TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The $i$ -string decomposition of the negative half

Statement

Let $U_{q}^{-}$ be the negative half of the Drinfeld–Jimbo algebra with its skew derivations $e_{i}^{'}, e_{i}^{''} : U_{q}^{-} \to U_{q}^{-}$ (Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half, Positive, negative and toral quantum subalgebras and their root gradings), fix $i \in I$ , and write $f_{i}^{(n)} = F_{i}^{n} / [n]_{i}!$ for the divided powers (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ). Then:

(i) $e_{i}^{'}$ is locally nilpotent on $U_{q}^{-}$ , and the endomorphism $P_{i} : = \sum_{m \geq 0} (- 1)^{m} q_{i}^{- m (m - 1) / 2} F_{i}^{(m)} e_{i}^{' m} \in {End}_{Q (q)} (U_{q}^{-})$ is well defined (each sum is finite on each element) and is the projection onto $Ker e_{i}^{'}$ along $f_{i} U_{q}^{-}$ . In particular $e_{i}^{'} P_{i} = 0$ , $P_{i} F_{i} = 0$ , $P_{i}^{2} = P_{i}$ , $P_{i} u = u$ for $u \in Ker e_{i}^{'}$ , and $U_{q}^{-} = Ker e_{i}^{'} \oplus f_{i} U_{q}^{-} .$

(ii) Every $u \in U_{q}^{-}$ has a unique $i$ -string decomposition $u = \sum_{n \geq 0} f_{i}^{(n)} u_{n}, u_{n} \in Ker e_{i}^{'}, u_{n} = 0 for n ≫ 0,$ and the components are given by $u_{n} = q_{i}^{n (n - 1) / 2} P_{i} e_{i}^{' n} u$ .

(iii) The Kashiwara operators on the negative half, defined on the string decomposition by ${\tilde{e}}_{i} u : = \sum_{n \geq 1} f_{i}^{(n - 1)} u_{n}, {\tilde{f}}_{i} u : = \sum_{n \geq 0} f_{i}^{(n + 1)} u_{n},$ are $Q (q)$ -linear endomorphisms of $U_{q}^{-}$ with ${\tilde{e}}_{i} {\tilde{f}}_{i} = {id}_{U_{q}^{-}}$ ; ${\tilde{e}}_{i} u = 0$ if and only if $u \in Ker e_{i}^{'}$ ; ${\tilde{f}}_{i}$ is injective and ${\tilde{f}}_{i} (1) = F_{i}$ . Both operators preserve the root grading and shift its degree by $\mp α_{i}$ , that is ${\tilde{e}}_{i} (U_{q}^{-} [- ν]) \subseteq U_{q}^{-} [- ν + α_{i}]$ and ${\tilde{f}}_{i} (U_{q}^{-} [- ν]) \subseteq U_{q}^{-} [- ν - α_{i}]$ .

No choice principle is used. Here $e_{i}^{' n}$ denotes the ordinary $n$ -fold power of the derivation, not its divided power. The same projector and every string coefficient preserve the $Z [q, q^{- 1}]$ divided-power lattice $U_{A}^{-}$ (The Lusztig divided-power integral form $U_{A}$ ), as proved by the integral skew-string supplier; this does not assert that lattice's freeness.

Facts & Assumptions

Given: The negative half $U_{q}^{-}$ , the skew derivations $e_{i}^{'}, e_{i}^{''}$ , a fixed $i$ , and the divided powers $f_{i}^{(n)} = F_{i}^{n} / [n]_{i}!$ .

[F1]

$U_{q}^{-} = ⨁_{ν \in Q_{+}} U_{q}^{-} [- ν]$ is graded by the positive root lattice, with $\deg F_{j} = - α_{j}$ and $U_{q}^{-} [0] = Q (q) \cdot 1$ ; the grading is the restriction of the root grading of $U_{q} (g)$ (Positive, negative and toral quantum subalgebras and their root gradings).

[F2]

The maps $e_{i}^{'}, e_{i}^{''} : U_{q}^{-} \to U_{q}^{-}$ are $Q (q)$ -linear, vanish on $1$ , satisfy $e_{i}^{'} (F_{j}) = δ_{i j}$ , the Leibniz rule $e_{i}^{'} (x y) = e_{i}^{'} (x) y + q_{i}^{⟨ w t (x), h_{i} ⟩} x e_{i}^{'} (y)$ on homogeneous $x$ of actual weight $w t (x)$ , and similarly for $e_{i}^{''}$ with the opposite twist; they satisfy $e_{i}^{'} F_{j} = q_{i}^{- a_{i j}} F_{j} e_{i}^{'} + δ_{i j}$ and decrease positive height by one, so $e_{i}^{'} (U_{q}^{-} [- ν]) \subseteq U_{q}^{-} [- ν + α_{i}]$ (zero when $ν - α_{i} \notin Q_{+}$ ) and $e_{i}^{'}$ is locally nilpotent (Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half).

[F3]

$[m]_{i} = (q_{i}^{m} - q_{i}^{- m}) / (q_{i} - q_{i}^{- 1}) \in Q (q)^{\times}$ for $m \geq 1$ , $f_{i}^{(m)} = F_{i}^{m} / [m]_{i}!$ , and $f_{i}^{(m)} f_{i}^{(n)} = {(\binom{m + n}{m})}_{i} f_{i}^{(m + n)}$ with $F_{i} f_{i}^{(n - 1)} = [n]_{i} f_{i}^{(n)}$ ; all displayed scalars are nonzero (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F4]

Gaussian identity: for $N \geq 0$ , $\sum_{r = 0}^{N} (- 1)^{r} q_{i}^{- r (N - 1)} {(\binom{N}{r})}_{i}$ equals $1$ for $N = 0$ and $0$ for $N \geq 1$ ; the Gaussian quotient is invariant under $q_{i} \mapsto q_{i}^{- 1}$ and the identity is the inversion of the Gauss product formula (The quantum Pascal recurrences, the Gauss product formula and Gaussian integrality, parts (iii)-(iv)).

[F5]

Integral skew-string supplier: the operator $P_{i}$ of (i) is finite on each vector and projects $U_{q}^{-}$ onto $Ker e_{i}^{'}$ along $F_{i} U_{q}^{-}$ ; every $x \in U_{A}^{-}$ has a unique finite expansion $x = \sum_{n} F_{i}^{(n)} x_{n}$ with $e_{i}^{'} x_{n} = 0$ and $x_{n} = q_{i}^{n (n - 1) / 2} P_{i} (e_{i}^{'})^{n} x \in U_{A}^{-}$ ; the projector and coefficients preserve $U_{A}^{-}$ , with no global basis or freeness assumed (Integral skew-string coefficients on the divided-power negative half).

Proof

technique · direct. The projector and its finite-sum properties are those of the integral skew-string supplier [F5]; the negative-half string decomposition, the operator identities and the grading shifts are then read off string by string, with the Gaussian identity [F4] supplying the only arithmetic input

1.1F2F3algebra

The $q$ -boson relation. By [F2] with $j = i$ and $a_{i i} = 2$ , $e_{i}^{'} F_{i} = q_{i}^{- 2} F_{i} e_{i}^{'} + 1$ ; applying the Leibniz rule to $F_{i}^{r}$ and summing the geometric series gives $e_{i}^{'} (F_{i}^{r}) = q_{i}^{- (r - 1)} [r]_{i} F_{i}^{r - 1}$ , and dividing by $[r]_{i}!$ yields $e_{i}^{'} F_{i}^{(r)} = q_{i}^{- (r - 1)} F_{i}^{(r - 1)} + q_{i}^{- 2 r} F_{i}^{(r)} e_{i}^{'}$ for $r \geq 1$ .

2.1step 1.1F3algebra

Iterated deletion. If $e_{i}^{'} u = 0$ , then induction on $n$ using step 1.1 gives $(e_{i}^{'})^{n} F_{i}^{(m)} u = q_{i}^{- n m + n (n + 1) / 2} F_{i}^{(m - n)} u$ for $0 \leq n \leq m$ , while $(e_{i}^{'})^{n} F_{i}^{(m)} u = (e_{i}^{'})^{n - m} u = 0$ for $n > m$ .

2.2F1F2step 1.1

Termination. By [F2] the derivation $e_{i}^{'}$ lowers positive height by one, so for homogeneous $x \in U_{q}^{-} [- ν]$ one has $(e_{i}^{'})^{n} x = 0$ whenever $n > \sum_{j} ν_{j} = ht (ν)$ ; hence on each $x$ only finitely many terms of the sum defining $P_{i}$ are nonzero, and $P_{i}$ is a well-defined $Q (q)$ -linear endomorphism of $U_{q}^{-}$ .

3.1F3F4step 1.1step 2.1step 2.2F5

The projection. For $u \in Ker e_{i}^{'}$ and $M \geq 0$ , steps 2.1 and [F3] give $P_{i} F_{i}^{(M)} u = \sum_{m = 0}^{M} (- 1)^{m} q_{i}^{- m (m - 1) / 2} q_{i}^{- m M + m (m + 1) / 2} {(\binom{M}{m})}_{i} F_{i}^{(M)} u = (\sum_{m = 0}^{M} (- 1)^{m} q_{i}^{- m (M - 1)} {(\binom{M}{m})}_{i}) F_{i}^{(M)} u$ , which by the Gaussian identity [F4] equals $u$ for $M = 0$ and $0$ for $M \geq 1$ . Hence $P_{i} F_{i} = 0$ , $P_{i}$ fixes $Ker e_{i}^{'}$ , and $P_{i}$ kills $f_{i} U_{q}^{-}$ ; moreover $e_{i}^{'} P_{i} = 0$ by the cancellation of consecutive terms in $e_{i}^{'} F_{i}^{(m)} e_{i}^{' m}$ using step 1.1, so $P_{i}^{2} = P_{i}$ and $P_{i}$ is the projection onto $Ker e_{i}^{'}$ along $f_{i} U_{q}^{-}$ , with $U_{q}^{-} = Ker e_{i}^{'} \oplus f_{i} U_{q}^{-}$ because $u = P_{i} u + (u - P_{i} u)$ lies in the two subspaces.

4.1F1F3F5step 2.2step 3.1algebra

Existence of the string decomposition. The algebra $U_{q}^{-}$ is spanned over $Q (q)$ by finite words in the $F_{j}$ , and each such word belongs to $U_{A}^{-}$ because $F_{j} = F_{j}^{(1)}$ . Thus write $u = \sum_{a = 1}^{t} c_{a} x_{a}$ with $c_{a} \in Q (q)$ and $x_{a} \in U_{A}^{-}$ . Apply [F5] to each $x_{a}$ and use the $Q (q)$ -linearity of $P_{i}$ and $e_{i}^{'}$ to obtain $u = \sum_{n} f_{i}^{(n)} u_{n}$ , where $u_{n} : = \sum_{a} c_{a} (x_{a})_{n} = q_{i}^{n (n - 1) / 2} P_{i} e_{i}^{' n} u \in Ker e_{i}^{'}$ . This sum is finite by step 2.2. The field-valued decomposition therefore follows from the integral one without assuming an integral basis or freeness.

5.1step 2.1step 3.1step 4.1

Uniqueness and the coefficient formula. Suppose $u = \sum_{m} f_{i}^{(m)} v_{m}$ with $v_{m} \in Ker e_{i}^{'}$ and $v_{m} = 0$ for $m$ large. Applying $e_{i}^{' n}$ and then $P_{i}$ for fixed $n$ : by step 2.1, $e_{i}^{' n} f_{i}^{(m)} v_{m} = q_{i}^{- n m + n (n + 1) / 2} f_{i}^{(m - n)} v_{m}$ for $m \geq n$ and $0$ for $m < n$ , and by step 3.1, $P_{i} f_{i}^{(m - n)} v_{m} = 0$ for $m > n$ while $P_{i} v_{n} = v_{n}$ ; hence $P_{i} e_{i}^{' n} u = q_{i}^{- n (n - 1) / 2} v_{n}$ , that is $v_{n} = q_{i}^{n (n - 1) / 2} P_{i} e_{i}^{' n} u = u_{n}$ . Thus the components of step 4.1 are determined by $u$ , giving uniqueness and the displayed coefficient formula.

6.1step 4.1step 5.1

The Kashiwara operators. Given the unique decomposition $u = \sum_{n} f_{i}^{(n)} u_{n}$ of steps 4.1-5.1, the prescriptions ${\tilde{e}}_{i} u : = \sum_{n \geq 1} f_{i}^{(n - 1)} u_{n}$ and ${\tilde{f}}_{i} u : = \sum_{n \geq 0} f_{i}^{(n + 1)} u_{n}$ are well-defined $Q (q)$ -linear maps by uniqueness; shifting the index shows ${\tilde{e}}_{i} {\tilde{f}}_{i} u = \sum_{n \geq 0} f_{i}^{(n)} u_{n} = u$ , so ${\tilde{e}}_{i} {\tilde{f}}_{i} = {i d}_{U_{q}^{-}}$ .

7.1step 4.1step 5.1step 6.1

Kernel, injectivity and value at $1$ . For $u = \sum_{n} f_{i}^{(n)} u_{n}$ one has ${\tilde{e}}_{i} u = 0$ exactly when $u_{n} = 0$ for every $n \geq 1$ , that is $u = u_{0} \in Ker e_{i}^{'}$ ; hence $\ker {\tilde{e}}_{i} = Ker e_{i}^{'}$ . If ${\tilde{f}}_{i} u = 0$ , then $\sum_{n} f_{i}^{(n + 1)} u_{n} = 0$ is a decomposition with all top coefficients $u_{n}$ in $Ker e_{i}^{'}$ , so uniqueness in step 5.1 forces $u_{n} = 0$ for all $n$ , that is $u = 0$ ; hence ${\tilde{f}}_{i}$ is injective. Finally $1 \in Ker e_{i}^{'}$ has decomposition $1 = f_{i}^{(0)} \cdot 1$ , so ${\tilde{f}}_{i} (1) = f_{i}^{(1)} = F_{i}$ .

7.2F1step 6.1

Root grading. Let $u \in U_{q}^{-} [- ν]$ be homogeneous. Since $f_{i}^{(n)}$ raises the degree by $- n α_{i}$ , the components $u_{n}$ of steps 4.1-5.1 may be taken homogeneous of degree $- ν + n α_{i}$ (decompose $u$ and each $u_{n}$ into finitely many homogeneous pieces and compare degrees), so $f_{i}^{(n - 1)} u_{n}$ has degree $- ν + α_{i}$ and $f_{i}^{(n + 1)} u_{n}$ has degree $- ν - α_{i}$ ; hence ${\tilde{e}}_{i} (U_{q}^{-} [- ν]) \subseteq U_{q}^{-} [- ν + α_{i}]$ and ${\tilde{f}}_{i} (U_{q}^{-} [- ν]) \subseteq U_{q}^{-} [- ν - α_{i}]$ .

8.1F5step 6.1∎

Integrality and choice. By [F5] the projector $P_{i}$ and every coefficient $u_{n} = q_{i}^{n (n - 1) / 2} P_{i} e_{i}^{' n} u$ preserve the divided-power lattice $U_{A}^{-}$ , and the shifts of step 6.1 preserve it as well; no freeness of $U_{A}^{-}$ is asserted. All constructions used unique decompositions, finite sums and the explicit operator $P_{i}$ ; no choice principle is used.

Depends on

Used by

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465-516 (standard reference, not scraped)
Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)
Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin and Vera Serganova, Berkeley Lectures on Lie Groups and Quantum Groups (book-length lecture notes, last updated 18 January 2024) (standard reference, not scraped)

The ii-string decomposition of the negative half