is the same as module simplicity. Choose a highest-weight vector of weight $λ$ in $S$ . The Verma universal property yields a nonzero map $M (λ) \to S$ , necessarily surjective. Its unique simple quotient identifies $S$ with $L (λ)$ .

F1F2F3F4

2.1

Conversely $L (λ)$ belongs to $O$ and is simple as a module, hence as an object of this full subcategory. Its highest line survives the Verma quotient: killing that generator kills the whole quotient. Its other weights are below $λ$ . These weight facts also follow directly from the induced Verma construction.

F4F5step 1.1

3.1

An isomorphism $L (λ) ≅ L (μ)$ preserves weights, so the two highest weights give $λ \leq μ$ and $μ \leq λ$ . The positive root cone is pointed, so $λ = μ$ . Conversely equal labels give the same quotient up to its defining isomorphism.

algebrastep 2.1∎

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Finite filtrations by highest-weight quotients

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Every $M \in O$ is a quotient of a module with a finite Verma flag. Consequently it has a finite filtration whose nonzero factors are quotients of Verma modules, and is finitely generated over $U (n^{-})$ . The empty filtration is allowed for zero. No truncation hypothesis is needed, and a Verma flag of $M$ itself is not asserted.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every $M \in O$ has a finite-dimensional, $b$ -stable, $h$ -semisimple generating subspace $E$ . There is a flag $0 = E_{0} \subset E_{1} \subset \dots \subset E_{r} = E$ of $b$ -submodules whose quotients are one dimensional and annihilated by $n^{+}$ . For $M = 0$ take $E = 0$ and the empty flag. (Finite Borel-stable generators and weight flags)

[F2]

Let $x_{1}, \dots, x_{r}$ be an ordered basis of a finite-dimensional complex Lie algebra $g$ . Then the monomials $x_{1}^{a_{1}} x_{2}^{a_{2}} \dots x_{r}^{a_{r}} (a_{i} \in N_{0})$ form a basis of $U (g)$ . In particular, multiplication identifies $gr U (g)$ with the symmetric algebra $S (g)$ on the symbols of the $x_{i}$ . (PBW gives an ordered monomial basis for the enveloping algebra)

[F3]

For a $g$ -module $V$ , sending a homomorphism $T : M (λ) \to V$ to $T (v_{λ})$ is a bijection onto the vectors $v \in V$ of weight $λ$ annihilated by $n^{+}$ . Here $M (λ)$ is def-verma-module. The nonzero vectors in this target are precisely the highest-weight vectors of weight $λ$ from def-highest-weight-vector-and-cyclic-highest-weight-module; the zero vector corresponds to the zero homomorphism. (The universal property of Verma modules)

[F4]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The category $O$ is closed under submodules, quotients and finite direct sums and is an abelian category. If $0 \to A \to E \to B \to 0$ is exact, $A, B \in O$ , and $E$ is $h$ -semisimple, then $E \in O$ . The middle-term weight hypothesis is essential. (Category O is abelian and extension closed among weight modules)

Proof

1.1

Choose the finite $b$ -stable generating subspace $E$ and its one-dimensional weight flag. PBW, with negative roots ordered first, makes $U (g)$ free as a right $U (b)$ -module. Tensoring with this right free module preserves injections and surjections, since it is a direct sum of copies of the input vector spaces.

F1F2

2.1

Inducing the flag therefore gives a filtration of $P = U (g) \otimes_{U (b)} E$ with factors $M (λ_{i})$ . The map $u \otimes e \mapsto u e$ is well defined and surjective onto $M$ . Images of the flag give a filtration of $M$ whose factors are quotients of the corresponding Vermas; delete repeated images to retain only nonzero factors. These are subquotients in the abelian category.

F3F4constructstep 1.1

3.1

PBW also identifies $P$ with $U (n^{-}) \otimes E$ as a left $U (n^{-})$ -module. A basis of $E$ is a finite generating set for this free module, and its image generates $M$ over $U (n^{-})$ . For $M = 0$ , choose $E = P = 0$ and no factors.

F2algebrastep 2.1∎

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

The center has finite-dimensional image on each O-object

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

For $M \in O$ and $Z = Z (U (g))$ , the image algebra $Z / {Ann}_{Z} (M)$ is finite dimensional over $C$ .

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Let $M$ be a finitely generated $h$ -semisimple $g$ -module. Then $M \in O$ if and only if $supp M \subset ⋃_{i = 1}^{r} (λ_{i} - Q^{+})$ for some finite list of weights. In either case every $M_{μ}$ is finite dimensional. The list may be empty for $M = 0$ ; finite generation is an independent hypothesis. (The support description of category O with finite generation)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The BGG category $O$ is the full subcategory of left $U (g)$ -modules $M$ satisfying all three conditions: $M$ is finitely generated; $M = ⨁_{μ \in h^{*}} M_{μ}$ where $M_{μ} = {v : h v = μ (h) v for all h \in h}$ ; and $U (n^{+}) v$ is finite dimensional for each $v \in M$ . Its morphisms are all $g$ -linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

Proof

1.1

Choose finitely many weight generators of $M$ and let $E$ be the sum of the full weight spaces at their distinct weights. Finite generation permits this choice and the support theorem makes $E$ finite dimensional. Every central element commutes with $h$ , hence preserves $E$ .

F1F2choose

2.1

Restriction gives an algebra homomorphism $Z \to {End}_{C} (E)$ . If $z$ kills $E$ , then $z (u e) = u (z e) = 0$ for every $u \in U (g)$ , so $z$ kills $M$ . Its kernel is exactly ${Ann}_{Z} (M)$ . The image algebra therefore embeds in a finite-dimensional endomorphism space. For $M = 0$ the quotient is the zero algebra.

algebrastep 1.1∎

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Generalized central-character subcategories

Definition

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Let $Z = Z (U (g))$ and let $χ : Z \to C$ be a unital complex-algebra character, as in Central character of a Lie algebra module. Put $m_{χ} = \ker χ$ and, for $M$ in The classical BGG category O, define

$M_{χ} = {v \in M : m_{χ}^{N} v = 0 for some integer N \geq 1} .$

Here $m_{χ}^{N} v = 0$ means every element of that ideal kills $v$ ; the exponent may initially depend on $v$ . The full subcategory $O_{χ}$ consists of the objects with $M = M_{χ}$ . This is a generalized central-character condition, weaker than scalar central action. It does not by definition assert that $O_{χ}$ is an indecomposable block.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Generalized central-character summands

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Every $M \in O$ decomposes canonically into finitely many nonzero generalized central-character submodules:

$M = ⨁_{χ} M_{χ} .$

For each summand there is a single $N \geq 1$ such that $m_{χ}^{N} M_{χ} = 0$ . The decomposition of zero is empty.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . For $M \in O$ and $Z = Z (U (g))$ , the image algebra $Z / {Ann}_{Z} (M)$ is finite dimensional over $C$ . (The center has finite-dimensional image on each O-object)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Let $Z = Z (U (g))$ and let $χ : Z \to C$ be a unital complex-algebra character, as in def-central-character-of-a-lie-algebra-module. Put $m_{χ} = \ker χ$ and, for $M$ in def-bgg-category-o, define $M_{χ} = {v \in M : m_{χ}^{N} v = 0 for some integer N \geq 1} .$ Here $m_{χ}^{N} v = 0$ means every element of that ideal kills $v$ ; the exponent may initially depend on $v$ . The full subcategory $O_{χ}$ consists of the objects with $M = M_{χ}$ . This is a generalized central-character condition, weaker than scalar central action. It does not by definition assert that $O_{χ}$ is an indecomposable block. (Generalized central-character subcategories)

[F3]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The category $O$ is closed under submodules, quotients and finite direct sums and is an abelian category. If $0 \to A \to E \to B \to 0$ is exact, $A, B \in O$ , and $E$ is $h$ -semisimple, then $E \in O$ . The middle-term weight hypothesis is essential. (Category O is abelian and extension closed among weight modules)

Proof

1.1

If $M = 0$ there is nothing to split. Otherwise let $R = Z / {Ann}_{Z} (M)$ , a nonzero finite-dimensional commutative algebra. Choose a vector-space basis $a_{1}, \dots, a_{d}$ of $R$ and consider commuting multiplication operators on its regular representation $R$ .

F1construct

2.1

For each $a_{i}$ , its minimal polynomial splits over $C$ into powers of distinct linear factors. Bezout identities for these relatively prime factors produce polynomial projections onto their generalized eigenspaces: the polynomials are 1 modulo one factor and 0 modulo every other, hence are orthogonal idempotents summing to 1 when evaluated at $a_{i}$ . Multiply these commuting projections for all $i$ and omit zero products. We obtain nonzero orthogonal elements $e_{t} \in R$ summing to 1 and ideals $R_{t} = e_{t} R$ .

algebrastep 1.1

3.1

On $R_{t}$ , multiplication by each $a_{i}$ has one eigenvalue $c_{i t}$ and $a_{i} e_{t} - c_{i t} e_{t}$ is nilpotent. The ideal $J_{t}$ generated by these finitely many commuting nilpotents is nilpotent: if their nilpotence exponents are $n_{i}$ , any product of more than $\sum_{i} (n_{i} - 1)$ generators vanishes. Since the $a_{i}$ span $R$ , $R_{t} / J_{t}$ is spanned by $e_{t}$ ; it is nonzero because a nilpotent ideal cannot contain its nonzero unit. Thus this quotient is $C$ and gives a unique character $χ_{t} : R \to C$ on that factor.

algebrastep 2.1

4.1

The $e_{t}$ act centrally on $M$ , so $M = ⨁_{t} e_{t} M$ as $g$ -modules, with each summand in $O$ . A fixed power of $\ker χ_{t}$ kills $e_{t} M$ by the nilpotence just proved, where $χ_{t}$ is composed with $Z \to R$ . For a different character $ψ$ of $Z$ , choose $z$ with $ψ (z) \neq χ_{t} (z)$ . On $e_{t} M$ , $z - ψ (z)$ is a nonzero scalar plus a nilpotent operator, hence is invertible by a finite geometric series. It cannot kill a nonzero vector to any power.

F2F3algebrastep 3.1

5.1

It follows that the intrinsic submodule $M_{χ_{t}}$ is exactly $e_{t} M$ and all other $M_{χ}$ vanish. This proves independence from the chosen basis and projections, as well as the common annihilating power on each summand.

F2algebrastep 4.1∎

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Generalized central-character decomposition of O

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

The category is the categorical direct sum $O = ⨁_{χ} O_{χ}$ : objects have finite support in the index $χ$ , morphisms between distinct components vanish, and the canonical component projections are exact.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every $M \in O$ decomposes canonically into finitely many nonzero generalized central-character submodules: $M = ⨁_{χ} M_{χ} .$ For each summand there is a single $N \geq 1$ such that $m_{χ}^{N} M_{χ} = 0$ . The decomposition of zero is empty. (Generalized central-character summands)

Proof

1.1

Use the finite intrinsic decomposition of each object. If $f : M \to N$ is a module map, $m_{χ}^{N} v = 0$ implies $m_{χ}^{N} f (v) = 0$ , so $f (M_{χ}) \subset N_{χ}$ . Consequently all off-diagonal components of a map vanish, and maps between objects decompose uniquely into their same-character components.

F1

2.1

For an exact sequence $0 \to A \to B \to C \to 0$ , the image and kernel equalities restrict to each character. In particular, to lift $c \in C_{χ}$ , lift it to $b \in B$ and decompose $b = \sum_{ψ} b_{ψ}$ . Preservation of characters and the direct decomposition of $C$ imply that $b_{χ}$ maps to $c$ . This proves surjectivity and hence exactness of every projection.

F1algebrastep 1.1

3.1

The functor taking a finitely supported family to its direct sum and the functor $M \mapsto (M_{χ})_{χ}$ are inverse up to the canonical isomorphisms. An empty family gives zero, and a single nonzero component is fixed by its projection.

F1algebrastep 2.1∎

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Finite weight-space detection of subquotients

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Suppose $M \in O_{χ_{λ}}$ . Every nonzero subquotient $T$ of $M$ has $T_{μ} \neq 0$ for some $μ \in W \cdot λ$ . In particular the number of strict inclusions in any finite chain of submodules of $M$ is at most

$d_{λ} (M) = \sum_{μ \in W \cdot λ} \dim M_{μ},$

where distinct weights in the orbit are counted once.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The category $O$ is closed under submodules, quotients and finite direct sums and is an abelian category. If $0 \to A \to E \to B \to 0$ is exact, $A, B \in O$ , and $E$ is $h$ -semisimple, then $E \in O$ . The middle-term weight hypothesis is essential. (Category O is abelian and extension closed among weight modules)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every nonzero $M \in O$ contains a nonzero weight vector killed by $n^{+}$ . (A nonzero O-object has a highest-weight vector)

[F3]

Let $χ_{λ}$ and $χ_{μ}$ be the central characters obtained from highest weights $λ$ and $μ$ . Then $χ_{λ} = χ_{μ} if and only if μ \in W \cdot λ,$ where $W \cdot λ : = {w (λ + ρ) - ρ : w \in W}$ . (Central characters are dot-Weyl orbits)

[F4]

Every central element acts on a cyclic highest-weight module by a scalar. In particular, each cyclic highest-weight module has a well-defined central character in the sense of def-central-character-of-a-lie-algebra-module. (Central elements act by scalars on cyclic highest-weight modules)

[F5]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every $M \in O$ decomposes canonically into finitely many nonzero generalized central-character submodules: $M = ⨁_{χ} M_{χ} .$ For each summand there is a single $N \geq 1$ such that $m_{χ}^{N} M_{χ} = 0$ . The decomposition of zero is empty. (Generalized central-character summands)

[F6]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Let $M$ be a finitely generated $h$ -semisimple $g$ -module. Then $M \in O$ if and only if $supp M \subset ⋃_{i = 1}^{r} (λ_{i} - Q^{+})$ for some finite list of weights. In either case every $M_{μ}$ is finite dimensional. The list may be empty for $M = 0$ ; finite generation is an independent hypothesis. (The support description of category O with finite generation)

[F7]

Let a cyclic highest-weight module have highest vector of weight $μ$ . Then every $z \in Z (U (g))$ acts by the scalar $pr (z) (μ) = χ_{μ} (z)$ . (The Harish-Chandra projection computes the highest-weight scalar)

Proof

1.1

By closure, a nonzero subquotient $T$ is in $O$ . Choose a nonzero highest-weight vector $v \in T_{μ}$ . A common power of $m_{χ_{λ}}$ kills $M$ and hence $T$ . On the cyclic highest-weight module $U (g) v$ , F4 gives scalar central action and F7 identifies its scalar as $χ_{μ} (z)$ . Thus $(z - χ_{λ} (z))^{N} v = 0$ forces $χ_{μ} (z) = χ_{λ} (z)$ for each $z \in Z$ .

F1F2F4F5F7

2.1

The exact central-character criterion now gives $μ \in W \cdot λ$ . The Weyl group is finite, and all weight spaces of an $O$ object are finite dimensional, so the displayed detector is finite. For a short exact sequence of weight modules, taking any fixed weight is exact (decompose a lift into weight components). Thus $d_{λ}$ is additive on subquotients of $M$ .

F6F3algebrastep 1.1

3.1

Every nonzero factor of a strict chain has detector at least one by the first two steps. Additivity bounds the number of strict inclusions by $d_{λ} (M)$ , whether the chain is written ascending or descending. If the detector is zero there is no nonzero subquotient; in particular $M = 0$ , with no strict inclusions.

algebrastep 2.1∎

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Every object of O has finite length

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Every object of $O$ has a finite composition series and is both Noetherian and Artinian. The length of zero is zero.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Suppose $M \in O_{χ_{λ}}$ . Every nonzero subquotient $T$ of $M$ has $T_{μ} \neq 0$ for some $μ \in W \cdot λ$ . In particular the number of strict inclusions in any finite chain of submodules of $M$ is at most $d_{λ} (M) = \sum_{μ \in W \cdot λ} \dim M_{μ},$ where distinct weights in the orbit are counted once. (Finite weight-space detection of subquotients)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The category is the categorical direct sum $O = ⨁_{χ} O_{χ}$ : objects have finite support in the index $χ$ , morphisms between distinct components vanish, and the canonical component projections are exact. (Generalized central-character decomposition of O)

[F3]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Let $M$ be a finitely generated $h$ -semisimple $g$ -module. Then $M \in O$ if and only if $supp M \subset ⋃_{i = 1}^{r} (λ_{i} - Q^{+})$ for some finite list of weights. In either case every $M_{μ}$ is finite dimensional. The list may be empty for $M = 0$ ; finite generation is an independent hypothesis. (The support description of category O with finite generation)

[F4]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every nonzero $M \in O$ contains a nonzero weight vector killed by $n^{+}$ . (A nonzero O-object has a highest-weight vector)

[F5]

Every central element acts on a cyclic highest-weight module by a scalar. In particular, each cyclic highest-weight module has a well-defined central character in the sense of def-central-character-of-a-lie-algebra-module. (Central elements act by scalars on cyclic highest-weight modules)

Proof

1.1

Split $M$ into its finitely many nonzero generalized central-character summands. Each such summand has a highest-weight vector: the finite-cone support of F3 has a maximal weight above any chosen weight, since simple-root coefficients in the interval are bounded. Central action on this highest line defines a character $χ_{λ}$ : a central element preserves the highest line of its cyclic module and commutes with the generator action. The generalized character on this summand must equal that scalar character, since a scalar with a zero power is zero. Thus each summand is indexed by some highest weight $λ$ .

F5F4F2F3algebra

2.1

Apply F1 to each summand and add the bounds. Character projections are exact by F2, so every strict submodule factor has a nonzero projection and contributes at least one to this sum of detectors. All finite strict submodule chains in $M$ consequently have a common finite integer bound. For $M = 0$ the sum and bound are zero.

F1F2step 1.1

3.1

Start with $0 \subset M$ , omitting the inclusion when $M = 0$ . If a nonzero factor is not simple, insert the inverse image of a nonzero proper submodule of that factor. Each insertion increases the number of strict inclusions. The bound forces termination and all resulting factors are simple. An infinite ascending or descending chain would have finite initial portions exceeding the same bound. Thus both chain conditions hold.

algebrastep 2.1∎

PropositionStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-07Open item page →

Finite-dimensional Hom spaces in O

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

For $M, N \in O$ , ${Hom}_{O} (M, N)$ is finite dimensional. For every weight $λ$ , ${End}_{O} (L (λ)) = C id$ .

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The BGG category $O$ is the full subcategory of left $U (g)$ -modules $M$ satisfying all three conditions: $M$ is finitely generated; $M = ⨁_{μ \in h^{*}} M_{μ}$ where $M_{μ} = {v : h v = μ (h) v for all h \in h}$ ; and $U (n^{+}) v$ is finite dimensional for each $v \in M$ . Its morphisms are all $g$ -linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Let $M$ be a finitely generated $h$ -semisimple $g$ -module. Then $M \in O$ if and only if $supp M \subset ⋃_{i = 1}^{r} (λ_{i} - Q^{+})$ for some finite list of weights. In either case every $M_{μ}$ is finite dimensional. The list may be empty for $M = 0$ ; finite generation is an independent hypothesis. (The support description of category O with finite generation)

[F3]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The simple objects of $O$ are exactly the modules $L (λ)$ , $λ \in h^{*}$ , and $L (λ) ≅ L (μ)$ if and only if $λ = μ$ . Simplicity here excludes zero. (The simple objects of O)

[F4]

For a $g$ -module $V$ , sending a homomorphism $T : M (λ) \to V$ to $T (v_{λ})$ is a bijection onto the vectors $v \in V$ of weight $λ$ annihilated by $n^{+}$ . Here $M (λ)$ is def-verma-module. The nonzero vectors in this target are precisely the highest-weight vectors of weight $λ$ from def-highest-weight-vector-and-cyclic-highest-weight-module; the zero vector corresponds to the zero homomorphism. (The universal property of Verma modules)

Proof

1.1

Choose finitely many weight generators $v_{i} \in M_{μ_{i}}$ . A map $f : M \to N$ is determined by the tuple $(f (v_{i}))_{i}$ , and $f (v_{i}) \in N_{μ_{i}}$ . Evaluation is therefore an injection into the finite-dimensional space $⨁_{i} N_{μ_{i}}$ . If $M = 0$ there are no generators; if $N = 0$ the target is zero.

F1F2

2.1

The module $L (λ)$ is generated by its one-dimensional highest line (the image of the Verma generator). Every endomorphism preserves that line and is scalar there, hence is the same scalar on its entire generated module. Each scalar multiple of the identity is an endomorphism and these are distinct because $L (λ) \neq 0$ .

F3F4algebrastep 1.1∎

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Restricted Chevalley dual

Definition

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

For an $h$ -semisimple module $M$ with finite-dimensional weight spaces, its restricted Chevalley dual is

$D (M) = ⨁_{μ \in h^{*}} M_{μ}^{*}, (x φ) (m) = φ (τ (x) m) (x \in U (g)) .$

Here each functional is extended by zero on the other weight spaces and $τ$ is the fixed anti-involution of Chevalley-contravariant forms. In particular $τ (h) = h$ and $D (M)_{μ} = M_{μ}^{*}$ . A map $f : M \to N$ induces $D (f) : D (N) \to D (M)$ by precomposition. The action law follows from $τ (x y) = τ (y) τ (x)$ ; a root vector of weight $α$ sends $M_{μ}^{*}$ to $M_{μ + α}^{*}$ , so the restricted sum is stable. This is a complex-linear algebraic dual, with no conjugation. Ordinary Lie-module duality has a minus sign and reverses weights; twisting that dual by the Lie automorphism $x \mapsto - τ (x)$ gives the convention used here.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Restricted self-duality of simple highest-weight modules

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

For every highest weight $λ$ , $D (L (λ)) ≅ L (λ)$ as $g$ -modules.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . For an $h$ -semisimple module $M$ with finite-dimensional weight spaces, its restricted Chevalley dual is $D (M) = ⨁_{μ \in h^{*}} M_{μ}^{*}, (x φ) (m) = φ (τ (x) m) (x \in U (g)) .$ Here each functional is extended by zero on the other weight spaces and $τ$ is the fixed anti-involution of def-chevalley-contravariant-form. In particular $τ (h) = h$ and $D (M)_{μ} = M_{μ}^{*}$ . A map $f : M \to N$ induces $D (f) : D (N) \to D (M)$ by precomposition. The action law follows from $τ (x y) = τ (y) τ (x)$ ; a root vector of weight $α$ sends $M_{μ}^{*}$ to $M_{μ + α}^{*}$ , so the restricted sum is stable. This is a complex-linear algebraic dual, with no conjugation. Ordinary Lie-module duality has a minus sign and reverses weights; twisting that dual by the Lie automorphism $x \mapsto - τ (x)$ gives the convention used here. (Restricted Chevalley dual)

[F2]

The proper submodule $J (λ)$ which is the sum of all proper submodules is the unique maximal submodule of $M (λ)$ . The quotient $L (λ) : = M (λ) / J (λ)$ is simple and is its unique simple quotient. (A Verma module has a unique simple quotient)

[F3]

The weights of $M (λ)$ are exactly $λ - β$ for $β \in Q^{+}$ ; every weight space is finite dimensional, and $M (λ)_{λ} = C v_{λ}$ . (Weights of a Verma module lie below lambda)

[F4]

For a $g$ -module $V$ , sending a homomorphism $T : M (λ) \to V$ to $T (v_{λ})$ is a bijection onto the vectors $v \in V$ of weight $λ$ annihilated by $n^{+}$ . Here $M (λ)$ is def-verma-module. The nonzero vectors in this target are precisely the highest-weight vectors of weight $λ$ from def-highest-weight-vector-and-cyclic-highest-weight-module; the zero vector corresponds to the zero homomorphism. (The universal property of Verma modules)

Proof

1.1

The simple Verma quotient $L = L (λ)$ has finite-dimensional weight spaces, support in $λ - Q^{+}$ , and a nonzero one-dimensional highest line. These follow from the Verma weight formula and the fact that a proper submodule cannot contain its generating top vector. Therefore $D (L)$ has the same weight dimensions, and its top line is killed by $n^{+}$ .

F1F2F3

2.1

If $S$ were a nonzero proper submodule of $D (L)$ , it would be a weight submodule. Finite-dimensionality of each weight space implies that its annihilator $S^{⊥} \subset L$ is nonzero (some weight component of $S$ is proper) and proper (some functional in $S$ is nonzero). It is a $g$ -submodule, since $φ (x m) = (τ (x) φ) (m)$ and $S$ is stable. This contradicts simplicity of $L$ . Hence $D (L)$ is simple without first presuming it is finitely generated.

F1F2algebrastep 1.1

3.1

A nonzero vector of the top line gives a nonzero Verma map $M (λ) \to D (L)$ by the universal property. It is surjective by simplicity, and the unique simple quotient identifies its target with $L (λ)$ .

F4F2step 2.1∎

PropositionStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Restricted duality is exact and involutive on O

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Restricted Chevalley duality is an exact contravariant equivalence $D : O \to O^{o p}$ , with a natural isomorphism $D^{2} ≅ id$ . It preserves each weight-space dimension, the formal character, and every simple composition multiplicity.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . For an $h$ -semisimple module $M$ with finite-dimensional weight spaces, its restricted Chevalley dual is $D (M) = ⨁_{μ \in h^{*}} M_{μ}^{*}, (x φ) (m) = φ (τ (x) m) (x \in U (g)) .$ Here each functional is extended by zero on the other weight spaces and $τ$ is the fixed anti-involution of def-chevalley-contravariant-form. In particular $τ (h) = h$ and $D (M)_{μ} = M_{μ}^{*}$ . A map $f : M \to N$ induces $D (f) : D (N) \to D (M)$ by precomposition. The action law follows from $τ (x y) = τ (y) τ (x)$ ; a root vector of weight $α$ sends $M_{μ}^{*}$ to $M_{μ + α}^{*}$ , so the restricted sum is stable. This is a complex-linear algebraic dual, with no conjugation. Ordinary Lie-module duality has a minus sign and reverses weights; twisting that dual by the Lie automorphism $x \mapsto - τ (x)$ gives the convention used here. (Restricted Chevalley dual)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . For every highest weight $λ$ , $D (L (λ)) ≅ L (λ)$ as $g$ -modules. (Restricted self-duality of simple highest-weight modules)

[F3]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every object of $O$ has a finite composition series and is both Noetherian and Artinian. The length of zero is zero. (Every object of O has finite length)

[F4]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The category $O$ is closed under submodules, quotients and finite direct sums and is an abelian category. If $0 \to A \to E \to B \to 0$ is exact, $A, B \in O$ , and $E$ is $h$ -semisimple, then $E \in O$ . The middle-term weight hypothesis is essential. (Category O is abelian and extension closed among weight modules)

[F5]

If an object $A$ in an abelian category has two composition series, then the two series have the same length and the same composition factors up to permutation and isomorphism. (Jordan-Holder theorem in an abelian category)

[F6]

The simple objects of $O$ are exactly the modules $L (λ)$ , $λ \in h^{*}$ , and $L (λ) ≅ L (μ)$ if and only if $λ = μ$ . (The simple objects of O)

Proof

1.1

First work in the larger category of weight modules with finite-dimensional weight spaces. A short exact sequence is exact at each weight; its finite-dimensional vector-space dual sequence is exact with arrows reversed. Their direct sum is exact. Evaluation $m \mapsto (φ \mapsto φ (m))$ identifies $M$ with $D^{2} (M)$ weightwise, is natural, and is $g$ -linear because $τ^{2} = 1$ . Also $\dim D (M)_{μ} = \dim M_{μ}$ .

F1algebra

2.1

For $M \in O$ , choose a finite composition series. By F6 every simple factor is some $L (λ)$ . Apply the exact functor just constructed to obtain the reversed filtration of $D (M)$ by annihilators of the original filtration terms. Each resulting factor is $D (L (λ)) ≅ L (λ)$ by F2 and therefore lies in $O$ .

F2F3F6step 1.1

3.1

Starting at zero, the qualified extension closure puts each term of this finite dual filtration in $O$ : all terms are already weight modules with finite-dimensional weights. Thus $D (M) \in O$ ; finite generation has been proved rather than assumed. Evaluation and the dual map functor now restrict to an exact contravariant equivalence on $O$ .

F4step 2.1

4.1

The weight equality from the first step proves character preservation. The reversed series has the same simple factors, and Jordan–Hölder makes their multiplicities independent of the series. The zero series dualizes to zero, so these assertions include the zero object.

F5algebrastep 3.1∎

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Standard and costandard objects

Definition

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

For $λ \in h^{*}$ , the standard and costandard objects are

$Δ (λ) = M (λ), \nabla (λ) = D (M (λ)) .$

The Verma module is defined in Verma modules, and Restricted duality is exact and involutive on O supplies the duality on $O$ . These symbols name the two objects; no projectivity or highest-weight-category axiom is part of this definition.

PropositionStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-07Open item page →

The simple socle of a costandard object

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

The costandard object $\nabla (λ)$ has a unique simple submodule, isomorphic to $L (λ)$ . Its socle, the sum of all simple submodules, is that submodule.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Restricted Chevalley duality is an exact contravariant equivalence $D : O \to O^{o p}$ , with a natural isomorphism $D^{2} ≅ id$ . It preserves each weight-space dimension, the formal character, and every simple composition multiplicity. (Restricted duality is exact and involutive on O)

[F2]

The proper submodule $J (λ)$ which is the sum of all proper submodules is the unique maximal submodule of $M (λ)$ . The quotient $L (λ) : = M (λ) / J (λ)$ is simple and is its unique simple quotient. (A Verma module has a unique simple quotient)

[F3]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . For $λ \in h^{*}$ , the standard and costandard objects are $Δ (λ) = M (λ), \nabla (λ) = D (M (λ)) .$ The Verma module is defined in def-verma-module, and prop-restricted-duality-is-an-exact-involution-on-category-o supplies the duality on $O$ . These symbols name the two objects; no projectivity or highest-weight-category axiom is part of this definition. (Standard and costandard objects)

Proof

1.1

Let $J (λ)$ be the unique maximal proper submodule of $M (λ)$ . Dualize $M (λ) ↠ L (λ)$ to obtain an injection $L (λ) ≅ D (L (λ)) ↪ D (M (λ)) = \nabla (λ)$ . Its image is the annihilator of $J (λ)$ .

F1F2F3

2.1

If $S \subset \nabla (λ)$ is any simple submodule, duality gives a simple quotient $M (λ) ↠ D (S)$ . Its kernel must be $J (λ)$ by uniqueness of the maximal submodule. Finite-dimensional weightwise biduality says $S$ is exactly that kernel annihilator. Hence all simple submodules have the image already constructed, and their sum equals it.

F1F2step 1.1∎

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

The integral Weyl group of a weight

Definition

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

For a weight $λ$ , define

$Φ_{λ} = {α \in Φ : ⟨ λ + ρ, α^{\lor} ⟩ \in Z}, W_{λ} = ⟨ s_{α} : α \in Φ_{λ} ⟩ \subset W .$

Reflections and coroots are those of Root reflections and the Weyl group action, with the shift from The Weyl vector rho for a chosen positive system. The integral-reflection linkage class through $λ$ is $W_{λ} \cdot λ$ . The word integral includes zero and negative integral pairings. If $Φ_{λ}$ is empty the generated group is ${1}$ . This definition uses generating reflections; no identification with a root-lattice-coset stabilizer is assumed.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Integral reflection linkage is an equivalence relation

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

The set $Φ_{λ}$ is preserved by its root reflections. If $μ \in W_{λ} \cdot λ$ , then $Φ_{μ} = Φ_{λ}$ and $W_{μ} = W_{λ}$ . The equivalence classes generated by moves $η \mapsto s_{α} \cdot η$ with $α \in Φ_{η}$ are exactly $W_{λ} \cdot λ$ . Every strong-linkage chain stays in one such class.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . For a weight $λ$ , define $Φ_{λ} = {α \in Φ : ⟨ λ + ρ, α^{\lor} ⟩ \in Z}, W_{λ} = ⟨ s_{α} : α \in Φ_{λ} ⟩ \subset W .$ Reflections and coroots are those of def-root-reflections-and-the-weyl-group-action, with the shift from def-weyl-vector-rho-for-a-chosen-positive-system. The integral-reflection linkage class through $λ$ is $W_{λ} \cdot λ$ . The word integral includes zero and negative integral pairings. If $Φ_{λ}$ is empty the generated group is ${1}$ . This definition uses generating reflections; no identification with a root-lattice-coset stabilizer is assumed. (The integral Weyl group of a weight)

[F2]

For the shifted (dot) action $s_{α} \cdot η : = s_{α} (η + ρ) - ρ,$ write $μ ↑ λ$ if there are weights $λ = η_{0} ≻ η_{1} ≻ \dots ≻ η_{r} = μ$ and positive roots $α_{j}$ such that $η_{j} = s_{α_{j}} \cdot η_{j - 1} and ⟨ η_{j - 1} + ρ, α_{j}^{\lor} ⟩ \in Z_{> 0}$ for every $j$ . The empty chain is allowed, so $λ ↑ λ$ . (The strong linkage order on weights)

Proof

1.1

Put $a = λ + ρ$ and let $α \in Φ_{λ}$ . For any root $β$ , reflection invariance of the root-coroot pairing gives $⟨ s_{α} a, β^{\lor} ⟩ = ⟨ a, s_{α} β^{\lor} ⟩ = ⟨ a, β^{\lor} ⟩ - ⟨ α, β^{\lor} ⟩ ⟨ a, α^{\lor} ⟩$ . The subtracted term is an integer by crystallographic integrality. Thus the left pairing is integral if and only if the first pairing on the right is integral.

F1algebra

2.1

Since $s_{α} β$ is a root with coroot $s_{α} β^{\lor}$ , the same identity says $β \in Φ_{λ}$ if and only if $s_{α} β \in Φ_{λ}$ . It also says $Φ_{s_{α} \cdot λ} = Φ_{λ}$ . Iteration along any generating word proves the claimed equality of root sets and therefore of the generated groups at every point in the orbit.

F1algebrastep 1.1

3.1

Every allowed sequence of moves now uses generators of the original $W_{λ}$ , so ends in its orbit. Conversely every word in those generators is an allowed sequence, since the root set stays unchanged at each intermediate weight. Each move reverses itself, and concatenation and the empty word give symmetry, transitivity and reflexivity. Strong-linkage moves use positive integral pairings, a subset of these allowed moves. A zero pairing fixes the weight.

F2algebrastep 2.1∎

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-07Open item page →

Simple extensions cannot cross linkage classes

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

A short exact sequence $0 \to L (μ) \to E \to L (λ) \to 0$ in $O$ splits whenever $λ$ and $μ$ belong to distinct integral-reflection linkage classes.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Restricted Chevalley duality is an exact contravariant equivalence $D : O \to O^{o p}$ , with a natural isomorphism $D^{2} ≅ id$ . It preserves each weight-space dimension, the formal character, and every simple composition multiplicity. (Restricted duality is exact and involutive on O)

[F2]

For a $g$ -module $V$ , sending a homomorphism $T : M (λ) \to V$ to $T (v_{λ})$ is a bijection onto the vectors $v \in V$ of weight $λ$ annihilated by $n^{+}$ . Here $M (λ)$ is def-verma-module. The nonzero vectors in this target are precisely the highest-weight vectors of weight $λ$ from def-highest-weight-vector-and-cyclic-highest-weight-module; the zero vector corresponds to the zero homomorphism. (The universal property of Verma modules)

[F3]

If $[M (λ) : L (μ)] \neq 0$ , then $μ ↑ λ$ . (The strong linkage principle for Verma modules)

[F4]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The set $Φ_{λ}$ is preserved by its root reflections. If $μ \in W_{λ} \cdot λ$ , then $Φ_{μ} = Φ_{λ}$ and $W_{μ} = W_{λ}$ . The equivalence classes generated by moves $η \mapsto s_{α} \cdot η$ with $α \in Φ_{η}$ are exactly $W_{λ} \cdot λ$ . Every strong-linkage chain stays in one such class. (Integral reflection linkage is an equivalence relation)

[F5]

The proper submodule $J (λ)$ which is the sum of all proper submodules is the unique maximal submodule of $M (λ)$ . The quotient $L (λ) : = M (λ) / J (λ)$ is simple and is its unique simple quotient. (A Verma module has a unique simple quotient)

Proof

1.1

If $λ < μ$ , dualize the sequence; exact self-duality of simples reverses the two labels, and splitting of the dual implies splitting of the original by biduality. Thus it is enough to treat the orientation $λ ≮ μ$ , which includes incomparable labels.

F1

2.1

Lift the highest vector of $L (λ)$ to a vector $v \in E_{λ}$ . Such a weight lift exists because the sequence consists of weight modules. If a positive-root operator acted nontrivially on $v$ , its image would lie in the submodule $L (μ)$ at weight $λ + α$ , forcing $λ + α \leq μ$ and hence $λ < μ$ . Therefore $v$ is singular, and the Verma universal property gives a map $M (λ) \to E$ whose image $H$ surjects onto $L (λ)$ . Here the support bound for each simple follows from its being the highest-weight Verma quotient.

F2F5algebrastep 1.1

3.1

The intersection $H \cap L (μ)$ is either zero or all of the simple submodule. In the latter case $H = E$ , so $E$ is a length-two quotient of $M (λ)$ and $L (μ)$ is a composition factor of that Verma. Strong linkage would imply $μ ↑ λ$ , and hence equality of integral-reflection classes, contrary to the hypothesis. Thus the intersection is zero, and $H \to L (λ)$ is an isomorphism whose inverse is a section.

F3F4algebrastep 2.1∎

LemmaStatement: Literature-sourcedProof: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Splitting finite-length modules across separated simple classes

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Partition the isomorphism classes of simple objects of $O$ into parts $P_{t}$ . Suppose every extension of two simples from different parts splits, in either order. Then each $M \in O$ has a unique decomposition $M = ⨁_{t} M_{t}$ into submodules whose composition factors lie in $P_{t}$ , with finitely many nonzero terms. This decomposition is functorial, and maps between modules supported on disjoint collections of parts are zero.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every object of $O$ has a finite composition series and is both Noetherian and Artinian. The length of zero is zero. (Every object of O has finite length)

[F2]

If an object $A$ in an abelian category has two composition series, then the two series have the same length and the same composition factors up to permutation and isomorphism. (Jordan-Holder theorem in an abelian category)

[F3]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The category $O$ is closed under submodules, quotients and finite direct sums and is an abelian category. If $0 \to A \to E \to B \to 0$ is exact, $A, B \in O$ , and $E$ is $h$ -semisimple, then $E \in O$ . The middle-term weight hypothesis is essential. (Category O is abelian and extension closed among weight modules)

Proof

1.1

Finite length and Jordan–Hölder make the set of composition factors of every object intrinsic. In any exact sequence the multiset of factors of the middle object is the union of those of the ends: concatenate a series in the subobject with the inverse images of a series in the quotient and use Jordan–Hölder. Thus a nonzero image of a map between objects with disjoint collections of parts would have a simple factor in both collections. Such maps are zero.

F1F2F3

2.1

First fix a simple $T$ in one collection of parts and an object $A$ supported in disjoint parts. We prove every extension $0 \to A \to E \to T \to 0$ splits by induction on the length of $A$ . For $A = 0$ this is immediate, and for $A$ simple it is the hypothesis. Otherwise choose a maximal proper submodule $A_{0}$ so $S = A / A_{0}$ is simple. The quotient $E / A_{0}$ is the pushout along $A \to S$ , explicitly $(E \oplus S) / {(a, - \overset{ˉ}{a}) : a \in A}$ . The extension of $T$ by $S$ splits by hypothesis. The inverse image in $E$ of a chosen section image is an extension of $T$ by $A_{0}$ ; induction splits it, supplying a section into $E$ .

givenalgebrastep 1.1

3.1

For general $B$ in parts disjoint from those of $A$ , induct on its length in $0 \to A \to E \overset{p}{\to} B \to 0$ . The case $B = 0$ is immediate and a simple $B$ was just handled. Choose a maximal submodule $B_{0} \subset B$ , with simple quotient $T$ . The pullback is ${(e, b) \in E \oplus B_{0} : p (e) = b}$ , equivalently $p^{- 1} (B_{0})$ . Induction splits it, giving a copy ${\tilde{B}}_{0} \subset E$ disjoint from $A$ . Now $0 \to A \to E / {\tilde{B}}_{0} \to T \to 0$ splits by the preceding step. Its retraction onto $A$ , composed with $E \to E / {\tilde{B}}_{0}$ , is a retraction of $E$ onto $A$ . Its kernel is a complementary copy of $B$ , proving the required splitting for all lengths.

algebrastep 2.1

4.1

Construct the decomposition by induction on the length of $M$ , with the empty decomposition for zero. Choose a maximal proper submodule $N$ and write its already constructed decomposition as $N = N_{t} \oplus N_{o u t}$ , where the simple quotient $M / N$ belongs to part $t$ . The extension $0 \to N_{o u t} \to M / N_{t} \to M / N \to 0$ splits by the preceding argument. Compose its retraction onto $N_{o u t}$ with $M \to M / N_{t}$ . The resulting retraction gives $M = N_{o u t} \oplus K$ , where $0 \to N_{t} \to K \to M / N \to 0$ . All factors of $K$ lie in part $t$ . This constructs finitely many summands.

F1algebrastep 3.1

5.1

For two such decompositions, the composite of the inclusion of a part- $t$ summand with projection onto any part- $u$ summand for $u \neq t$ vanishes by the first step. Therefore that part- $t$ submodule is contained in the other part- $t$ submodule, and reversing the decompositions gives equality. The same argument for any map proves preservation of parts and functoriality.

algebrastep 4.1∎

Notes

The source leaves the formal finite-length decomposition to the reader. The local proof supplies the pushout, pullback, retraction and uniqueness arguments explicitly; its proof provenance is therefore ai-generated rather than literature-derived or a claimed transcription.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Central-character summands refine into linkage blocks

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

For a linkage class $C = W_{λ} \cdot λ$ , let $O_{C}$ be the full subcategory of objects all of whose simple composition factors have labels in $C$ . Then $O = ⨁_{C} O_{C}$ , and each nonzero $O_{C}$ is indecomposable as a categorical direct summand. These are precisely the blocks. Each $O_{C}$ lies in $O_{χ_{λ}}$ ; a central-character summand can contain several blocks. Independently, grouping weights by cosets of the root lattice $Q$ gives a canonical coarser decomposition by weight cosets.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . For a weight $λ$ , define $Φ_{λ} = {α \in Φ : ⟨ λ + ρ, α^{\lor} ⟩ \in Z}, W_{λ} = ⟨ s_{α} : α \in Φ_{λ} ⟩ \subset W .$ Reflections and coroots are those of def-root-reflections-and-the-weyl-group-action, with the shift from def-weyl-vector-rho-for-a-chosen-positive-system. The integral-reflection linkage class through $λ$ is $W_{λ} \cdot λ$ . The word integral includes zero and negative integral pairings. If $Φ_{λ}$ is empty the generated group is ${1}$ . This definition uses generating reflections; no identification with a root-lattice-coset stabilizer is assumed. (The integral Weyl group of a weight)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The set $Φ_{λ}$ is preserved by its root reflections. If $μ \in W_{λ} \cdot λ$ , then $Φ_{μ} = Φ_{λ}$ and $W_{μ} = W_{λ}$ . The equivalence classes generated by moves $η \mapsto s_{α} \cdot η$ with $α \in Φ_{η}$ are exactly $W_{λ} \cdot λ$ . Every strong-linkage chain stays in one such class. (Integral reflection linkage is an equivalence relation)

[F3]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . A short exact sequence $0 \to L (μ) \to E \to L (λ) \to 0$ in $O$ splits whenever $λ$ and $μ$ belong to distinct integral-reflection linkage classes. (Simple extensions cannot cross linkage classes)

[F4]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Partition the isomorphism classes of simple objects of $O$ into parts $P_{t}$ . Suppose every extension of two simples from different parts splits, in either order. Then each $M \in O$ has a unique decomposition $M = ⨁_{t} M_{t}$ into submodules whose composition factors lie in $P_{t}$ , with finitely many nonzero terms. This decomposition is functorial, and maps between modules supported on disjoint collections of parts are zero. (Splitting finite-length modules across separated simple classes)

[F5]

For $α \in Φ^{+}$ , if $⟨ λ + ρ, α^{\lor} ⟩ \in Z_{> 0}$ , then $M (s_{α} \cdot λ) ↪ M (λ)$ . (Verma embedding for an arbitrary positive root)

[F6]

The proper submodule $J (λ)$ which is the sum of all proper submodules is the unique maximal submodule of $M (λ)$ . The quotient $L (λ) : = M (λ) / J (λ)$ is simple and is its unique simple quotient. (A Verma module has a unique simple quotient)

[F7]

Let $χ_{λ}$ and $χ_{μ}$ be the central characters obtained from highest weights $λ$ and $μ$ . Then $χ_{λ} = χ_{μ} if and only if μ \in W \cdot λ,$ where $W \cdot λ : = {w (λ + ρ) - ρ : w \in W}$ . (Central characters are dot-Weyl orbits)

Proof

1.1

Integral-reflection orbits partition the simple labels, and extensions between simples in distinct parts split. The finite-length splitting lemma therefore gives the stated functorial decomposition with no cross-part morphisms. Each part contains its simple $L (λ)$ and is nonzero.

F1F2F3F4

2.1

A Verma module is indecomposable: if it were a sum of two nonzero submodules, neither could be the whole module, and their sum would lie in the unique maximal proper submodule, a contradiction. Thus the decomposition just constructed places $M (η)$ entirely in the part containing its simple quotient $L (η)$ .

F6step 1.1

3.1

Let $α \in Φ_{η}$ and replace it by its positive choice of sign, which leaves the reflection unchanged. Set $k = ⟨ η + ρ, α^{\lor} ⟩ \in Z$ . If $k > 0$ , F5 embeds $M (s_{α} \cdot η)$ in $M (η)$ . If $k < 0$ , the pairing at $s_{α} \cdot η$ is $- k > 0$ , so F5 gives the reverse embedding. If $k = 0$ , the labels agree. In any categorical refinement the containing indecomposable Verma stays in one summand, so its two simple subquotients $L (η)$ and $L (s_{α} \cdot η)$ stay together.

F1F5F6algebrastep 2.1

4.1

Every pair of labels in $C$ is joined by a finite word of these moves. Hence all its simples stay together under any categorical refinement. A nonzero object of a putative second summand has a simple composition factor, which is impossible. Therefore $O_{C}$ is indecomposable, and the displayed direct sum lists all blocks.

F2F4step 3.1

5.1

Every label in $C$ lies in the full dot orbit, so its simple module has character $χ_{λ}$ . If a module has $r$ such composition factors, each element of $m_{χ_{λ}}$ lowers its composition filtration by at least one step. Therefore $m_{χ_{λ}}^{r}$ annihilates the module; for zero use exponent 1. This proves $O_{C} \subset O_{χ_{λ}}$ .

F7algebrastep 4.1

6.1

For each coset $c \in h^{*} / Q$ the sum $M^{(c)} = ⨁_{μ \in c} M_{μ}$ is a submodule, because root operators shift weights by roots and Cartan operators preserve weights. Their sum is direct and exhausts $M$ ; only finitely many occur, since finitely many weight generators occupy finitely many cosets. An integral-reflection move changes a label by an integral multiple of a root. Thus each linkage part occupies a single coset, but no converse identification with a coset stabilizer was used.

F1algebrastep 5.1∎

CorollaryStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Duality preserves linkage blocks and block orthogonality

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Restricted duality preserves every linkage block. If $B$ is a Chevalley-contravariant bilinear form on $M \in O$ , then $B (M_{C}, M_{C^{'}}) = 0$ for distinct block summands $C \neq C^{'}$ . No nondegeneracy of $B$ is required.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . For a linkage class $C = W_{λ} \cdot λ$ , let $O_{C}$ be the full subcategory of objects all of whose simple composition factors have labels in $C$ . Then $O = ⨁_{C} O_{C}$ , and each nonzero $O_{C}$ is indecomposable as a categorical direct summand. These are precisely the blocks. Each $O_{C}$ lies in $O_{χ_{λ}}$ ; a central-character summand can contain several blocks. Independently, grouping weights by cosets of the root lattice $Q$ gives a canonical coarser decomposition by weight cosets. (Central-character summands refine into linkage blocks)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Restricted Chevalley duality is an exact contravariant equivalence $D : O \to O^{o p}$ , with a natural isomorphism $D^{2} ≅ id$ . It preserves each weight-space dimension, the formal character, and every simple composition multiplicity. (Restricted duality is exact and involutive on O)

[F3]

Fix simple roots ${α_{i}}$ in the chosen positive system and normalized Chevalley generators $e_{i} \in g_{α_{i}}$ , $f_{i} \in g_{- α_{i}}$ . Let $τ : U (g) \to U (g)$ be the Chevalley anti-involution determined by $τ (e_{i}) = f_{i}$ , $τ (f_{i}) = e_{i}$ , and $τ (h) = h$ for $h \in h$ . A bilinear form $B$ on a $g$ -module is Chevalley-contravariant when $B (x u, v) = B (u, τ (x) v) (x \in U (g)) .$ This is a bilinear condition, not a Hermitian or positivity condition. (Chevalley-contravariant forms)

Proof

1.1

Duality preserves each simple composition multiplicity. Hence the list of factor labels of a dualized block object remains in the same class; the block characterization gives $D (O_{C}) = O_{C}$ .

F1F2

2.1

For weight vectors $u \in M_{μ}$ and $v \in M_{ν}$ , contravariance and $τ (h) = h$ give $(μ (h) - ν (h)) B (u, v) = 0$ for all $h$ . If $μ \neq ν$ choose $h$ separating them, and obtain $B (u, v) = 0$ . Each $u$ has only finitely many weight components, so $v \mapsto B (u, v)$ belongs to the restricted dual. The assignment $u \mapsto B (u, -)$ is $g$ -linear by the defining contravariance identity.

F3algebrastep 1.1

3.1

Restrict this map to $M_{C}$ and project to $D (M_{C^{'}})$ . Its source and target are in different blocks by the first step, so it is zero by the block decomposition. This says exactly $B (M_{C}, M_{C^{'}}) = 0$ , including zero summands and the zero form.

F1F2step 2.1∎

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Verma self-extensions in O split

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

Every short exact sequence $0 \to M (λ) \to E \overset{p}{\to} M (λ) \to 0$ in $O$ splits.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The category $O$ is closed under submodules, quotients and finite direct sums and is an abelian category. If $0 \to A \to E \to B \to 0$ is exact, $A, B \in O$ , and $E$ is $h$ -semisimple, then $E \in O$ . The middle-term weight hypothesis is essential. (Category O is abelian and extension closed among weight modules)

[F2]

For a $g$ -module $V$ , sending a homomorphism $T : M (λ) \to V$ to $T (v_{λ})$ is a bijection onto the vectors $v \in V$ of weight $λ$ annihilated by $n^{+}$ . Here $M (λ)$ is def-verma-module. The nonzero vectors in this target are precisely the highest-weight vectors of weight $λ$ from def-highest-weight-vector-and-cyclic-highest-weight-module; the zero vector corresponds to the zero homomorphism. (The universal property of Verma modules)

[F3]

The weights of $M (λ)$ are exactly $λ - β$ for $β \in Q^{+}$ ; every weight space is finite dimensional, and $M (λ)_{λ} = C v_{λ}$ . (Weights of a Verma module lie below lambda)

Proof

1.1

Weightwise exactness and the Verma support formula imply that $E$ has no weights outside $λ - Q^{+}$ . Lift the highest vector to an actual vector $v \in E_{λ}$ , using the surjection on that weight space. Every positive-root operator kills $v$ since its target weight is absent.

F1F3choose

2.1

The universal property supplies $s : M (λ) \to E$ taking its highest vector to $v$ . The composite $p s$ fixes the highest generator, hence equals the identity on the cyclic module. Thus $s$ is a section. No integrality or regularity condition on $λ$ was used.

F2algebrastep 1.1∎

PropositionStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Finite-dimensional tensoring preserves O

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

If $E$ is a finite-dimensional $h$ -semisimple $g$ -module, the functor $M \mapsto E \otimes_{C} M$ with diagonal action is exact and maps $O$ into itself.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The BGG category $O$ is the full subcategory of left $U (g)$ -modules $M$ satisfying all three conditions: $M$ is finitely generated; $M = ⨁_{μ \in h^{*}} M_{μ}$ where $M_{μ} = {v : h v = μ (h) v for all h \in h}$ ; and $U (n^{+}) v$ is finite dimensional for each $v \in M$ . Its morphisms are all $g$ -linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Let $M$ be a finitely generated $h$ -semisimple $g$ -module. Then $M \in O$ if and only if $supp M \subset ⋃_{i = 1}^{r} (λ_{i} - Q^{+})$ for some finite list of weights. In either case every $M_{μ}$ is finite dimensional. The list may be empty for $M = 0$ ; finite generation is an independent hypothesis. (The support description of category O with finite generation)

[F3]

Let $x_{1}, \dots, x_{r}$ be an ordered basis of a finite-dimensional complex Lie algebra $g$ . Then the monomials $x_{1}^{a_{1}} x_{2}^{a_{2}} \dots x_{r}^{a_{r}} (a_{i} \in N_{0})$ form a basis of $U (g)$ . In particular, multiplication identifies $gr U (g)$ with the symmetric algebra $S (g)$ on the symbols of the $x_{i}$ . (PBW gives an ordered monomial basis for the enveloping algebra)

Proof

1.1

Choose a basis $e_{1}, \dots, e_{d}$ of $E$ and finitely many weight generators $v_{1}, \dots, v_{s}$ of $M$ . Let $T$ be the $g$ -submodule generated by all $e_{i} \otimes v_{j}$ . It contains $e \otimes v_{j}$ for every $e \in E$ . Induct on the length of a word $u$ in Lie algebra elements. If $e \otimes u v_{j}$ is already in $T$ for every $e$ , then $e \otimes x u v_{j} = x (e \otimes u v_{j}) - (x e) \otimes u v_{j}$ lies in $T$ . Since such words span $U (g)$ , $T = E \otimes M$ , proving finite generation. Empty bases or generators give the zero tensor product.

F1F3algebra

2.1

The tensor weight decomposition is $(E \otimes M)_{μ} = ⨁_{ν} E_{ν} \otimes M_{μ - ν}$ , a finite sum over weights of $E$ . Its support is a finite union of translates by those weights of the finitely many support cones for $M$ . The support criterion gives $E \otimes M \in O$ .

F2algebrastep 1.1

3.1

Tensoring a vector-space exact sequence with $E$ gives a direct sum of $d$ copies of that exact sequence after choosing a basis. The resulting maps commute with the diagonal $g$ -action, so this is exact in $O$ . The assertion includes $E = 0$ and one-dimensional $E$ .

algebrastep 2.1∎

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

The Grothendieck group and character of O

Definition

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

For the abelian category Category O is abelian and extension closed among weight modules, define $K_{0} (O)$ as the free abelian group on isomorphism classes $[M]$ , modulo $[B] = [A] + [C]$ for every short exact sequence $0 \to A \to B \to C \to 0$ . A set of representatives suffices: every finitely generated $U (g)$ -module is a quotient of some $U (g)^{n}$ , and those quotients form a set up to isomorphism.

Define the formal character by $ch M = \sum_{μ} (\dim M_{μ}) e^{μ}$ . By The support description of category O with finite generation, its integer coefficients are finite and supported in finitely many downward cones. Let $R$ be the group of all such integer coefficient families, with pointwise addition. It is a ring with $e^{μ} e^{ν} = e^{μ + ν}$ : at a fixed resulting weight, in any pair of cones the equation $β + γ = η$ with $β, γ \in Q^{+}$ has finitely many solutions, since every simple-root coefficient is bounded. Taking weight spaces is exact, so character gives a well-defined homomorphism $K_{0} (O) \to R$ . The zero object's class and character are zero.

PropositionStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Simple and standard bases of K0(O)

Statement

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

The classes $[L (λ)]$ , and separately the classes $[M (λ)]$ , form $Z$ -bases of $K_{0} (O)$ . For a fixed finite central-character label set $Λ = W \cdot λ$ , the transition between its standard and simple classes is unitriangular in any linear order extending $\leq$ . This remains true on a downward-closed subset of that finite poset. It is not a claim about finite downward ideals of all of $h^{*}$ .

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . For the abelian category thm-category-o-is-abelian-and-extension-closed, define $K_{0} (O)$ as the free abelian group on isomorphism classes $[M]$ , modulo $[B] = [A] + [C]$ for every short exact sequence $0 \to A \to B \to C \to 0$ . A set of representatives suffices: every finitely generated $U (g)$ -module is a quotient of some $U (g)^{n}$ , and those quotients form a set up to isomorphism. Define the formal character by $ch M = \sum_{μ} (\dim M_{μ}) e^{μ}$ . By prop-equivalent-support-description-of-category-o, its integer coefficients are finite and supported in finitely many downward cones. Let $R$ be the group of all such integer coefficient families, with pointwise addition. It is a ring with $e^{μ} e^{ν} = e^{μ + ν}$ : at a fixed resulting weight, in any pair of cones the equation $β + γ = η$ with $β, γ \in Q^{+}$ has finitely many solutions, since every simple-root coefficient is bounded. Taking weight spaces is exact, so character gives a well-defined homomorphism $K_{0} (O) \to R$ . The zero object's class and character are zero. (The Grothendieck group and character of O)

[F2]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every object of $O$ has a finite composition series and is both Noetherian and Artinian. The length of zero is zero. (Every object of O has finite length)

[F3]

If an object $A$ in an abelian category has two composition series, then the two series have the same length and the same composition factors up to permutation and isomorphism. (Jordan-Holder theorem in an abelian category)

[F4]

Let $χ_{λ}$ and $χ_{μ}$ be the central characters obtained from highest weights $λ$ and $μ$ . Then $χ_{λ} = χ_{μ} if and only if μ \in W \cdot λ,$ where $W \cdot λ : = {w (λ + ρ) - ρ : w \in W}$ . (Central characters are dot-Weyl orbits)

[F5]

The proper submodule $J (λ)$ which is the sum of all proper submodules is the unique maximal submodule of $M (λ)$ . The quotient $L (λ) : = M (λ) / J (λ)$ is simple and is its unique simple quotient. (A Verma module has a unique simple quotient)

[F6]

The weights of $M (λ)$ are exactly $λ - β$ for $β \in Q^{+}$ ; every weight space is finite dimensional, and $M (λ)_{λ} = C v_{λ}$ . (Weights of a Verma module lie below lambda)

[F7]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The simple objects of $O$ are exactly the modules $L (λ)$ , $λ \in h^{*}$ , and $L (λ) ≅ L (μ)$ if and only if $λ = μ$ . Simplicity here excludes zero. (The simple objects of O)

[F8]

Every central element acts on a cyclic highest-weight module by a scalar. In particular, each cyclic highest-weight module has a well-defined central character in the sense of def-central-character-of-a-lie-algebra-module. (Central elements act by scalars on cyclic highest-weight modules)

Proof

1.1

Finite length gives $[X] = \sum_{μ} [X : L (μ)] [L (μ)]$ with a finite sum. Multiplicities are well defined by Jordan–Hölder and additive on exact sequences, by concatenating composition series and then applying that theorem. Each multiplicity therefore defines a homomorphism on $K_{0}$ , taking the simple classes to coordinate vectors. This proves spanning and independence of the simple classes.

F7F1F2F3

2.1

The highest weight of $M (η)$ has multiplicity one as a weight and survives in its unique simple quotient. All other simple factor labels $μ$ satisfy $μ < η$ : weightwise additivity and the support formula give $μ \leq η$ , while the top weight dimension excludes a second factor with label $η$ . The scalar central character of a Verma passes to each factor. Hence every such label is in $W \cdot η$ by the exact character criterion. Here scalar central action follows directly because the center preserves the one-dimensional highest line and commutes with its cyclic generator action.

F8F4F5F6algebrastep 1.1

3.1

On the finite set $Λ$ , choose a linear extension of the positive-root order. The expansion $[M (η)] = [L (η)] + \sum_{μ < η} a_{η μ} [L (μ)]$ has an integral triangular matrix $I + N$ with $N$ strictly triangular. If $r = ∣ Λ ∣$ , then $N^{r} = 0$ , so the inverse is the finite integral sum $I - N + \dots + (- N)^{r - 1}$ . Thus standard classes form a basis of the subgroup on those simple labels.

algebrastep 2.1

4.1

A downward-closed subset contains every smaller factor label of its standards, so the restricted matrix has the same property; for the empty subset the group and basis are zero and empty. Finally the full label set is partitioned into finite dot orbits. Taking the direct sum of their basis changes proves the global standard basis, with every element still a finite linear combination.

algebrastep 3.1∎

Category O Finiteness Duality and Blocks

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

The classical BGG category O

Definition

Noetherianity of the enveloping algebra

Statement

Facts & Assumptions

Proof

Finite Borel-stable generators and weight flags

Statement

Facts & Assumptions

Proof

The support description of category O with finite generation

Statement

Facts & Assumptions

Proof

Verma and finite-dimensional weight modules belong to O

Statement

Facts & Assumptions

Proof

Category O is abelian and extension closed among weight modules

Statement

Facts & Assumptions

Proof

A nonzero O-object has a highest-weight vector

Statement

Facts & Assumptions

Proof

The simple objects of O

Statement

Facts & Assumptions

Proof

Finite filtrations by highest-weight quotients

Statement

Facts & Assumptions

Proof

The center has finite-dimensional image on each O-object

Statement

Facts & Assumptions

Proof

Generalized central-character subcategories

Definition

Generalized central-character summands

Statement

Facts & Assumptions

Proof

Generalized central-character decomposition of O

Statement

Facts & Assumptions

Proof

Finite weight-space detection of subquotients

Statement

Facts & Assumptions

Proof

Every object of O has finite length

Statement

Facts & Assumptions

Proof

Finite-dimensional Hom spaces in O

Statement

Facts & Assumptions

Proof

Restricted Chevalley dual

Definition

Restricted self-duality of simple highest-weight modules

Statement

Facts & Assumptions

Proof

Restricted duality is exact and involutive on O

Statement

Facts & Assumptions

Proof

Standard and costandard objects

Definition

The simple socle of a costandard object

Statement

Facts & Assumptions