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Mixed relations, local nilpotence and weights give the integrable quantum action

Statement

Let $V = ⨁_{μ \in P} V_{μ}$ be a label-weight-graded $Q (q)$ -vector space with toral operators $K_{h}$ ( $h \in P^{\lor}$ ) acting on $V_{μ}$ by $q^{⟨ μ, h ⟩}$ , and operators $E_{i}, F_{i}$ of weights $α_{i}, - α_{i}$ mapping $V_{μ}$ to $V_{μ + α_{i}}$ and $V_{μ - α_{i}}$ . Assume the toral conjugation and mixed relations, and local nilpotence of each $E_{i}$ and $F_{i}$ ; vectors have finite label support. Then these operators uniquely define an integrable type-one $U_{q} (g)$ -module structure on $V$ for EVERY real symmetrizable Cartan datum. Every vector has a finite rank-one $U_{i}$ orbit and the admissible Kashiwara $i$ -string decomposition applies. Finite weight spaces do not imply finite-dimensional full- $g$ orbits; no such conclusion is asserted.

Facts & Assumptions

Given: A symmetrizable Cartan datum $(I, A, D, P, P^{\lor}, q)$ with $a_{i i} = 2$ , over $Q (q)$ , and a label-graded space $V = ⨁_{μ \in P} V_{μ}$ with operators $K_{h}$ ( $h \in P^{\lor}$ ), $E_{i}$ , $F_{i}$ satisfying:

[A1]

(toral and toral conjugation) $K_{0} = 1$ , $K_{h} K_{h^{'}} = K_{h + h^{'}}$ , $K_{h}$ acts on $V_{μ}$ by $q^{⟨ μ, h ⟩}$ , $K_{h} E_{i} K_{h}^{- 1} = q^{⟨ α_{i}, h ⟩} E_{i}$ and $K_{h} F_{i} K_{h}^{- 1} = q^{- ⟨ α_{i}, h ⟩} F_{i}$ ; write $K_{i} : = K_{d_{i} h_{i}}$ , so $K_{i}$ acts on $V_{μ}$ by $q_{i}^{⟨ μ, h_{i} ⟩}$ with $q_{i} = q^{d_{i}}$ .

[A2]

(mixed relations) $E_{i} F_{j} - F_{j} E_{i} = δ_{i j} (K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1})$ for all $i, j$ ; in particular $E_{i} F_{j} = F_{j} E_{i}$ for $i \neq j$ .

[A3]

(local nilpotence) for every $v \in V$ and every $i$ there are $N_{e} (v, i), N_{f} (v, i)$ with $E_{i}^{N_{e} (v, i)} v = F_{i}^{N_{f} (v, i)} v = 0$ ; every vector is a finite sum of label components.

[F1]

In $U_{q} (g)$ the displayed toral, toral-action, mixed and Serre relations hold, and a linear map on the generators satisfying all of them extends uniquely to a $U_{q} (g)$ -module structure (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations, Symmetrizable Cartan data for quantum groups, A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring).

[F2]

$[m]_{i} = (q_{i}^{m} - q_{i}^{- m}) / (q_{i} - q_{i}^{- 1})$ and $[m]_{i}! \neq 0$ for $m \geq 0$ , and $[m]_{i} = 0$ exactly when $m = 0$ , because $q_{i}$ is transcendental over $Q$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F3]

In the rank-one algebra generated by $E_{i}, F_{i}, K_{i}^{\pm 1}$ , with $F_{i}^{(b)} = F_{i}^{b} / [b]_{i}!$ , one has $E_{i} F_{i}^{(b)} = F_{i}^{(b)} E_{i} + F_{i}^{(b - 1)} Θ_{i, b}, Θ_{i, b} = \frac{1}{q_{i}^{b} - q_{i}^{- b}} \sum_{s = 0}^{b - 1} (q_{i}^{- 2 s} K_{i} - q_{i}^{2 s} K_{i}^{- 1})$ for $b \geq 1$ , and $K_{i} E_{i} K_{i}^{- 1} = q_{i}^{2} E_{i}$ , $K_{i} F_{i} K_{i}^{- 1} = q_{i}^{- 2} F_{i}$ (Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules, parts (i) and (ii)).

[F4]

The Gaussian quotients satisfy the Pascal recurrence $C_{m, r} = q_{i}^{r} C_{m - 1, r} + q_{i}^{r - m} C_{m - 1, r - 1}$ with $C_{m, r} = 0$ outside $0 \leq r \leq m$ (The quantum Pascal recurrences, the Gauss product formula and Gaussian integrality).

[F5]

Once $V$ is a chosen $P$ -graded type-one quantum weight module on which $E_{i}$ and $F_{i}$ are locally nilpotent, the $i$ -string decomposition theorem applies to $V$ (Uniqueness of the $i$ -string decomposition of an integrable module, Quantum weight modules, weights in $P$ , and integrability).

Proof

technique · finite rank-one reductions: enclose each vector in a finite-dimensional rank-one module, use the rank-one adjoint action on a Hom space to turn $F_j$ into a highest vector, bound its string by the divided-power commutation formula, and read the Serre sum off the surviving coefficient

1.1A1A2A3

Finite rank-one orbits. Fix $i$ and $v \in V$ . Then $W : = span {F_{i}^{a} E_{i}^{b} K_{i}^{c} v : a, b \geq 0, c \in Z}$ is a finite-dimensional $A_{i}$ -stable subspace, where $A_{i}$ is the algebra generated by $E_{i}, F_{i}, K_{i}^{\pm 1}$ : the toral operator $K_{i}$ acts diagonally on the finitely many nonzero label components of $v$ , so the vectors $K_{i}^{c} v$ span a finite-dimensional subspace; by [A3] there is $B$ with $E_{i}^{b} v = 0$ for $b \geq B$ ; the mixed relation of [A2] rewrites any word in $E_{i}, F_{i}$ as a linear combination of words $F_{i}^{a} E_{i}^{b}$ with coefficients in the algebra generated by $K_{i}$ ; and one may choose finitely many $K_{i}^{c} v$ spanning the toral orbit and use local nilpotence of $F_{i}$ on their finitely many $E_{i}^{b}$ -images to bound $a$ . The span of the resulting finite set is stable under $E_{i}, F_{i}, K_{i}^{\pm 1}$ .

2.1step 1.1A1A2A3

The second finite module. Fix $i \neq j$ and set $W^{'} : = span {F_{i}^{a} E_{i}^{b} K_{i}^{c} F_{j} w : w \in W, a, b \geq 0, c \in Z}$ . Since $W$ is finite-dimensional, step 1.1 applied to each of the finitely many vectors $F_{j} w$ exhibits $W^{'}$ as a finite union of finite-dimensional $A_{i}$ -stable subspaces; hence $W^{'}$ is a finite-dimensional $A_{i}$ -stable subspace of $V$ containing $F_{j} W$ .

3.1A1A2step 2.1

The rank-one adjoint module. On the finite-dimensional space $Hom (W, W^{'})$ define operators by $A d (K_{i}) (T) = K_{i} T K_{i}^{- 1}$ , $A d (E_{i}) (T) = (E_{i} T - T E_{i}) K_{i}$ and $A d (F_{i}) (T) = F_{i} T - K_{i} T K_{i}^{- 1} F_{i}$ . Substituting these formulas, using $K_{i} E_{i} = q_{i}^{2} E_{i} K_{i}$ , $K_{i} F_{i} = q_{i}^{- 2} F_{i} K_{i}$ and $E_{i} F_{i} - F_{i} E_{i} = (K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1})$ on $W, W^{'}$ , gives $A d (K_{i}) A d (E_{i}) A d (K_{i})^{- 1} = q_{i}^{2} A d (E_{i})$ , $A d (K_{i}) A d (F_{i}) A d (K_{i})^{- 1} = q_{i}^{- 2} A d (F_{i})$ and $[A d (E_{i}), A d (F_{i})] = (A d (K_{i}) - A d (K_{i})^{- 1}) / (q_{i} - q_{i}^{- 1})$ . Thus these formulas give a rank-one module directly. Moreover $A d (K_{i})$ is diagonalizable because $K_{i}$ acts diagonally on $W$ and $W^{'}$ , and $A d (E_{i}), A d (F_{i})$ are locally nilpotent: $A d (E_{i})$ raises and $A d (F_{i})$ lowers the $A d (K_{i})$ -weight by $2$ , the weights of $Hom (W, W^{'})$ form a finite set, and decomposing any $T$ into weight components each orbit is supported on distinct weights and hence terminates.

4.1A1A2step 3.1

$F_{j}$ is a highest vector of weight $m$ . Since $F_{j} W \subseteq W^{'}$ , the restriction $T : = F_{j} ∣_{W}$ lies in $Hom (W, W^{'})$ . By [A2], $E_{i} F_{j} = F_{j} E_{i}$ for $i \neq j$ , so $A d (E_{i}) (T) = (E_{i} T - T E_{i}) K_{i} = 0$ ; and by [A1], $A d (K_{i}) (T) = K_{i} F_{j} K_{i}^{- 1} = q_{i}^{- a_{i j}} T$ , because $⟨ α_{j}, d_{i} h_{i} ⟩ = d_{i} a_{i j}$ . Put $m : = - a_{i j} \geq 0$ , the off-diagonal Cartan entries being nonpositive. Thus $T$ is annihilated by $A d (E_{i})$ and has $A d (K_{i})$ -weight $q_{i}^{m}$ with $m \geq 0$ .

5.1F2F3step 4.1

The rank-one string bound. Assume $T \neq 0$ . In the rank-one module $Hom (W, W^{'})$ of steps 3.1-4.1 let $t \geq 1$ be minimal with $A d (F_{i})^{t} T = 0$ ; it exists by the local nilpotence of step 3.1. For $b \geq 1$ the divided-power formula of [F3] gives $A d (E_{i}) A d (F_{i})^{(b)} (T) = A d (F_{i})^{(b - 1)} Θ_{i, b} (T)$ , and $Θ_{i, b}$ acts on $T$ by the scalar $\frac{1}{q_{i}^{b} - q_{i}^{- b}} \sum_{s = 0}^{b - 1} (q_{i}^{m - 2 s} - q_{i}^{- m + 2 s}) = \frac{q_{i}^{m - b + 1} - q_{i}^{- m + b - 1}}{q_{i} - q_{i}^{- 1}} = [m - b + 1]_{i}$ , by summing the two geometric progressions. Taking $b = t$ and using $A d (F_{i})^{t} T = 0$ together with $A d (F_{i})^{(t - 1)} T = [t - 1]_{i}!^{- 1} A d (F_{i})^{t - 1} T \neq 0$ gives $[m - t + 1]_{i} = 0$ , hence $m - t + 1 = 0$ by [F2], that is $t = m + 1$ ; therefore $A d (F_{i})^{m + 1} T = 0$ .

6.1F4step 1.1step 5.1

The Serre sum. By induction on $n \geq 0$ , using $A d (K_{i}) (T) = q_{i}^{m} T$ and the Pascal recurrence of [F4], the powers of the adjoint action expand as $A d (F_{i})^{n} (T) = \sum_{r = 0}^{n} (- 1)^{r} q_{i}^{r (m - n + 1)} {(\binom{n}{r})}_{i} F_{i}^{n - r} T F_{i}^{r}$ : the case $n = 0$ is clear, and applying $A d (F_{i})$ to the formula for $n$ , conjugating $K_{i} F_{i}^{n - r} T F_{i}^{r} K_{i}^{- 1} = q_{i}^{m - 2 n} F_{i}^{n - r} T F_{i}^{r}$ and comparing the coefficient of $F_{i}^{n + 1 - r} T F_{i}^{r}$ uses exactly $C_{n + 1, r} = q_{i}^{r} C_{n, r} + q_{i}^{r - n - 1} C_{n, r - 1}$ , the second form of [F4]. At $n = m + 1 = 1 - a_{i j}$ every Laurent exponent $r (m - n + 1)$ is zero, so step 5.1 gives $\sum_{r = 0}^{1 - a_{i j}} (- 1)^{r} {(\binom{1 - a_{i j}}{r})}_{i} F_{i}^{1 - a_{i j} - r} F_{j} F_{i}^{r} = 0$ on $W$ . Since every $v \in V$ lies in such a $W$ by step 1.1, the negative Serre relation holds as an operator on all of $V$ .

7.1A1A2A3step 6.1

The positive Serre relation. Put $E_{i}^{'} : = F_{i}$ , $F_{i}^{'} : = E_{i}$ and $K_{h}^{'} : = K_{- h}$ for all $i$ and $h$ . The primed data again satisfies the toral, toral-action, mixed and local-nilpotence hypotheses [A1]-[A3]: for instance $K_{h}^{'} E_{i}^{'} K_{h}^{' - 1} = K_{- h} F_{i} K_{h} = q^{⟨ α_{i}, h ⟩} F_{i} = q^{⟨ α_{i}, h ⟩} E_{i}^{'}$ , while $E_{i}^{'} F_{i}^{'} - F_{i}^{'} E_{i}^{'} = F_{i} E_{i} - E_{i} F_{i} = - (K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1}) = (K_{i}^{'} - K_{i}^{' - 1}) / (q_{i} - q_{i}^{- 1})$ . Applying steps 1.1-6.1 to the primed data gives $\sum_{r} (- 1)^{r} {(\binom{1 - a_{i j}}{r})}_{i} E_{i}^{1 - a_{i j} - r} E_{j} E_{i}^{r} = 0$ on $V$ , which is the positive Serre relation for the original data.

8.1A3F1F5step 1.1step 6.1step 7.1∎

Conclusion. By [A1]-[A3] the operators satisfy all the defining relations of $U_{q} (g)$ : the toral and toral-action relations, the mixed commutators, and both Serre families by steps 6.1 and 7.1. Hence the universal property [F1] equips $V$ with a $U_{q} (g)$ -module structure; it is unique because $U_{q} (g)$ is generated by these operators, and it is a chosen $P$ -graded type-one weight module by construction. Local nilpotence is exactly [A3], so the module is integrable, and by [F5] the admissible $i$ -string decomposition applies to every vector; step 1.1 supplies the finite rank-one $U_{i}$ orbit. No finite-dimensionality of the full weight spaces is asserted or used.

Depends on

Used by

Normalized cyclotomic functors give an integrable quantum action Lemma

Dependency tree · two levels

32 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

Seok-Jin Kang and Masaki Kashiwara, Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras, arXiv:1102.4677v4 (standard reference, not scraped)
Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991) 465-516 (standard reference, not scraped)
Kashiwara, Miwa, Petersen and Yung, Perfect crystals and q-deformed Fock spaces, arXiv:q-alg/9603025 (standard reference, not scraped)