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The dominant quantum Serre quotient is nonzero and integrable

Statement

Let $λ \in P$ be dominant integral, so $m_{i} : = ⟨ λ, h_{i} ⟩ \geq 0$ for all $i$ , and let $M (λ)$ be the Verma module with highest vector $v_{λ}$ (Quantum Verma modules $M (λ)$ , highest weight vectors, and the simple highest weight quotient $L (λ)$ ). Put $s_{i} : = F_{i}^{(m_{i} + 1)} v_{λ}$ .

(i) Every $s_{i}$ is a singular (highest weight) vector of weight $λ - (m_{i} + 1) α_{i}$ .

(ii) The graded submodule $S : = \sum_{i} U_{q} (g) s_{i} = \sum_{i} U_{q}^{-} s_{i}$ is proper, and the quotient $\tilde{L} (λ) : = M (λ) / S$ is nonzero, has finite-dimensional label-weight spaces, and is integrable (Quantum weight modules, weights in $P$ , and integrability).

(iii) Every integrable cyclic highest weight module $V$ of highest weight $λ$ is a quotient of $\tilde{L} (λ)$ .

No commutation between divided powers $F_{i}^{(a)}, F_{j}^{(b)}$ of distinct indices with $a_{i j} \neq 0$ is assumed anywhere.

Facts & Assumptions

Given: A dominant integral weight $λ \in P$ , the Verma module $M (λ) = U_{q}^{-} v_{λ}$ with its label grading, and the elements $s_{i} = F_{i}^{(m_{i} + 1)} v_{λ}$ .

[F1]

$M (λ)$ is a $P$ -graded type-one module with $M (λ)_{λ - ν} = U_{q}^{-} [- ν] v_{λ}$ , finite-dimensional graded pieces, $K_{h} v_{λ} = q^{⟨ λ, h ⟩} v_{λ}$ and $E_{i} v_{λ} = 0$ ; the labels of all nonzero vectors lie in $λ - Q_{+}$ (Quantum Verma modules $M (λ)$ , highest weight vectors, and the simple highest weight quotient $L (λ)$ , Quantum weight modules, weights in $P$ , and integrability).

[F2]

$E_{i}^{(a)}$ and $F_{j}^{(b)}$ commute for $i \neq j$ , while for every $b \geq 1$ the rank-one reordering gives $E_{i} F_{i}^{(b)} = F_{i}^{(b)} E_{i} + F_{i}^{(b - 1)} T_{i, 1 - b, 1}$ with $T_{i, 1 - b, 1} v_{λ} = [m_{i} - b + 1]_{i} v_{λ}$ (Divided-power multiplication, toral binomials and the $E_{i}^{(a)} F_{i}^{(b)}$ reordering identity (iii), Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules (ii)).

[F3]

For every $j \neq i$ with $N : = 1 - a_{i j} \geq 1$ , the quantum Serre relation $\sum_{r = 0}^{N} (- 1)^{r} {(\binom{N}{r})}_{i} F_{i}^{N - r} F_{j} F_{i}^{r} = 0$ holds, monic in $F_{j}$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F4]

$[n]_{i} = 0$ if and only if $n = 0$ (for integers $n$ ), and $[n]_{i}! \neq 0$ for all $n \geq 0$ ; consequently $F_{i}^{(t)} \neq 0$ whenever $F_{i}^{t} \neq 0$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

Verification

technique · compute the action of the raising generators on the divided-power string vectors, prove the generated submodule is proper by its label support, and prove local nilpotence of $F_i$ by a word induction that uses the quantum Serre relation instead of the false commutation of adjacent generators

1.1givenF1F2algebra

The vector $s_{i}$ is singular. For the raising operator $E_{i}$ : by [F2], $E_{i} F_{i}^{(m_{i} + 1)} v_{λ} = F_{i}^{(m_{i})} T_{i, - m_{i}, 1} v_{λ} = [m_{i} - (m_{i} + 1) + 1]_{i} F_{i}^{(m_{i})} v_{λ} = [0]_{i} F_{i}^{(m_{i})} v_{λ} = 0$ . For $j \neq i$ : $E_{j}$ commutes with $F_{i}^{(m_{i} + 1)}$ by [F2], so $E_{j} s_{i} = F_{i}^{(m_{i} + 1)} E_{j} v_{λ} = 0$ . Also $K_{h} s_{i} = q^{⟨ λ - (m_{i} + 1) α_{i}, h ⟩} s_{i}$ by [F1], so $s_{i}$ is a highest weight vector of weight $λ - (m_{i} + 1) α_{i}$ .

1.2givenF3algebra

Local nilpotence of $F_{i}$ , first the Serre rewrite. Claim: for every $R \geq N : = 1 - a_{i j}$ ( $j \neq i$ ) one has $F_{i}^{R} F_{j} = \sum_{t = 0}^{N - 1} c_{t} (R) F_{i}^{t} F_{j} F_{i}^{R - t}$ with coefficients $c_{t} (R) \in Q (q)$ that may depend on $R$ . For $R = N$ this is [F3] solved for $F_{i}^{N} F_{j}$ : the $F_{i}^{N} F_{j}$ coefficient is ${(\binom{N}{0})}_{i} = 1$ , giving $F_{i}^{N} F_{j} = \sum_{r = 1}^{N} (- 1)^{r + 1} {(\binom{N}{r})}_{i} F_{i}^{N - r} F_{j} F_{i}^{r}$ . If the claim holds for $R$ , then $F_{i}^{R + 1} F_{j} = F_{i} (\sum_{t} c_{t} (R) F_{i}^{t} F_{j} F_{i}^{R - t}) = \sum_{t \leq N - 2} c_{t} (R) F_{i}^{t + 1} F_{j} F_{i}^{R - t} + c_{N - 1} (R) F_{i}^{N} F_{j} F_{i}^{R - N + 1}$ , and substituting the $R = N$ expansion into the last term expresses it in the required form with $t = N - r \leq N - 1$ .

2.1step 1.1F1algebra

The submodule $S$ is proper. Since each $s_{i}$ is a highest weight vector killed by all $E_{j}$ and acted on by the torus by scalars, the triangular decomposition gives $U_{q} (g) s_{i} = U_{q}^{-} s_{i}$ . Hence $S = \sum_{i} U_{q}^{-} s_{i}$ is spanned by vectors whose labels lie in $λ - (m_{i} + 1) α_{i} - Q_{+}$ , all of which are strictly below $λ$ in the order of [F1]; therefore the top component of $S$ vanishes, $v_{λ} \notin S$ , and $S \neq M (λ)$ : the quotient $\tilde{L} (λ) = M (λ) / S$ is nonzero with ${\overset{ˉ}{v}}_{λ} \neq 0$ . Being a quotient of $M (λ)$ by a graded submodule, $\tilde{L} (λ)$ is graded with label-weight spaces that are quotients of the finite-dimensional spaces $M (λ)_{μ}$ ; its support lies in $λ - Q_{+}$ and its top space is $Q (q) {\overset{ˉ}{v}}_{λ}$ .

2.2step 1.2F1F4algebra

Local nilpotence of $F_{i}$ on every vector. We prove by induction on the word length $m$ that for every word $w = F_{j_{1}} \dots F_{j_{m}}$ there is an integer $K (w)$ with $F_{i}^{K} w {\overset{ˉ}{v}}_{λ} = 0$ for $K \geq K (w)$ . For $m = 0$ , $F_{i}^{m_{i} + 1} {\overset{ˉ}{v}}_{λ} = 0$ in the quotient by its defining relation, and therefore $F_{i}^{t} {\overset{ˉ}{v}}_{λ} = 0$ for $t \geq m_{i} + 1$ . For the step, write $w = F_{j} w^{'}$ . If $j = i$ then $F_{i}^{K} w {\overset{ˉ}{v}}_{λ} = F_{i}^{K + 1} w^{'} {\overset{ˉ}{v}}_{λ} = 0$ for $K + 1 \geq K (w^{'})$ . If $j \neq i$ , let $N = 1 - a_{i j}$ and let $S : = K (w^{'})$ ; for $R \geq S + N - 1$ the rewrite of step 1.2 gives $F_{i}^{R} w {\overset{ˉ}{v}}_{λ} = \sum_{t = 0}^{N - 1} c_{t} (R) F_{i}^{t} F_{j} F_{i}^{R - t} w^{'} {\overset{ˉ}{v}}_{λ}$ with $R - t \geq S$ , so each $F_{i}^{R - t} w^{'} {\overset{ˉ}{v}}_{λ} = 0$ by induction and $F_{i}^{R} w {\overset{ˉ}{v}}_{λ} = 0$ . Thus $K (w) : = S + N - 1$ works. Since $\tilde{L} (λ)$ is spanned by the finitely many word classes $w {\overset{ˉ}{v}}_{λ}$ in each label piece, a general vector is a finite combination of such classes and is killed by a large power of $F_{i}$ : $F_{i}$ is locally nilpotent on $\tilde{L} (λ)$ .

3.1step 2.1F1algebra

Local nilpotence of $E_{i}$ on $\tilde{L} (λ)$ : if $x$ is a nonzero vector of label $μ$ , then $E_{i}^{n} x$ has label $μ + n α_{i}$ , and by the support bound of step 2.1 a nonzero vector of that label requires $λ - μ - n α_{i} \in Q_{+}$ , i.e. $n \leq ht (λ - μ) < \infty$ . Hence $E_{i}^{n} x = 0$ for $n > ht (λ - μ)$ , uniformly for $x$ of that label: $E_{i}$ is locally nilpotent.

4.1step 1.1step 2.1step 3.1step 2.2F1F2F3F4algebra∎

The universal property and conclusion. Let $V$ be an integrable cyclic highest weight module of weight $λ$ with highest vector $v \neq 0$ ( $E_{i} v = 0$ , $K_{h} v = q^{⟨ λ, h ⟩} v$ , $V = U_{q} (g) v$ ). By the universal property of the Verma module there is a surjection $M (λ) ↠ V$ , $v_{λ} \mapsto v$ . Let $t_{i}$ be minimal with $F_{i}^{t_{i}} v = 0$ (it exists since $V$ is integrable). If $t_{i} \geq 1$ then $F_{i}^{t_{i} - 1} v \neq 0$ ; applying $E_{i}$ and using the rank-one reordering of [F2] gives $0 = E_{i} F_{i}^{(t_{i})} v = [m_{i} - t_{i} + 1]_{i} F_{i}^{(t_{i} - 1)} v$ , so $[m_{i} - t_{i} + 1]_{i} = 0$ and hence $t_{i} = m_{i} + 1$ by [F4]. Therefore the surjection kills every $s_{i} = F_{i}^{(m_{i} + 1)} v_{λ}$ , hence kills $S$ , and factors through $\tilde{L} (λ) ↠ V$ : every integrable cyclic highest weight module of weight $λ$ is a quotient of $\tilde{L} (λ)$ . Together with the preceding construction this proves (i), (ii) and (iii); no commutation between distinct adjacent-root negative divided powers was used, only the quantum Serre relation and the cross-half commutation.

Depends on

Used by

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Sources

Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)
Math 263C Quantum Groups (Stanford, 2022), lecture slides 'Lusztig's quantum group. Tilting modules.' (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)