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Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules

Statement

Fix a symmetrizable Cartan datum $(I, A, D, P, P^{\lor}, q)$ , an index $i \in I$ , and put $q_{i} = q^{d_{i}}$ . Let $U_{q} (g)$ be the Drinfeld–Jimbo algebra and let $A_{i}$ be the subalgebra generated by $E_{i}, F_{i}, K_{i}^{\pm 1}$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations, Positive, negative and toral quantum subalgebras and their root gradings); write $x^{(m)} : = x^{m} / [m]_{i}!$ for the divided powers of Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ .

(i) The assignment $e \mapsto E_{i}$ , $f \mapsto F_{i}$ , $k \mapsto K_{i}$ , $k^{- 1} \mapsto K_{i}^{- 1}$ defines an algebra isomorphism from the standard rank-one algebra $A$ over $Q (q)$ , presented by $e, f, k^{\pm 1}$ with $k e k^{- 1} = q_{i}^{2} e$ , $k f k^{- 1} = q_{i}^{- 2} f$ and $[e, f] = (k - k^{- 1}) / (q_{i} - q_{i}^{- 1})$ onto $A_{i}$ ; the images of the monomials $F_{i}^{a} K_{i}^{m} E_{i}^{b}$ ( $a, b \geq 0$ , $m \in Z$ ) are linearly independent in $U_{q} (g)$ , being images of the PBW monomials of the triangular decomposition (Triangular decomposition of a quantized enveloping algebra).

(ii) For all $a, b \geq 0$ , $K_{i} E_{i}^{(a)} K_{i}^{- 1} = q_{i}^{2 a} E_{i}^{(a)}$ and $K_{i} F_{i}^{(b)} K_{i}^{- 1} = q_{i}^{- 2 b} F_{i}^{(b)}$ , and for $b \geq 1$ the divided powers satisfy the exact commutation formula $E_{i} F_{i}^{(b)} = F_{i}^{(b)} E_{i} + \frac{F_{i}^{(b - 1)}}{q_{i}^{b} - q_{i}^{- b}} \sum_{s = 0}^{b - 1} (q_{i}^{- 2 s} K_{i} - q_{i}^{2 s} K_{i}^{- 1});$ iterating it computes $E_{i}^{(a)} F_{i}^{(b)}$ as a finite sum of monomials $F_{i}^{(b - t)} K_{i}^{m} E_{i}^{(a - t)}$ ( $0 \leq t \leq \min (a, b)$ ) with coefficients in $Q (q)$ .

(iii) For every $N \geq 0$ there is a simple $A_{i}$ -module $L_{i} (N)$ of dimension $N + 1$ : it is the quotient of the cyclic module $\tilde{M} (N) : = A_{i} / (A_{i} E_{i} + A_{i} (K_{i} - q_{i}^{N}))$ by the submodule generated by $F_{i}^{(N + 1)} w_{N}$ , where $w_{N}$ is the image of $1$ , with basis the images $v_{t}$ of $F_{i}^{(t)} w_{N}$ , $0 \leq t \leq N$ , on which $K_{i} v_{t} = q_{i}^{N - 2 t} v_{t}$ , $F_{i} v_{t} = [t + 1]_{i} v_{t + 1}$ and $E_{i} v_{t} = [N - t + 1]_{i} v_{t - 1}$ (with $v_{- 1} = v_{N + 1} = 0$ ); $L_{i} (N)$ is the unique simple $A_{i}$ -module generated by a vector $v \neq 0$ with $E_{i} v = 0$ , $K_{i} v = q_{i}^{N} v$ and $F_{i}^{N + 1} v = 0$ .

Facts & Assumptions

Given: A symmetrizable Cartan datum, an index $i$ , and the subalgebra $A_{i}$ generated by $E_{i}, F_{i}, K_{i}^{\pm 1}$ .

[F1]

$E_{i} F_{i} - F_{i} E_{i} = (K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1})$ ; $K_{i} E_{i} K_{i}^{- 1} = q_{i}^{2} E_{i}$ and $K_{i} F_{i} K_{i}^{- 1} = q_{i}^{- 2} F_{i}$ ; $K_{i}^{- 1} = K_{- d_{i} h_{i}}$ is the inverse of $K_{i}$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F2]

$A_{i}$ is the subalgebra generated by $E_{i}, F_{i}, K_{i}^{\pm 1}$ ; the universal property of $U_{q} (g)$ assigns a homomorphism to every assignment satisfying the defining relations; the same quotient universal property applies to the explicit rank-one presentation (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations, Positive, negative and toral quantum subalgebras and their root gradings, A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring).

[F3]

$[m]_{i} = (q_{i}^{m} - q_{i}^{- m}) / (q_{i} - q_{i}^{- 1})$ , $[m]_{i}! = \prod_{k = 1}^{m} [k]_{i}$ , $x^{(m)} = x^{m} / [m]_{i}!$ , and $q_{i} - q_{i}^{- 1} \neq 0$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F4]

Triangular decomposition: the multiplication map $U_{q}^{-} \otimes U_{q}^{0} \otimes U_{q}^{+} \to U_{q} (g)$ is an isomorphism, so the images of the monomials $F_{i}^{a} K_{i}^{m} E_{i}^{b}$ are linearly independent and span the rank-one subalgebra $A_{i}$ (Triangular decomposition of a quantized enveloping algebra); only the unconditional tensor-decomposition clauses (i)–(ii) of that supplier are needed here.

[F5]

The independent coroots and $d_{i} > 0$ make $m d_{i} h_{i}$ distinct for distinct integers $m$ (Symmetrizable Cartan data for quantum groups). The finite geometric-sum identities $\sum_{s = 0}^{b - 1} q_{i}^{\pm 2 s} = (q_{i}^{\pm 2 b} - 1) / (q_{i}^{\pm 2} - 1)$ hold because $q$ is transcendental and $d_{i} > 0$ ; likewise $[m]_{i} \neq 0$ for $m \geq 1$ by [F3].

Proof

technique · prove the divided-power identities by a two-term induction on the exponent, then build the cyclic module $\widetilde M(N)$ and its quotient and compute the action on the string vectors; the rank-one normal basis identifies the cyclic quotient and proves its string vectors are independent. In the infinite cyclic module, extend the integer notation to $s\in\mathbb Z$ by $[s]_i=(q_i^s-q_i^{-s})/(q_i-q_i^{-1})$; this agrees with [F3] for $s\ge0$ and gives $[-s]_i=-[s]_i$

1.1F1F3algebra

Weight rules. By [F1], $K_{i} E_{i}^{a} K_{i}^{- 1} = q_{i}^{2 a} E_{i}^{a}$ and $K_{i} F_{i}^{b} K_{i}^{- 1} = q_{i}^{- 2 b} F_{i}^{b}$ for all $a, b \geq 0$ by induction on the exponent via multiplicativity of conjugation; dividing by the nonzero scalars $[a]_{i}!$ and $[b]_{i}!$ of [F3] gives $K_{i} E_{i}^{(a)} K_{i}^{- 1} = q_{i}^{2 a} E_{i}^{(a)}$ and $K_{i} F_{i}^{(b)} K_{i}^{- 1} = q_{i}^{- 2 b} F_{i}^{(b)}$ .

1.2F1F3algebra

Commutation induction. Put $c : = 1 / (q_{i} - q_{i}^{- 1})$ and $T_{b} : = c \sum_{s = 0}^{b - 1} (q_{i}^{- 2 s} K_{i} - q_{i}^{2 s} K_{i}^{- 1})$ for $b \geq 1$ . We prove $E_{i} F_{i}^{b} = F_{i}^{b} E_{i} + F_{i}^{b - 1} T_{b}$ by induction on $b$ . For $b = 1$ this is [F1]. For the induction step, $E_{i} F_{i}^{b} = (E_{i} F_{i}^{b - 1}) F_{i} = (F_{i}^{b - 1} E_{i} + F_{i}^{b - 2} T_{b - 1}) F_{i} = F_{i}^{b - 1} (F_{i} E_{i} + c (K_{i} - K_{i}^{- 1})) + F_{i}^{b - 2} T_{b - 1} F_{i}$ , and [F1] gives $T_{b - 1} F_{i} = F_{i} c \sum_{t = 1}^{b - 1} (q_{i}^{- 2 t} K_{i} - q_{i}^{2 t} K_{i}^{- 1}) = F_{i} (T_{b} - c (K_{i} - K_{i}^{- 1}))$ after reindexing $s = t - 1$ ; the two correction terms cancel and the identity follows.

2.1F3step 1.1step 1.2algebra

Divided-power formula. Dividing the identity of 1.2 by $[b]_{i}!$ gives $E_{i} F_{i}^{(b)} = F_{i}^{(b)} E_{i} + F_{i}^{(b - 1)} \frac{T_{b}}{[b]_{i}}, \frac{T_{b}}{[b]_{i}} = \frac{1}{q_{i}^{b} - q_{i}^{- b}} \sum_{s = 0}^{b - 1} (q_{i}^{- 2 s} K_{i} - q_{i}^{2 s} K_{i}^{- 1}),$ because $F_{i}^{b - 1} / [b]_{i}! = F_{i}^{(b - 1)} / [b]_{i}$ and $[b]_{i} = (q_{i}^{b} - q_{i}^{- b}) / (q_{i} - q_{i}^{- 1})$ ; this is the displayed formula of (ii). Multiplying the formula for $E_{i}$ and for $E_{i}^{(a - 1)}$ on the left and using $E_{i} F_{i}^{(b)} = F_{i}^{(b)} E_{i} + \dots$ repeatedly computes $E_{i}^{(a)} F_{i}^{(b)}$ as a finite sum of monomials $F_{i}^{(b - t)} K_{i}^{m} E_{i}^{(a - t)}$ with $0 \leq t \leq \min (a, b)$ , since each application of the formula moves one $E_{i}$ to the right at the cost of one $F_{i}$ -power.

3.1F1F3F4F5step 1.1step 2.1construct

The cyclic module. Let $\tilde{M} (N) : = A_{i} / (A_{i} E_{i} + A_{i} (K_{i} - q_{i}^{N}))$ and $w_{N} : = 1 + (A_{i} E_{i} + A_{i} (K_{i} - q_{i}^{N}))$ . Every element of $A_{i}$ is a finite $Q (q)$ -linear combination of monomials $F_{i}^{t} K_{i}^{m} E_{i}^{a}$ with $t, a \geq 0$ and $m \in Z$ : words are rearranged with [F1] and 1.1, and each application of the identity of 2.1 moves one $E_{i}$ to the right of an $F_{i}$ -power at the cost of one $F_{i}$ -power and finitely many $K$ -factors, so the rearrangement terminates. By [F4] these normal monomials are a basis. The left ideal $A_{i} E_{i}$ is their span with $a > 0$ . Modulo that span, a product from $A_{i} (K_{i} - q_{i}^{N})$ with positive $E_{i}$ -exponent still has positive exponent after toral crossing, and its terms with exponent zero are exactly $F_{i}^{t} p (K_{i}) (K_{i} - q_{i}^{N})$ for Laurent polynomials $p$ . Thus the remaining toral factor is quotiented by evaluation $K_{i} = q_{i}^{N}$ , and the cyclic quotient has basis the images of $F_{i}^{t}$ , $t \geq 0$ ; a toral monomial $F_{i}^{t} K_{i}^{m}$ maps to $q_{i}^{N m} F_{i}^{t}$ , not to zero. Writing $v_{t} : = F_{i}^{(t)} w_{N}$ for $t \geq 0$ , the relations [F1] and 1.1 give $K_{i} v_{t} = q_{i}^{N - 2 t} v_{t}$ (using $K_{i} = K_{d_{i} h_{i}}$ and $α_{i} (d_{i} h_{i}) = 2 d_{i}$ ), $F_{i} v_{t} = F_{i} F_{i}^{(t)} w_{N} = [t + 1]_{i} F_{i}^{(t + 1)} w_{N} = [t + 1]_{i} v_{t + 1}$ by [F3], and, using 2.1 with $a = 1$ , $E_{i} v_{t} = F_{i}^{(t - 1)} \frac{1}{q_{i}^{t} - q_{i}^{- t}} \sum_{s = 0}^{t - 1} (q_{i}^{N - 2 s} - q_{i}^{2 s - N}) w_{N},$ for $t \geq 1$ , and $E_{i} v_{0} = 0$ ; the bracket equals $[N - t + 1]_{i}$ by the geometric-sum identity [F5].

4.1F3F4step 3.1algebra

The quotient and its action. Let $L_{i} (N) : = \tilde{M} (N) / A_{i} F_{i}^{(N + 1)} w_{N}$ . The submodule $A_{i} F_{i}^{(N + 1)} w_{N}$ is spanned by the $F_{i}^{(t)} w_{N}$ with $t \geq N + 1$ : it is contained in that span by the action formulas of 3.1, and it contains them because $F_{i}^{(t)} w_{N} = F_{i}^{(t - N - 1)} F_{i}^{(N + 1)} w_{N}$ up to the nonzero scalar $[t]_{i}! / [t - N - 1]_{i}! [N + 1]_{i}!$ by [F3]. Hence in $L_{i} (N)$ the vectors $v_{0}, \dots, v_{N}$ satisfy the displayed action formulas with $v_{- 1} = v_{N + 1} = 0$ , and they span $L_{i} (N)$ ; their nonvanishing and independence follow from the cyclic-module basis in step 3.1, since the specified tail submodule is spanned by the disjoint basis indices $t \geq N + 1$ .

5.1F3step 4.1algebra

Simplicity and uniqueness. Let $W \subseteq L_{i} (N)$ be a nonzero submodule and pick $0 \neq w = \sum_{t \leq T} c_{t} v_{t}$ with $T$ maximal and $c_{T} \neq 0$ . By 3.1, $E_{i}^{T} w = [N]_{i} [N - 1]_{i} \dots [N - T + 1]_{i} c_{T} v_{0}$ (each application of $E_{i}$ lowers the index by one with the stated nonzero scalar; all displayed scalars are nonzero because $[m]_{i} \neq 0$ for $m \geq 1$ by [F3] and the nonvanishing of $q_{i}^{2 m} - 1$ ), so $v_{0} \in W$ ; then $F_{i}^{t} v_{0} = [t]_{i}! v_{t}$ puts every $v_{t}$ in $W$ , so $W = L_{i} (N)$ and $L_{i} (N)$ is simple and generated by $w_{N}$ with $E_{i} w_{N} = 0$ , $K_{i} w_{N} = q_{i}^{N} w_{N}$ and $F_{i}^{N + 1} w_{N} = 0$ . Conversely, if $V$ is any simple $A_{i}$ -module generated by $v \neq 0$ with these three properties, then the assignment $1 \mapsto v$ defines a surjection $\tilde{M} (N) \to V$ whose kernel contains $F_{i}^{(N + 1)} w_{N}$ by the third property, hence factors through $L_{i} (N)$ ; as $L_{i} (N)$ is simple and $V \neq 0$ , this map is an isomorphism, which gives the asserted uniqueness.

6.1F1F2F4F5algebra∎

Part (i). The generators $E_{i}, F_{i}, K_{i}^{\pm 1}$ satisfy precisely the explicit rank-one relations by [F1], so sending $e, f, k$ to them gives a surjective map $A \to A_{i}$ by the quotient universal property in [F2]. The same finite rearrangement used in step 3.1 shows that $f^{a} k^{m} e^{b}$ spans this presented rank-one algebra; its torus here is generated by $k$ alone, rather than by additional lattice roots of $k$ . In the full algebra, the pure color- $i$ component of either separate half has one word and no Serre relation, so its powers are nonzero and independent. The toral powers $K_{i}^{m} = K_{m d_{i} h_{i}}$ are distinct because the datum's coroots are independent and $d_{i} > 0$ . By the tensor decomposition [F4], the images $F_{i}^{a} K_{i}^{m} E_{i}^{b}$ are therefore independent. The spanning rank-one monomials have independent images, proving injectivity and the claimed isomorphism.

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Sources

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)
Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)

Divided-power commutation and the simple Uqi(sl2)Uqi​​(sl2​) string modules