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The negative-half bilinear form is symmetric, nondegenerate and integral

Statement

Let $U_{q}^{-} = ⨁_{ν \in Q_{+}} U_{q}^{-} [- ν]$ be the negative half of the Drinfeld-Jimbo algebra of a finite symmetrizable Cartan datum with $a_{i i} = 2$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations, Positive, negative and toral quantum subalgebras and their root gradings), with skew derivations $e_{i}^{'}, e_{i}^{''}$ (Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half), and let $(,)$ be a bilinear form on $U_{q}^{-}$ .

(i) Existence and symmetry. A bilinear form with $(1, 1) = 1$ and $(F_{i} P, Q) = (P, e_{i}^{'} (Q))$ for all $i, P, Q$ exists on $U_{q}^{-}$ ; every normalized homogeneous form satisfying these conditions is symmetric; and by the uniqueness of The Lusztig-Kashiwara bilinear form on the negative half: the recursion there is exactly one such form. No separate right-adjunction formula for $(P, Q F_{i})$ is asserted: in the rank-one subalgebra a formula $(P, Q F_{i}) = c (e_{i}^{''} P, Q)$ with constant $c$ is false, as the explicit values in (iv) show.

(ii) Orthogonality. $(U_{q}^{-} [- ν], U_{q}^{-} [- μ]) = 0$ whenever $ν \neq μ$ .

(iii) Nondegeneracy. For every $0 \neq x \in U_{q}^{-}$ there is $y \in U_{q}^{-}$ with $(x, y) \neq 0$ ; equivalently, the restriction of the form to each finite-dimensional graded piece $U_{q}^{-} [- ν]$ is nondegenerate.

(iv) Integrality on the word lattice. Let $A = Z [q, q^{- 1}]$ (The Laurent polynomial ring as the principal localisation of Z[t] at t) and let $W \subseteq U_{A}^{-}$ be the $A$ -span of the monomials $F_{i_{1}} \dots F_{i_{k}}$ ( $k \geq 0$ ). Then $W \subseteq U_{A}^{-}$ (because $F_{i}^{k} = [k]_{i}! F_{i}^{(k)}$ and $[k]_{i}! \in A$ ), and $(W, W) \subseteq A$ . On the larger divided-power lattice $U_{A}^{-}$ the form is not $A$ -valued: in the rank-one subalgebra generated by a single $F_{i}$ one has, for all $a, b \geq 0$ , $(F_{i}^{(a)}, F_{i}^{(b)}) = δ_{a b} \frac{q_{i}^{- a (a - 1) / 2}}{[a]_{i}!} = δ_{a b} \frac{1}{[a]_{q_{i}^{2}}!},$ in particular $(F_{i}^{(2)}, F_{i}^{(2)}) = 1 / (1 + q_{i}^{2}) \notin A$ ; here $[a]_{q_{i}^{2}}! = \prod_{m = 1}^{a} (1 + q_{i}^{2} + \dots + q_{i}^{2 (m - 1)})$ . Globally, with $A_{l o c} = A [s^{- 1} : s \in S]$ , the localization of $A$ inverting the multiplicative set $S$ generated by ${[r]_{i}! : i \in I, r \geq 1}$ (equivalently inverting the Gaussian factorials $[r]_{q_{i}^{2}}!$ ), one has $(U_{A}^{-}, U_{A}^{-}) \subseteq A_{l o c}$ : divided-power words are word vectors divided by finite products of these factorials. This is an exact denominator bound, not an $A$ -integrality assertion on $U_{A}^{-}$ in this normalization.

Facts & Assumptions

Given: The negative half with its grading, the derivations $e_{i}^{'}, e_{i}^{''}$ and the integral form $U_{A}$ with $W \subseteq U_{A}^{-} \subseteq U_{q}^{-}$ (The Lusztig divided-power integral form $U_{A}$ ), and the axioms (H), (N), (R) of The Lusztig-Kashiwara bilinear form on the negative half: the recursion.

[F1]

The generic Serre halves $H^{+}, H^{-}$ over $Q (q)$ have primitive generators $e_{i}, f_{i}$ , braided tensor multiplication $(u \otimes v) (u^{'} \otimes v^{'}) = q^{- ⟨ \deg u^{'}, \deg v ⟩} u u^{'} \otimes v v^{'}$ , and a nondegenerate pairing $⟨, ⟩$ zero on unequal opposite degrees, with $⟨ e_{i}, f_{j} ⟩ = δ_{i j} / (q_{i} - q_{i}^{- 1})$ , $⟨ 1, 1 ⟩ = 1$ and both Hopf adjunctions. Opposite finite-dimensional components are dual. This is Generic quantum Serre halves have classical PBW ranks and a nondegenerate Hopf pairing (ii), with its declared AC use. The actual negative half has the separate symmetric Serre presentation by Triangular decomposition of a quantized enveloping algebra (ii), so replacing its $F_{i}$ by $e_{i}$ , respectively $f_{i}$ , gives algebra isomorphisms $φ_{+} : U_{q}^{-} \to H^{+}$ and $φ_{-} : U_{q}^{-} \to H^{-}$ , with opposite root degrees. The symmetric color form is $⟨ α_{i}, α_{j} ⟩ = d_{i} a_{i j}$ (Symmetrizable Cartan data for quantum groups).

[F2]

Axioms (H) and (N) together with (R) determine a form uniquely: peeling first letters writes every positive-height $P$ as $P = \sum_{i} F_{i} P_{i}$ and evaluates $B (P, Q) = \sum_{i} B (P_{i}, e_{i}^{'} Q)$ , reducing the height, with base $B (1, Q) = ε (Q)$ (The Lusztig-Kashiwara bilinear form on the negative half: the recursion).

[F3]

$e_{i}^{'}$ is $Q (q)$ -linear, and on words $e_{i}^{'} (F_{j_{1}} \dots F_{j_{m}}) = \sum_{t : j_{t} = i} q_{i}^{- a_{i j_{1}} - \dots - a_{i j_{t - 1}}} F_{j_{1}} \dots \hat{F_{j_{t}}} \dots F_{j_{m}}$ with coefficients in $A$ ; also $e_{i}^{'} (F_{i}^{(n)}) = q_{i}^{- (n - 1)} F_{i}^{(n - 1)}$ and $e_{i}^{'} F_{j} = q_{i}^{- a_{i j}} F_{j} e_{i}^{'} + δ_{i j}$ (Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half).

[F4]

$F_{i}^{(n)} = F_{i}^{n} / [n]_{i}!$ , $F_{i}^{n} = [n]_{i}! F_{i}^{(n)}$ , and $[m]_{i} = q_{i}^{- (m - 1)} [m]_{q_{i}^{2}}$ with $[a]_{q_{i}^{2}}! = \prod_{m = 1}^{a} (1 + q_{i}^{2} + \dots + q_{i}^{2 (m - 1)})$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F5]

$1 + q_{i}^{2} \notin A^{\times}$ in $A = Z [q, q^{- 1}]$ : its value at $q = 1$ is $2$ , and no Laurent polynomial of the form $(1 + q_{i}^{2})^{- 1}$ lies in $A$ (an inverse in the domain $Q [q, q^{- 1}]$ would force $1 = (1 + q_{i}^{2}) \cdot Laurent$ ). The Laurent polynomial ring as the principal localisation of Z[t] at t is used for the Laurent-ring arithmetic.

Verification

technique · construct the form by transporting and rescaling the generic braided pairing; derive its skew adjunction and symmetry on words, and recover uniqueness by first-letter peeling; and compute integrality by height induction on word vectors, with the rank-one recursion for the exact divided-power denominators

1.1F1algebra

Construction, homogeneity and nondegeneracy. For $ν = \sum_{i} n_{i} α_{i}$ put $c_{ν} = \prod_{i} (q_{i} - q_{i}^{- 1})^{n_{i}} \neq 0$ and, for $P, Q \in U_{q}^{-} [- ν]$ , define $B (P, Q) = c_{ν} ⟨ φ_{+} (P), φ_{-} (Q) ⟩$ using [F1]. Set $B$ to zero on unequal degrees and extend bilinearly over the algebraic direct sum. The isomorphisms ensure this is defined on the actual Serre quotient. The form is homogeneous and has $B (1, 1) = 1$ ; each component is nondegenerate because its pairing is dual and $c_{ν}$ is nonzero. Thus (ii) and (iii) hold, including the global assertion by selecting a nonzero component of a vector with finite support. The rescaled pairing still satisfies both Hopf adjunctions: in each nonzero summand the two factor degrees add to $ν$ , and $c_{ν_{1}} c_{ν_{2}} = c_{ν}$ . Its generator values are now $δ_{i j}$ .

2.1step 1.1F1F2F3algebra

Adjunction, symmetry and uniqueness. In the braided coproduct of a word $w = F_{j_{1}} \dots F_{j_{m}}$ , the component with first factor of degree $- α_{i}$ is $\sum_{t : j_{t} = i} q^{- \sum_{s < t} ⟨ α_{j_{s}}, α_{i} ⟩} f_{i} \otimes φ_{-} (w ∖ t)$ : the letter at position $t$ crosses precisely the earlier letters assigned to the second factor. The rescaled Hopf adjunction of step 1.1 therefore gives $B (F_{i} P, w) = \sum_{t : j_{t} = i} q_{i}^{- \sum_{s < t} a_{i j_{s}}} B (P, w ∖ t) = B (P, e_{i}^{'} w)$ by [F3]; unequal degrees give zero on both sides, so this proves (R) for all $P, Q$ by linearity. Repeated deletion gives, for equal-length words $u = F_{i_{1}} \dots F_{i_{m}}$ and $v = F_{j_{1}} \dots F_{j_{m}}$ , the formula $B (u, v) = \sum_{π} q^{- \sum_{a < b, π (a) > π (b)} ⟨ α_{i_{a}}, α_{i_{b}} ⟩}$ , where $π$ ranges over bijections with $j_{π (a)} = i_{a}$ ; if their color degrees differ the value is zero. Indeed the first deletion chooses $π (1)$ , its crossing exponent records exactly the inversions involving that first letter, and induction accounts for the remaining inversions, with empty-word value $1$ . The map $π \mapsto π^{- 1}$ preserves the exponent after interchanging the two words because the color form is symmetric, so $B$ is symmetric. Finally [F2] identifies every normalized homogeneous form satisfying (R) with $B$ . This proves (i), without attributing existence or symmetry to the Statement of the separation lemma.

3.1step 2.1F2F3F4algebra

Integrality on the word lattice. We show $B (u, v) \in A$ for classes of words $u, v$ by induction on the total height of $u$ plus $\deg v$ (in fact on the height of the first argument): if $u = 1$ , then $B (1, v) = ε (v) \in {0, 1} \subseteq A$ ; if $u = F_{i} w$ then $B (u, v) = B (w, e_{i}^{'} v)$ , and $e_{i}^{'} v$ is an $A$ -linear combination of classes of words of strictly smaller height by [F3], so the induction hypothesis applies. By bilinearity $B (W, W) \subseteq A$ , and $W \subseteq U_{A}^{-}$ because $F_{i}^{k} = [k]_{i}! F_{i}^{(k)}$ with $[k]_{i}! \in A$ by [F4].

3.2step 1.1step 2.1F3F4F5algebra

The rank-one divided-power values. Fix $i$ and assume $I = {i}$ for the computation, so that the rank-one subalgebra is the free algebra on $F_{i}$ and $e_{i}^{'} (F_{i}^{m}) = q_{i}^{- (m - 1)} [m]_{i} F_{i}^{m - 1}$ by inducting on $m$ from $e_{i}^{'} F_{i}^{m} = q_{i}^{- 2} F_{i} e_{i}^{'} F_{i}^{m - 1} + F_{i}^{m - 1}$ . Dividing by $[m]_{i}!$ gives $e_{i}^{'} (F_{i}^{(m)}) = q_{i}^{- (m - 1)} F_{i}^{(m - 1)}$ . Put $b_{m} : = B (F_{i}^{(m)}, F_{i}^{(m)})$ . Using $F_{i}^{(m)} = [m]_{i}^{- 1} F_{i} F_{i}^{(m - 1)}$ and (R), $b_{m} = [m]_{i}^{- 1} B (F_{i}^{(m - 1)}, e_{i}^{'} (F_{i}^{(m)})) = q_{i}^{- (m - 1)} [m]_{i}^{- 1} b_{m - 1}$ with $b_{0} = 1$ , so $b_{m} = q_{i}^{- m (m - 1) / 2} / [m]_{i}!$ . By [F4], $[m]_{i}! = q_{i}^{- m (m - 1) / 2} [m]_{q_{i}^{2}}!$ , hence $b_{m} = 1 / [m]_{q_{i}^{2}}! = \prod_{j = 1}^{m} (1 + \dots + q_{i}^{2 (j - 1)})^{- 1}$ ; for $m \neq n$ the two divided powers have different heights, so $B (F_{i}^{(m)}, F_{i}^{(n)}) = 0$ by homogeneity. At $m = 2$ this is $1 / (1 + q_{i}^{2}) \notin A$ by [F5]. This proves the rank-one clause of (iv) and shows that the form is not $A$ -valued on $U_{A}^{-}$ .

4.1step 3.1step 3.2F1F3F4algebra∎

The right-adjunction failure and the global denominator bound. First, no constant $c$ makes $(P, Q F_{i}) = c (e_{i}^{''} P, Q)$ hold on the rank-one subalgebra: $(F_{i}, F_{i}) = B (F_{i}, F_{i}) = 1$ and $(F_{i}, F_{i} \cdot 1) = 1$ force $c = 1$ from $(e_{i}^{''} F_{i}, 1) = (1, 1) = 1$ , but $(F_{i}^{2}, F_{i}^{2}) = [2]_{i}^{2} b_{2} = (q_{i} + q_{i}^{- 1})^{2} / (1 + q_{i}^{2}) = 1 + q_{i}^{- 2}$ while $(e_{i}^{''} F_{i}^{2}, F_{i}) = (1 + q_{i}^{2}) (F_{i}, F_{i}) = 1 + q_{i}^{2}$ , which differs; this justifies the parenthetical in (i). Second, every element of $U_{A}^{-}$ is an $A$ -linear combination of divided-power words $\prod_{i} F_{i}^{(r_{i})}$ with coefficients in $A$ (the divided powers generate $U_{A}^{-}$ by definition), and each such word is its underlying ordinary word divided by the finite product $\prod_{i} [r_{i}]_{i}!$ ; by bilinearity of $B$ and $B (W, W) \subseteq A$ from step 3.1, $B (U_{A}^{-}, U_{A}^{-}) \subseteq A_{l o c} = A [s^{- 1} : s \in S]$ . Since $[r]_{i}! = q_{i}^{- r (r - 1) / 2} [r]_{q_{i}^{2}}!$ by [F4] and $q_{i}^{\pm 1} \in A^{\times}$ , inverting the $[r]_{i}!$ is the same as inverting the Gaussian factorials $[r]_{q_{i}^{2}}!$ ; the bound is exact in rank one by step 3.2. This completes (iv).

Depends on

Used by

Cited to discharge well-definedness by The Lusztig-Kashiwara bilinear form on the negative half: the recursion.

Dependency tree · two levels

47 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

Benjamin Enriquez, PBW and Duality Theorems for Quantum Groups and Quantum Current Algebras, Journal of Lie Theory 13 (2003), 21-64 (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)
George Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110, Birkhauser, 1993; second printing 1994; author-hosted scan) (standard reference, not scraped)
Masaki Kashiwara, On crystal bases of the q-analogue of the universal enveloping algebra (Duke Math. J. 63 (1991), 465-516) (standard reference, not scraped)