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Integrable rank-one actions force the quantum Serre sums

Statement

Work over $Q (q)$ with a symmetrizable Cartan datum of type $A$ as in The cyclic affine type A Cartan datum and its residues, put $q_{i} = q^{d_{i}}$ and let $B$ be a unital $Q (q)$ -algebra equipped with elements $E_{i}, F_{i}, K_{h}^{\pm 1}$ ( $i \in I$ , $h \in P^{\lor}$ ) satisfying the toral and toral-action relations and the mixed commutators $E_{i} F_{j} - F_{j} E_{i} = δ_{i j} (K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1})$ of The Drinfeld-Jimbo quantized enveloping algebra by generators and relations; let $W$ be a $B$ -module with a chosen integral label grading on which $K_{h}$ acts by $q^{⟨ μ, h ⟩}$ on $W_{μ}$ , and $E_{i}, F_{i}$ shift labels by $\pm α_{i}$ . Assume that $E_{i}$ and $F_{i}$ are locally nilpotent on $W$ for every $i$ (rank-one integrability). Then the negative quantum Serre sums

$\sum_{r = 0}^{1 - a_{i j}} (- 1)^{r} {(\binom{1 - a_{i j}}{r})}_{i} F_{i}^{1 - a_{i j} - r} F_{j} F_{i}^{r} = 0 (i \neq j)$

vanish as operators on $W$ , and, by the Chevalley interchange $E_{i} \leftrightarrow F_{i}$ , $K_{h} \leftrightarrow K_{- h}$ , so do the positive sums $\sum_{r} (- 1)^{r} {(\binom{1 - a_{i j}}{r})}_{i} E_{i}^{1 - a_{i j} - r} E_{j} E_{i}^{r}$ . In particular the $e = 2$ cubic sums with $a_{01} = a_{10} = - 2$ hold. No Serre relation is assumed.

Facts & Assumptions

Given: A symmetrizable Cartan datum whose symmetrizer is $D = d i a g (1, \dots, 1)$ with $q_{i} = q$ , a $Q (q)$ -algebra $B$ with elements $E_{i}, F_{i}, K_{h}^{\pm 1}$ satisfying the toral, toral-action and mixed relations, and the chosen weight module $W$ with both families locally nilpotent.

[F1]

$K_{i} F_{j} K_{i}^{- 1} = q^{- a_{i j}} F_{j}$ and $K_{i} E_{j} K_{i}^{- 1} = q^{a_{i j}} E_{j}$ ; hence $F_{j}$ has weight $m : = - a_{i j} \geq 0$ under conjugation by $K_{i}$ , and $[E_{i}, F_{j}] = 0$ for $i \neq j$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations, The cyclic affine type A Cartan datum and its residues).

[F2]

In the rank-one subalgebra generated by $E_{i}, F_{i}, K_{i}^{\pm 1}$ the commutator formula $E_{i} F_{i}^{(b)} = F_{i}^{(b)} E_{i} + F_{i}^{(b - 1)} (q_{i}^{b} - q_{i}^{- b})^{- 1} \sum_{s = 0}^{b - 1} (q_{i}^{- 2 s} K_{i} - q_{i}^{2 s} K_{i}^{- 1})$ holds, together with the explicit string modules $L_{i} (N)$ of dimension $N + 1$ on which $E_{i}, F_{i}$ are injective/surjective string maps (Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules).

[F3]

Every finite-dimensional integrable $U_{i}$ -module is a direct sum of the string modules $L_{i} (N)$ ; every weight vector has a unique finite admissible expansion $\sum_{n \geq 0} F_{i}^{(n)} u_{n}$ in divided-power descendants of highest vectors, with $E_{i} u_{n} = 0$ and $n \leq ⟨ w t (u_{n}), h_{i} ⟩$ for each nonzero term (Uniqueness of the $i$ -string decomposition of an integrable module).

[F4]

$(x + y)^{N} = \sum_{r = 0}^{N} q^{r (N - r)} {(\binom{N}{r})}_{i} x^{r} y^{N - r}$ whenever $y x = q^{2} x y$ , and the Gaussian binomials satisfy the Pascal recurrences ${(\binom{n}{r})}_{i} = q^{- r} {(\binom{n - 1}{r})}_{i} + q^{n - r} {(\binom{n - 1}{r - 1})}_{i}$ (The quantum binomial expansion for $q$ -commuting elements, The quantum Pascal recurrences, the Gauss product formula and Gaussian integrality).

[F5]

The assignment $E_{i} \mapsto F_{i}$ , $F_{i} \mapsto E_{i}$ , $K_{h} \mapsto K_{- h}$ preserves the toral, toral-action and mixed relations and interchanges the two Serre families, so a proof of one family gives the other (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

Proof

technique · compute the iterated adjoint action of $F_i$ on $F_j$ by the Gaussian binomial formula, show it vanishes beyond the rank-one string bound, and read the Serre sum off the surviving coefficient

1.1F1F4givenalgebra

The adjoint action of $F_{i}$ on operators is $ad (T) = F_{i} T - K_{i} T K_{i}^{- 1} F_{i}$ , and on $T = F_{j}$ one has ${ad}^{n} (F_{j}) = \sum_{r = 0}^{n} (- 1)^{r} q^{r (m - n + 1)} {(\binom{n}{r})}_{i} F_{i}^{n - r} F_{j} F_{i}^{r}$ for every $n \geq 0$ , where $m = - a_{i j}$ is the $K_{i}$ -weight of $F_{j}$ from [F1]. The case $n = 0$ is the definition; the induction step multiplies the formula for $n$ on the left by $F_{i}$ and conjugates the rightmost factor, using $K_{i} F_{j} K_{i}^{- 1} = q^{m} F_{j}$ and $K_{i} F_{i}^{r} K_{i}^{- 1} = q^{- 2 r} F_{i}^{r}$ to move the toral elements, and the coefficient identity ${(\binom{n + 1}{r})}_{i} = q^{r} {(\binom{n}{r})}_{i} + q^{r - n - 1} {(\binom{n}{r - 1})}_{i}$ of [F4] (with $q$ replaced by $q_{i} = q$ ); the powers of $q$ combine to $q^{r (m - n)}$ .

2.1F1F2F3step 1.1givenalgebra

The finite Hom-space string bound. Fix $w \in W$ and let $U$ be its finite-dimensional rank-one submodule: for a homogeneous vector take the span of $F_{i}^{a} E_{i}^{b} w$ , with finitely many $b$ by local nilpotence of $E_{i}$ and finitely many $a$ for each $b$ by local nilpotence of $F_{i}$ ; the rank-one commutator in [F2] makes this span stable. For a general vector sum the finitely many such spaces of its label components. Choose a finite basis of $U$ and let $U^{'}$ be the sum of the finite rank-one submodules generated by its $F_{j}$ -images. Then $F_{j} ∣_{U}$ belongs to the finite type-one rank-one module $Hom (U, U^{'})$ , with operators $K (T) = K_{i}^{'} T K_{i}^{- 1}$ , $E (T) = (E_{i}^{'} T - T E_{i}) K_{i}$ and $F (T) = F_{i}^{'} T - K (T) F_{i}$ . Substitution of the rank-one relations verifies $K E K^{- 1} = q^{2} E$ , $K F K^{- 1} = q^{- 2} F$ and $[E, F] = (K - K^{- 1}) / (q - q^{- 1})$ . Its integral weights lie in a finite set, so both weight-shifting operators are nilpotent. For $T = F_{j} ∣_{U}$ , [F1] gives $K (T) = q^{m} T$ and $E (T) = 0$ . The highest-vector endpoint calculation in [F2] (or the rank-one proof of [F3], which uses no Serre relations) therefore gives $F^{m + 1} T = 0$ . Its expansion is exactly step 1.1 restricted to $U$ , so ${ad}^{m + 1} (F_{j}) w = 0$ . No preservation of $U$ by $F_{j}$ is assumed.

3.1step 1.1step 2.1givenalgebra

The negative Serre sum vanishes. Put $n : = m + 1 = 1 - a_{i j}$ . At $n = m + 1$ the exponent in the expansion of step 1.1 is $r (m - n + 1) = 0$ , so step 2.1 gives $0 = {ad}^{n} (F_{j}) = \sum_{r = 0}^{n} (- 1)^{r} {(\binom{n}{r})}_{i} F_{i}^{n - r} F_{j} F_{i}^{r}$ ; as an operator identity on each $U$ containing $w$ , hence on every vector $w \in W$ this is exactly the negative Serre sum.

4.1step 3.1F5givenalgebra

The positive Serre sum vanishes. The Chevalley interchange of [F5] preserves all toral, toral-action and mixed relations used above and exchanges the roles of the $E$ 's and $F$ 's, so applying the argument of steps 1.1--3.1 to the image operators gives $\sum_{r = 0}^{1 - a_{i j}} (- 1)^{r} {(\binom{1 - a_{i j}}{r})}_{i} E_{i}^{1 - a_{i j} - r} E_{j} E_{i}^{r} = 0$ on every weight vector, hence as operators.

5.1step 3.1step 4.1∎

Conclusion. Under only the toral, toral-action, mixed-commutator and rank-one-integrability hypotheses, both quantum Serre sums of order $1 - a_{i j}$ vanish, for every pair $i \neq j$ of the cyclic affine type $A$ datum, the cubic case included; no Serre relation was assumed.

Remarks

The two finite rank-one spaces and their Hom-space in step 2.1 are produced by the rank-one commutator alone; no mixed Serre relation is used before the conclusion, so the implication is not circular.
Only the rank-one argument of Uniqueness of the $i$ -string decomposition of an integrable module is used in step 2.1; its finite reduction and highest-vector endpoint calculation require no Serre relations.

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Sources

Kashiwara-Miwa-Petersen-Yung, Perfect crystals and q-deformed Fock spaces (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)
Bernard Leclerc, Fock space representations (author-hosted complete 38-page treatment) (standard reference, not scraped)