DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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 Lusztig-Kashiwara bilinear form on the negative half: the recursion

Definition

Let $U_{q}^{-} = ⨁_{ν \in Q_{+}} U_{q}^{-} [- ν]$ be the negative half of the Drinfeld–Jimbo algebra with its skew derivations $e_{i}^{'}$ (Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half, Positive, negative and toral quantum subalgebras and their root gradings), and let $ε ⁣ : U_{q}^{-} \to Q (q)$ be the augmentation, the projection onto the degree-zero component $U_{q}^{-} [0] = Q (q) \cdot 1$ (so $ε (Q) = 0$ on all positive-height components and $ε (1) = 1$ ).

The normalized Lusztig–Kashiwara form on $U_{q}^{-}$ is a bilinear form $B ⁣ : U_{q}^{-} \times U_{q}^{-} \to Q (q)$ such that

(H) $B$ is homogeneous: $B (U_{q}^{-} [- ν], U_{q}^{-} [- μ]) = 0$ whenever $ν \neq μ$ ;

(N) $B (1, 1) = 1$ ;

(R) $B (F_{i} P, Q) = B (P, e_{i}^{'} Q)$ for all $i \in I$ and all $P, Q \in U_{q}^{-}$ .

Equivalently, axioms (H) and (N) may be replaced by the single counit normalization

in the presence of (R) (Verification, step 1.1).

Axioms (H) and (N) (equivalently (C)) are essential and not redundant: the recursion (R) together with $B (1, 1) = 1$ alone neither forces symmetry nor uniqueness, as the rank-one witness of step 2.2 shows.

The axioms determine $B$ uniquely: peeling first letters in the first argument writes every positive-height $P$ as a finite sum $P = \sum_{i} F_{i} P_{i}$ and evaluates $B (P, Q) = \sum_{i} B (P_{i}, e_{i}^{'} Q)$ , reducing the height; the base of the induction is (C) (Verification, step 2.1). The existence, symmetry and nondegeneracy of $B$ on $U_{q}^{-}$ for the actual quotient by the Serre ideal are not proved here: they are the content of the negative-half form theorem thm-the-negative-half-bilinear-form-is-symmetric-nondegenerate-and-integral, the declared justification of this definition, which also settles the exact integrality normalization ( $A$ -valuedness on the word lattice and the Gaussian-factorial denominators on the divided-power lattice). That theorem is proved on this page, and its statement is the declared supplier of existence, symmetry and nondegeneracy; this definition itself claims only that (H), (N) and (R) determine $B$ and that the normalization cannot be relaxed.

Facts & Assumptions

Given: The negative half with its root grading and the skew derivations $e_{i}^{'}$ , the augmentation $ε$ , and a bilinear form $B$ on $U_{q}^{-}$ .

[F1]

Each $U_{q}^{-} [- ν]$ is a $Q (q)$ -subspace and $U_{q}^{-} = ⨁_{ν} U_{q}^{-} [- ν]$ is a direct sum; every word in the $F_{i}$ lies in the component of its degree, and for $ν \neq 0$ every such word begins with some letter $F_{i}$ , so every $P \in U_{q}^{-} [- ν]$ is a finite sum $P = \sum_{i} F_{i} P_{i}$ with $P_{i} \in U_{q}^{-} [- (ν - α_{i})]$ (Positive, negative and toral quantum subalgebras and their root gradings).

[F2]

$e_{i}^{'} (U_{q}^{-} [- ν]) \subseteq U_{q}^{-} [- (ν - α_{i})]$ , and $e_{i}^{'}$ is $Q (q)$ -linear with $e_{i}^{'} (1) = 0$ (Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half).

[F3]

The augmentation $ε$ is $Q (q)$ -linear and vanishes on every $U_{q}^{-} [- ν]$ with $ν \neq 0$ ; $ε (1) = 1$ (Positive, negative and toral quantum subalgebras and their root gradings).

[F4]

The free algebra on one generator $F$ is the tensor algebra of a one-dimensional space; it has the word basis $1, F, F^{2}, \dots$ (Tensor algebra of a vector space).

Verification

technique · reduce the normalizations to each other by peeling, prove uniqueness by height induction, and exhibit the failure of the unnormalized recursion in rank one

1.1givenF1F2F3algebra

Equivalence of the normalizations. Assume (H) and (N); if $Q \in U_{q}^{-} [- μ]$ with $μ \neq 0$ then $Q = ε (Q) 1 + Q^{'}$ with $Q^{'} \in ⨁_{ν \neq 0} U_{q}^{-} [- ν]$ and $B (1, Q) = B (1, ε (Q) 1) + B (1, Q^{'}) = ε (Q) B (1, 1) + 0$ by (H), so (C) holds. Conversely assume (C) and (R); then (N) is immediate, and (H) is proved by induction on $ht (ν)$ for $P \in U_{q}^{-} [- ν]$ , $Q \in U_{q}^{-} [- μ]$ : for $ν = 0$ one gets $B (1, Q) = ε (Q)$ , which vanishes unless $μ = 0$ ; for $ν \neq 0$ write $P = \sum_{i} F_{i} P_{i}$ as in [F1] and compute $B (P, Q) = \sum_{i} B (P_{i}, e_{i}^{'} Q)$ by (R), where $e_{i}^{'} Q \in U_{q}^{-} [- (μ - α_{i})]$ by [F2]; by induction each term vanishes unless $ν - α_{i} = μ - α_{i}$ , i.e. unless $ν = μ$ . Hence (H), (N) and (R) hold.

2.1step 1.1F1F2algebra

Uniqueness: let $B, B^{'}$ satisfy (H), (N) and (R). We prove $B (P, Q) = B^{'} (P, Q)$ by induction on $ht (ν)$ for $P \in U_{q}^{-} [- ν]$ , all $Q$ simultaneously. For $ν = 0$ , $P = c 1$ and $B (P, Q) = c ε (Q) = B^{'} (P, Q)$ by step 1.1. For $ν \neq 0$ , choose a decomposition $P = \sum_{i} F_{i} P_{i}$ as in [F1]; then $B (P, Q) = \sum_{i} B (P_{i}, e_{i}^{'} Q) = \sum_{i} B^{'} (P_{i}, e_{i}^{'} Q) = B^{'} (P, Q)$ , the middle equality by the induction hypothesis applied to the lower-height elements $P_{i}$ and to the arguments $e_{i}^{'} Q$ . Hence at most one form satisfies the axioms, and it is uniquely reconstructed from $ε$ by the recursion.

2.2step 1.1F4algebra

The normalization is essential. Work in the free algebra $Q (q) ⟨ F ⟩$ on one generator [F4], and let $\tilde{e}^{'}$ be the word recursion of Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half transported to it, so $\tilde{e}^{'} (1) = 0$ and $\tilde{e}^{'} (F^{m}) = [m]_{q^{- 2}} F^{m - 1}$ (the free-algebra analogue of the deletion rule). Fix any sequence $(c_{m})_{m \geq 0}$ with $c_{0} = 1$ and define a bilinear form by $B (1, F^{m}) : = c_{m}$ , and recursively $B (F u, Q) : = B (u, \tilde{e}^{'} Q)$ . Then $B (1, 1) = 1$ and the recursion hold by construction. Taking $c_{1} = 1$ gives $B (1, F) = 1$ , while $B (F, 1) = B (1, \tilde{e}^{'} (1)) = B (1, 0) = 0$ ; hence this $B$ is not symmetric, and the values $c_{m}$ are exactly the freedom left by the recursion and $B (1, 1) = 1$ . Thus the counit normalization (C), equivalently the homogeneity (H), is what removes the freedom, and the normalized form is the case $c_{m} = 0$ for $m \geq 1$ .

3.1step 1.1step 2.1step 2.2given∎

The remaining claims are the declared content of the negative-half form theorem: on the quotient $U_{q}^{-}$ of the free algebra by the Serre ideal, thm-the-negative-half-bilinear-form-is-symmetric-nondegenerate-and-integral proves that the values forced by (C) and (R) descend (existence), that the resulting form is symmetric and nondegenerate, and it states the exact integrality normalization on the divided-power lattice; by step 2.1 this descended form is the unique one satisfying (H), (N) and (R). Accordingly this definition is complete modulo that supplier, and no transposed or rescaled variant of the recursion is used anywhere.

Depends on

Used by

Dependency tree · two levels

14 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

Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)
George Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110, Birkhauser, 1993; second printing 1994; author-hosted scan) (standard reference, not scraped)