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Divided-power multiplication, toral binomials and the $E_{i}^{(a)} F_{i}^{(b)}$ reordering identity

Statement

Let $i \in I$ and let $T_{i, c, t}$ be the toral quantum binomials of The Lusztig divided-power integral form $U_{A}$ ( $i \in I$ , $c \in Z$ , $t \geq 0$ , $T_{i, c, 0} = 1$ ), inside the Drinfeld–Jimbo algebra $U_{q} (g)$ of The Drinfeld-Jimbo quantized enveloping algebra by generators and relations with $U_{A}$ and its parts $U_{A}^{-}, U_{A}^{0}, U_{A}^{+}$ as in that definition. Write $E = E_{i}$ , $F = F_{i}$ , $E^{(a)} = E_{i}^{(a)}$ , $F^{(b)} = F_{i}^{(b)}$ for the divided powers of Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ and put $q = q_{i}$ inside the displayed formulas.

(i) Divided-power multiplication. For all $a, b \geq 0$ , $E^{(a)} E^{(b)} = {(\binom{a + b}{a})}_{i} E^{(a + b)}, F^{(a)} F^{(b)} = {(\binom{a + b}{a})}_{i} F^{(a + b)} .$

(ii) Toral recursion and commutation. For $t \geq 1$ , $T_{i, c, t} = \frac{K_{i} q_{i}^{c - t + 1} - K_{i}^{- 1} q_{i}^{- c + t - 1}}{q_{i}^{t} - q_{i}^{- t}} T_{i, c, t - 1} .$ Each $T_{i, c, t}$ is a Laurent polynomial in $K_{i}^{\pm 1}$ with coefficients in $Q (q)$ ; the toral elements commute with one another and with all $K_{h}$ . For all $i, j$ and all $m \in Z$ one has $K_{i}^{m} T_{j, c, t} = T_{j, c, t} K_{i}^{m}$ if $i \neq j$ , while $E_{i} T_{i, c, t} = T_{i, c - 2, t} E_{i}$ , $E_{j} T_{i, c, t} = T_{i, c - a_{i j}, t} E_{j}$ , $F_{i} T_{i, c, t} = T_{i, c + 2, t} F_{i}$ , $F_{j} T_{i, c, t} = T_{i, c + a_{i j}, t} F_{j}$ for all $j$ . In particular $E_{i} T_{i, c, t} \in U_{A}^{0} E_{i}$ , $F_{i} T_{i, c, t} \in U_{A}^{0} F_{i}$ , $T_{i, c, t} E_{i} = E_{i} T_{i, c + 2, t} \in E_{i} U_{A}^{0}$ and $T_{i, c, t} F_{i} = F_{i} T_{i, c - 2, t} \in F_{i} U_{A}^{0}$ : the toral factor can be moved to the middle at the cost of a shift of the binomial index.

(iii) Reordering identity. For all $a, b \geq 0$ , $E_{i}^{(a)} F_{i}^{(b)} = \sum_{t = 0}^{\min (a, b)} F_{i}^{(b - t)} T_{i, 2 t - a - b, t} E_{i}^{(a - t)},$ with every coefficient a toral quantum binomial in $U_{A}^{0}$ . In particular $E_{i} F_{i} - F_{i} E_{i} = T_{i, 0, 1}$ and $E_{i}^{(a)} F_{i}^{(a)} = \sum_{t = 0}^{a} F_{i}^{(a - t)} T_{i, 2 t - 2 a, t} E_{i}^{(a - t)}$ . For $i \neq j$ , $E_{i}^{(a)}$ and $F_{j}^{(b)}$ commute; the same-half divided powers $E_{i}^{(a)}, E_{j}^{(b)}$ (and likewise $F_{i}^{(a)}, F_{j}^{(b)}$ ) commute whenever $a_{i j} = 0$ , and for $a_{i j} \neq 0$ no commutation is asserted.

(iv) Triangular spanning. Every element of $U_{A}$ is an $A$ -linear combination of products $b_{-} u b_{+}$ with $b_{-} \in U_{A}^{-}$ , $u \in U_{A}^{0}$ and $b_{+} \in U_{A}^{+}$ ; equivalently $U_{A} = U_{A}^{-} U_{A}^{0} U_{A}^{+}$ as an $A$ -module. No independence of the spanning family is asserted.

Facts & Assumptions

Given: The Cartan datum, the algebra $U_{q} (g)$ , the toral binomials $T_{i, c, t}$ , and the divided powers $x^{(m)} = x^{m} / [m]_{i}!$ .

[F1]

$x^{(a)} x^{(b)} = \frac{x^{a + b}}{[a]_{i}! [b]_{i}!} = {(\binom{a + b}{a})}_{i} x^{(a + b)}$ , because ${(\binom{a + b}{a})}_{i} = \frac{[a + b]_{i}!}{[a]_{i}! [b]_{i}!}$ and $[m]_{i}! \neq 0$ for all $m$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F2]

The defining relations: $E_{i} F_{i} - F_{i} E_{i} = (K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1})$ ; $E_{i} F_{j} = F_{j} E_{i}$ for $i \neq j$ ; $E_{i} E_{j} - E_{j} E_{i} = 0 = F_{i} F_{j} - F_{j} F_{i}$ when $a_{i j} = 0$ ; and the general same-half relations are the quantum Serre relations (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F3]

Toral conjugation and shifts: $K_{i} E_{j} K_{i}^{- 1} = q_{i}^{a_{i j}} E_{j}$ and $K_{i} F_{j} K_{i}^{- 1} = q_{i}^{- a_{i j}} F_{j}$ , so for every Laurent polynomial $f (K_{i}^{\pm 1})$ one has $E_{j} f (K_{i}^{\pm 1}) = f (q_{i}^{- a_{i j}} K_{i}^{\pm 1}) E_{j}$ and $F_{j} f (K_{i}^{\pm 1}) = f (q_{i}^{a_{i j}} K_{i}^{\pm 1}) F_{j}$ (Positive, negative and toral quantum subalgebras and their root gradings).

[F4]

The published rank-one formula: for $b \geq 1$ , $E_{i} F_{i}^{(b)} = F_{i}^{(b)} E_{i} + F_{i}^{(b - 1)} T_{i, 1 - b, 1}$ , equivalently $E_{i} F_{i}^{(b)} = F_{i}^{(b)} E_{i} + F_{i}^{(b - 1)} \frac{1}{q_{i}^{b} - q_{i}^{- b}} \sum_{s = 0}^{b - 1} (q_{i}^{- 2 s} K_{i} - q_{i}^{2 s} K_{i}^{- 1})$ , and for $b = 0$ one has $E_{i} F_{i}^{(0)} = E_{i} = F_{i}^{(0)} E_{i}$ (Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules (ii) and The Lusztig divided-power integral form $U_{A}$ for the identification of the coefficient with $T_{i, 1 - b, 1}$ ).

[F5]

For use in this proof, extend the quantum integer to every $m \in Z$ by $[m]_{i} : = (q_{i}^{m} - q_{i}^{- m}) / (q_{i} - q_{i}^{- 1})$ , agreeing with Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ for $m \geq 0$ and giving $[- m]_{i} = - [m]_{i}$ . Direct expansion of the numerators gives $q_{i}^{- n} (q_{i}^{m} - q_{i}^{- m}) + q_{i}^{m} (q_{i}^{n} - q_{i}^{- n}) = q_{i}^{m + n} - q_{i}^{- m - n}$ ; replacing $q_{i}$ by $q_{i}^{- 1}$ gives the other version. Thus $[m + n]_{i} = q_{i}^{- n} [m]_{i} + q_{i}^{m} [n]_{i} = q_{i}^{n} [m]_{i} + q_{i}^{- m} [n]_{i}$ for all integers $m, n$ , without extending factorials or Gaussian binomials to negative indices.

[F6]

$U_{A}$ is the $A$ -algebra generated by the $E_{i}^{(r)}, F_{i}^{(r)}, K_{h}^{\pm 1}, T_{i, c, t}$ ; its elements are finite $A$ -linear combinations of finite products of these generators, and $E_{i}^{(r)} \in U_{A}^{+} [r α_{i}]$ , $F_{i}^{(r)} \in U_{A}^{-} [- r α_{i}]$ , $K_{h}^{\pm 1}, T_{i, c, t} \in U_{A}^{0} [0]$ (The Lusztig divided-power integral form $U_{A}$ ).

Verification

technique · the divided-power identities are factorial arithmetic; the toral identities are computed by conjugating $K_i$ through $E_j,F_j$; the reordering identity is an induction on $a$ whose induction step is the toral product identity proved in step 1.2; the triangular spanning is a terminating rewriting using the reordering identity

1.1givenF1algebra

Part (i): by [F1], $E^{(a)} E^{(b)} = E^{a + b} / ([a]_{i}! [b]_{i}!) = {(\binom{a + b}{a})}_{i} E^{a + b} / [a + b]_{i}! = {(\binom{a + b}{a})}_{i} E^{(a + b)}$ and likewise for $F$ .

1.2givenF4F5algebra

Auxiliary product identity. Fix integers $a, b$ and $t \geq 1$ , put $c : = 2 t - a - b - 2$ and let $G (x, t) : = \prod_{s = 1}^{t} (K_{i} q_{i}^{x - s + 1} - K_{i}^{- 1} q_{i}^{- x + s - 1})$ , so that $T_{i, x, t} = G (x, t) / D (t)$ with $D (t) = \prod_{s = 1}^{t} (q_{i}^{s} - q_{i}^{- s})$ ; we prove $[a + 1]_{i} T_{i, c + 1, t} = [a - t + 1]_{i} T_{i, c, t} + T_{i, t - b, 1} T_{i, c, t - 1}$ . Factoring $G (x, t) = q_{i}^{- t (t - 1) / 2} \prod_{s = 1}^{t} (K_{i} q_{i}^{x} - K_{i}^{- 1} q_{i}^{2 s - 2 - x})$ and putting $u : = K_{i} q_{i}^{c + 1}$ , so that $K_{i} q_{i}^{x} = u q_{i}^{x - c - 1}$ and $K_{i}^{- 1} q_{i}^{y} = u^{- 1} q_{i}^{y + c + 1}$ , the relevant factors become $G (c + 1, t) = q_{i}^{- t (t - 1) / 2} (u - u^{- 1}) W$ , $G (c, t) = q_{i}^{- t (t - 1) / 2} q_{i}^{- t} W (u - q_{i}^{2 t} u^{- 1})$ , $G (c, t - 1) = q_{i}^{- (t - 1) (t - 2) / 2} q_{i}^{- (t - 1)} W$ and $G (t - b, 1) = u q_{i}^{a - t + 1} - u^{- 1} q_{i}^{- a + t - 1}$ , where $W : = \prod_{s = 1}^{t - 1} (u - q_{i}^{2 s} u^{- 1})$ . Multiplying the target identity by $D (t) = D (t - 1) (q_{i}^{t} - q_{i}^{- t})$ , using $D (t) / (D (t - 1)) = [t]_{i} (q_{i} - q_{i}^{- 1})$ and $q_{i}^{t} - q_{i}^{- t} = [t]_{i} (q_{i} - q_{i}^{- 1})$ , and factoring out the common factor $q_{i}^{- t (t - 1) / 2} W$ , the two sides become $[a + 1]_{i} (u - u^{- 1})$ and $[a - t + 1]_{i} q_{i}^{- t} (u - q_{i}^{2 t} u^{- 1}) + [t]_{i} (u q_{i}^{a - t + 1} - u^{- 1} q_{i}^{- a + t - 1})$ ; comparing the coefficients of $u$ and $u^{- 1}$ gives the identities $[a + 1]_{i} = q_{i}^{- t} [a - t + 1]_{i} + q_{i}^{a - t + 1} [t]_{i}$ and $[a + 1]_{i} = q_{i}^{t} [a - t + 1]_{i} + q_{i}^{- a + t - 1} [t]_{i}$ , both instances of [F5] with $(m, n) = (a - t + 1, t)$ .

2.1step 1.1F3algebra

Part (ii): unrolling the product definition, $T_{i, c, t} = \prod_{s = 1}^{t} \frac{K_{i} q_{i}^{c - s + 1} - K_{i}^{- 1} q_{i}^{- c + s - 1}}{q_{i}^{s} - q_{i}^{- s}}$ equals the $s = t$ factor times $T_{i, c, t - 1}$ ; since both are Laurent polynomials in $K_{i}^{\pm 1}$ (The Lusztig divided-power integral form $U_{A}$ , Verification) they commute and the recursion follows. The commutativity statements are those of that step. For the shifts, [F3] gives $E_{j} f (K_{i}^{\pm 1}) = f (q_{i}^{- a_{i j}} K_{i}^{\pm 1}) E_{j}$ and $F_{j} f (K_{i}^{\pm 1}) = f (q_{i}^{a_{i j}} K_{i}^{\pm 1}) F_{j}$ ; applying this to the $t$ -fold product $f = T_{i, c, t}$ , the first factor of $T_{i, c, t}$ is multiplied by $q_{i}^{- a_{i j}}$ (first term) and $q_{i}^{a_{i j}}$ (second term), which is the same as replacing $c$ by $c - a_{i j}$ ; hence $E_{j} T_{i, c, t} = T_{i, c - a_{i j}, t} E_{j}$ (for $j = i$ , $a_{i i} = 2$ ), and likewise $F_{j} T_{i, c, t} = T_{i, c + a_{i j}, t} F_{j}$ . Rewriting gives $T_{i, c, t} E_{j} = E_{j} T_{i, c + a_{i j}, t}$ and $T_{i, c, t} F_{j} = F_{j} T_{i, c - a_{i j}, t}$ .

3.1step 1.2step 2.1F1F4algebra

Reordering by induction on $a$ for fixed $b$ . At $a = 0$ the identity is $F_{i}^{(b)} = F_{i}^{(b)} T_{i, - b, 0} E_{i}^{(0)}$ . Multiply the formula for $a$ on the left by $E_{i} / [a + 1]_{i}$ . For a target index $1 \leq t \leq \min (a + 1, b)$ put $c = 2 t - a - b - 2$ . The no-contraction term from index $t$ contributes $[a - t + 1]_{i} T_{i, c, t}$ to the coefficient of $F_{i}^{(b - t)} E_{i}^{(a + 1 - t)}$ , after shifting $E_{i}$ through the toral factor. The contraction term from index $t - 1$ contributes $T_{i, t - b, 1} T_{i, c, t - 1}$ by [F4]. Their sum is $[a + 1]_{i} T_{i, c + 1, t}$ by step 1.2, so division by the nonzero $[a + 1]_{i}$ gives the target coefficient. When $t = a + 1$ the absent no-contraction term is represented by $[0]_{i} = 0$ ; the $t = 0$ term is just $F_{i}^{(b)} E_{i}^{(a + 1)}$ . A contraction from $t = b$ is absent because $F_{i}^{(0)} = 1$ , so no target exceeds $b$ . These account for every term and prove the induction, including the stated special cases.

3.2step 2.1F2algebra

The remaining clauses of (iii): for $i \neq j$ the relation $E_{i} F_{j} = F_{j} E_{i}$ of [F2] gives $E_{i}^{(a)} F_{j}^{(b)} = F_{j}^{(b)} E_{i}^{(a)}$ after dividing by the nonzero factorials [F1]; if $a_{i j} = 0$ then $E_{i} E_{j} = E_{j} E_{i}$ and $F_{i} F_{j} = F_{j} F_{i}$ by [F2], hence $E_{i}^{(a)} E_{j}^{(b)} = E_{j}^{(b)} E_{i}^{(a)}$ and $F_{i}^{(a)} F_{j}^{(b)} = F_{j}^{(b)} F_{i}^{(a)}$ for all $a, b$ . No commutation is asserted when $a_{i j} \neq 0$ , where the quantum Serre relation is the only comparison used later.

4.1step 2.1step 3.1step 3.2F3F6construct∎

Part (iv): every element of $U_{A}$ is a finite $A$ -linear combination of finite products of generators by [F6]. Move every $K_{h}^{\pm 1}$ and every $T_{j, c, t}$ to the middle: $K_{h}$ commutes with the toral generators and crosses $E_{j}, F_{j}$ at the cost of the scalar $q^{\mp ⟨ α_{j}, h ⟩} \in A$ by [F3]; $T_{j, c, t}$ crosses $E_{i}, F_{i}$ at the cost of an index shift by step 2.1. These operations do not change the relative order of the positive and negative generators, so after finitely many of them we obtain a finite sum of terms $b_{-} u b_{+}$ with $u \in U_{A}^{0}$ and $b_{\pm}$ products of divided powers of a single sign, except that a product may still contain an $E$ -factor to the left of an $F$ -factor. Define the measure of such a product as the pair (total height $h$ = sum of the exponents of all its divided powers, number $ι$ of ordered pairs (positive factor, negative factor) in which the positive factor precedes the negative one), ordered lexicographically. An adjacent sign inversion, after moving toral factors aside, has the form $E_{i}^{(a)} F_{j}^{(b)}$ with $a, b \geq 1$ . If $i \neq j$ , step 3.2 swaps the factors, keeping $h$ and decreasing $ι$ . If $i = j$ , step 3.1 rewrites it as $\sum_{t = 0}^{\min (a, b)} F_{i}^{(b - t)} T_{i, 2 t - a - b, t} E_{i}^{(a - t)}$ : the terms with $t \geq 1$ strictly decrease $h$ , while the $t = 0$ term $F_{i}^{(b)} T_{i, - a - b, 0} E_{i}^{(a)}$ keeps $h$ but strictly decreases $ι$ (that inversion disappears and no other inversion is created, as toral factors are ignored by $ι$ ). By induction on $(h, ι)$ the rewriting terminates, producing the asserted span by triangular products in $U_{A}^{-} U_{A}^{0} U_{A}^{+}$ .

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Sources

Math 263C Quantum Groups (Stanford, 2022), lecture slides 'Lusztig's quantum group. Tilting modules. The Fusion Category in Action' (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)
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 multiplication, toral binomials and the Ei(a)Fi(b)Ei(a)​Fi(b)​ reordering identity