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The Chevalley involution, bar involution, and contravariant anti-involution preserve the Drinfeld–Jimbo ideal

Statement

Let $T$ and $I_{D J}$ be the free algebra and defining ideal of The Drinfeld-Jimbo quantized enveloping algebra by generators and relations, and put $m_{i j} = 1 - a_{i j}$ and ${S e r r e}_{i j}^{\pm}$ for its displayed Serre sums. Define $ω, ψ, τ : T \to T$ on generators by

$ω (E_{i}) = - F_{i}, ω (F_{i}) = - E_{i}, ω (K_{h}) = K_{- h},$

$ψ (E_{i}) = E_{i}, ψ (F_{i}) = F_{i}, ψ (K_{h}) = K_{- h},$

$τ (E_{i}) = F_{i}, τ (F_{i}) = E_{i}, τ (K_{h}) = K_{- h} .$

Here $ω$ fixes $Q (q)$ , while $ψ$ and $τ$ apply the field involution $q \mapsto q^{- 1}$ to coefficients; $ω$ and $ψ$ extend as algebra maps, and $τ$ extends as an anti-algebra map. Each map preserves $I_{D J}$ and descends to $U_{q} (g)$ . All three are involutions; $ω$ is a $Q (q)$ -algebra automorphism, $ψ$ a semilinear algebra automorphism, and $τ$ a semilinear algebra anti-involution. On the Serre sums,

$ω ({S e r r e}_{i j}^{\pm}) = (- 1)^{a_{i j}} {S e r r e}_{i j}^{\mp}, ψ ({S e r r e}_{i j}^{\pm}) = {S e r r e}_{i j}^{\pm}, τ ({S e r r e}_{i j}^{\pm}) = (- 1)^{1 - a_{i j}} {S e r r e}_{i j}^{\mp} .$

Facts & Assumptions

Given: The free algebra $T$ and the four families of generators of $I_{D J}$ in the Drinfeld–Jimbo presentation.

[F1]

The ideal is generated by the toral, toral-action, mixed $E_{i} F_{j}$ , and positive and negative Serre relations displayed in The Drinfeld-Jimbo quantized enveloping algebra by generators and relations.

[F2]

The symmetric Gaussian coefficients satisfy ${(\binom{m}{r})}_{i} = {(\binom{m}{m - r})}_{i}$ and are invariant under $q_{i} \mapsto q_{i}^{- 1}$ ; $[m]_{i}$ has the same inversion symmetry (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F3]

Assignments of the free generators extend uniquely to algebra maps on $T$ (Universal property of the tensor algebra); reversing the order of each word gives the corresponding anti-algebra extension.

Proof

technique · Check the four relation families and then pass to the quotient

1.1F3construct

The substitution $σ (q) = q^{- 1}$ is an involutive field automorphism of $Q (q)$ . By [F3], the displayed assignments extend to a $Q (q)$ -algebra endomorphism $ω$ , a $σ$ -semilinear algebra endomorphism $ψ$ , and a $σ$ -semilinear anti-algebra endomorphism $τ$ of $T$ . Each square fixes every generator and coefficient, so all three squares are the identity on $T$ .

1.2F1algebra

The maps send $K_{0} - 1$ to itself. They send $K_{h} K_{h^{'}} - K_{h + h^{'}}$ respectively to $K_{- h} K_{- h^{'}} - K_{- (h + h^{'})}$ , the same relation, and $K_{- h^{'}} K_{- h} - K_{- (h + h^{'})}$ , the same family with its two indices reversed. Thus each image lies in $I_{D J}$ .

1.3F1algebra

Write $a = ⟨ α_{i}, h ⟩$ , $R_{i, h}^{E} = K_{h} E_{i} - q^{a} E_{i} K_{h}$ , and $R_{i, h}^{F} = K_{h} F_{i} - q^{- a} F_{i} K_{h}$ . Then $ω (R_{i, h}^{E}) = - R_{i, - h}^{F}$ and $ω (R_{i, h}^{F}) = - R_{i, - h}^{E}$ ; $ψ (R_{i, h}^{E}) = R_{i, - h}^{E}$ and $ψ (R_{i, h}^{F}) = R_{i, - h}^{F}$ ; and $τ (R_{i, h}^{E}) = - q^{- a} R_{i, - h}^{F}$ and $τ (R_{i, h}^{F}) = - q^{a} R_{i, - h}^{E}$ . Each is a scalar multiple of a defining toral-action relation.

1.4F1F2algebra

Let $R_{i j} = E_{i} F_{j} - F_{j} E_{i} - δ_{i j} (K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1})$ . Direct substitution, including the semilinear inversion of the coefficient in $ψ$ and $τ$ , gives $ω (R_{i j}) = - R_{j i}$ , $ψ (R_{i j}) = R_{i j}$ , and $τ (R_{i j}) = R_{j i}$ . The ratio $(K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1})$ is unchanged when both numerator and denominator are inverted. Hence these images lie in $I_{D J}$ .

1.5F1F2algebra

Put $m = 1 - a_{i j}$ . Applying $ω$ to a positive Serre sum contributes $m + 1$ minus signs and preserves the order of factors, so $ω ({S e r r e}_{i j}^{+}) = (- 1)^{m + 1} {S e r r e}_{i j}^{-} = (- 1)^{a_{i j}} {S e r r e}_{i j}^{-}$ ; the negative case is symmetric. By [F2], $ψ$ fixes the coefficients and each $E_{i}, F_{i}$ , so it fixes both Serre sums. The anti-map $τ$ reverses each monomial; reindexing $r \mapsto m - r$ and using [F2] gives $τ ({S e r r e}_{i j}^{\pm}) = (- 1)^{m} {S e r r e}_{i j}^{\mp}$ . Thus all Serre images belong to $I_{D J}$ .

2.1F1step 1.1step 1.2step 1.3step 1.4step 1.5algebra∎

The preceding steps show that each map sends every generator of $I_{D J}$ into that ideal. Since $I_{D J}$ is two-sided, the algebra maps and the anti-algebra map send the whole ideal into itself. They therefore descend to the quotient; their squares remain the identity there, which makes the descended maps automorphisms or an anti-automorphism of the stated types. This proves the assertion.

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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 (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)