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The bar involution and the contravariant anti-involution preserve $U_{A}$

Statement

Let $ψ$ (the bar involution) and $τ$ (the contravariant anti-involution) be the semilinear maps of The Chevalley involution, bar involution, and contravariant anti-involution preserve the Drinfeld–Jimbo ideal, so that $ψ (q) = q^{- 1}$ , $ψ (E_{i}) = E_{i}$ , $ψ (F_{i}) = F_{i}$ , $ψ (K_{h}) = K_{h}^{- 1}$ and $τ (q) = q^{- 1}$ , $τ (E_{i}) = F_{i}$ , $τ (F_{i}) = E_{i}$ , $τ (K_{h}) = K_{h}^{- 1}$ , with $τ$ anti-multiplicative. Let $A = Z [q, q^{- 1}]$ and let $U_{A}$ be the Lusztig divided-power integral form with its generators $E_{i}^{(r)}$ , $F_{i}^{(r)}$ , $K_{h}^{\pm 1}$ and toral binomials $T_{i, c, t}$ (The Lusztig divided-power integral form $U_{A}$ ). Then $ψ (U_{A}) = U_{A}$ and $τ (U_{A}) = U_{A}$ . Explicitly, on the generators,

$ψ (E_{i}^{(r)}) = E_{i}^{(r)}, ψ (F_{i}^{(r)}) = F_{i}^{(r)}, ψ (K_{h}) = K_{h}^{- 1}, ψ (T_{i, c, t}) = T_{i, c, t},$

$τ (E_{i}^{(r)}) = F_{i}^{(r)}, τ (F_{i}^{(r)}) = E_{i}^{(r)}, τ (K_{h}) = K_{h}^{- 1}, τ (T_{i, c, t}) = T_{i, c, t} .$

In particular both maps restrict to bijections of $U_{A}$ : $ψ$ to a semilinear algebra involution of $U_{A}$ and $τ$ to a semilinear anti-involution of $U_{A}$ , while $q^{\pm 1} \in A$ keeps all coefficients in $A$ . These are the two involutions under which the bar lattice and the contravariant duality of the later constructions are taken; neither formula is the Chevalley involution $ω$ , which changes signs, exchanges the two halves and is treated in The Chevalley involution, bar involution, and contravariant anti-involution preserve the Drinfeld–Jimbo ideal. No choice is used.

Facts & Assumptions

Given: The Drinfeld–Jimbo algebra $U_{q} (g)$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations), the involution lemma defining $ψ, τ$ and their semilinearity (The Chevalley involution, bar involution, and contravariant anti-involution preserve the Drinfeld–Jimbo ideal), and the $A$ -algebra $U_{A}$ with its listed generators (The Lusztig divided-power integral form $U_{A}$ ).

[F1]

$ψ$ is an involutive semilinear algebra automorphism and $τ$ an involutive semilinear anti-automorphism of $U_{q} (g)$ with the stated values on generators; both apply the field involution $σ (q) = q^{- 1}$ (The Chevalley involution, bar involution, and contravariant anti-involution preserve the Drinfeld–Jimbo ideal).

[F2]

The field involution $σ$ preserves $A = Z [q, q^{- 1}]$ , since $σ (\sum_{k} a_{k} q^{k}) = \sum_{k} a_{k} q^{- k}$ ; and $σ$ fixes $[r]_{i}$ and $[r]_{i}!$ for every $r \geq 0$ , because $[r]_{i}$ is invariant under $q_{i} \mapsto q_{i}^{- 1}$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F3]

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

[F4]

$T_{i, c, t} = \prod_{s = 1}^{t} \frac{K_{i} q_{i}^{c - s + 1} - K_{i}^{- 1} q_{i}^{- c + s - 1}}{q_{i}^{s} - q_{i}^{- s}}$ with the empty product $T_{i, c, 0} = 1$ (The Lusztig divided-power integral form $U_{A}$ ).

Proof

Proof technique: compute both (anti-)maps on each family of generators of $U_{A}$ and conclude by semilinearity and [F3].

1.1givenF1F2F4algebra

For every $r \geq 0$ one has $ψ (E_{i}^{(r)}) = ψ (E_{i})^{r} / ψ ([r]_{i}!) = E_{i}^{r} / [r]_{i}! = E_{i}^{(r)}$ and $ψ (F_{i}^{(r)}) = F_{i}^{(r)}$ , since $ψ$ fixes $E_{i}, F_{i}$ and fixes the scalar $[r]_{i}!$ by [F2]; on the torus $ψ (K_{h}) = K_{h}^{- 1}$ . For the toral binomial, applying $ψ$ factor by factor and using $ψ (K_{i}^{\pm 1}) = K_{i}^{\mp 1}$ , $ψ (q_{i}^{\pm 1}) = q_{i}^{\mp 1}$ gives $ψ (K_{i} q_{i}^{c - s + 1} - K_{i}^{- 1} q_{i}^{- c + s - 1}) = - (K_{i} q_{i}^{c - s + 1} - K_{i}^{- 1} q_{i}^{- c + s - 1})$ and $ψ (q_{i}^{s} - q_{i}^{- s}) = - (q_{i}^{s} - q_{i}^{- s})$ ; the two signs cancel, so each factor and hence the product $T_{i, c, t}$ of [F4] is fixed: $ψ (T_{i, c, t}) = T_{i, c, t}$ .

2.1step 1.1F1F2F4algebra

For $τ$ , which is anti-multiplicative, $τ (E_{i}^{(r)}) = τ (E_{i}^{r}) / τ ([r]_{i}!) = τ (E_{i})^{r} / [r]_{i}! = F_{i}^{r} / [r]_{i}! = F_{i}^{(r)}$ by [F2], and symmetrically $τ (F_{i}^{(r)}) = E_{i}^{(r)}$ ; $τ (K_{h}) = K_{h}^{- 1}$ ; and on each factor of [F4], $τ (K_{i} q_{i}^{c - s + 1} - K_{i}^{- 1} q_{i}^{- c + s - 1}) = K_{i}^{- 1} q_{i}^{- (c - s + 1)} - K_{i} q_{i}^{c - s + 1} = - (K_{i} q_{i}^{c - s + 1} - K_{i}^{- 1} q_{i}^{- c + s - 1})$ while $τ (q_{i}^{s} - q_{i}^{- s}) = - (q_{i}^{s} - q_{i}^{- s})$ , so $τ$ fixes every factor; since the factors commute, $τ (T_{i, c, t}) = T_{i, c, t}$ as well.

3.1step 1.1step 2.1F1F2F3algebra∎

Consequently each of $ψ, τ$ sends every generator of $U_{A}$ into $U_{A}$ by steps 1.1 and 2.1, and sends scalars $a \in A$ into $A$ by [F2]. From [F3], an element of $U_{A}$ is a finite sum $\sum_{j} a_{j} g_{j, 1} \dots g_{j, m_{j}}$ with $a_{j} \in A$ and generators $g_{j, l}$ ; applying the semilinear algebra map $ψ$ gives $\sum_{j} σ (a_{j}) ψ (g_{j, 1}) \dots ψ (g_{j, m_{j}}) \in U_{A}$ , and applying the semilinear anti-algebra map $τ$ gives $\sum_{j} σ (a_{j}) τ (g_{j, m_{j}}) \dots τ (g_{j, 1}) \in U_{A}$ . Hence $ψ (U_{A}) \subseteq U_{A}$ and $τ (U_{A}) \subseteq U_{A}$ ; since $ψ, τ$ are involutions on $U_{q} (g)$ by [F1], the reverse inclusions hold as well, so $ψ (U_{A}) = U_{A}$ and $τ (U_{A}) = U_{A}$ and the restrictions are bijections of the stated (anti-)multiplicative types. No choice is used.

Depends on

Used by

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Sources

George Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110, Birkhauser, 1993; second printing 1994; author-hosted scan) (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)

The bar involution and the contravariant anti-involution preserve UAUA​