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The contravariant form on an integrable highest weight module

Definition

Let $(I, A, D, P, P^{\lor}, q)$ be a symmetrizable Cartan datum with $a_{i i} = 2$ for all $i$ (Symmetrizable Cartan data for quantum groups) and let $U_{q} (g)$ be its Drinfeld–Jimbo algebra (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations). The linear contravariant anti-involution $σ$ of $U_{q} (g)$ is defined on generators by

$σ (q) = q, σ (K_{h}) = K_{h}, σ (E_{i}) = q_{i} F_{i} K_{i}^{- 1}, σ (F_{i}) = q_{i}^{- 1} K_{i} E_{i},$

extended as an anti-algebra map, $σ (x y) = σ (y) σ (x)$ . It is $Q (q)$ -linear, preserves the torus (it does not invert it), and is an involution (Verification, step 1.1). It is distinct from the semilinear contravariant anti-involution $τ$ with $τ (q) = q^{- 1}$ , $τ (E_{i}) = F_{i}$ , $τ (F_{i}) = E_{i}$ , $τ (K_{h}) = K_{- h}$ of The Chevalley involution, bar involution, and contravariant anti-involution preserve the Drinfeld–Jimbo ideal.

Let $λ \in P$ be dominant integral (Kac moody integral and dominant integral weights), so $⟨ λ, h_{i} ⟩ \geq 0$ for all $i$ , and let $(M, v_{λ})$ be a normalized highest weight module: $M$ is a $P$ -graded type-one quantum weight module (Quantum weight modules, weights in $P$ , and integrability) with $v_{λ} \in M_{λ}$ , $E_{i} v_{λ} = 0$ for all $i$ , and $M = U_{q} (g) v_{λ}$ . (The normalized simple highest weight module $L (λ)$ is the companion construction later on this page; the present definition applies to any normalized highest weight module, in particular to the Verma module and to its simple quotient.)

A contravariant form on $(M, v_{λ})$ is a bilinear form $B ⁣ : M \times M \to Q (q)$ such that

(i) $B$ is symmetric: $B (u, v) = B (v, u)$ for all $u, v \in M$ ;

(ii) $B$ is homogeneous for the label grading: $B (M_{μ}, M_{τ}) = 0$ whenever the two labels have different toral characters, that is, whenever $⟨ μ, h ⟩ \neq ⟨ τ, h ⟩$ for some $h \in P^{\lor}$ ;

(iii) $B (v_{λ}, v_{λ}) = 1$ (the normalization);

(iv) $B$ is $σ$ -invariant: $B (x u, v) = B (u, σ (x) v)$ for all $x \in U_{q} (g)$ and all $u, v \in M$ .

Because $σ$ is $Q (q)$ -linear and fixes the torus, condition (iv) is equivalent to the adjunction displays

$B (K_{h} u, v) = B (u, K_{h} v), B (E_{i} u, v) = B (u, q_{i} F_{i} K_{i}^{- 1} v), B (F_{i} u, v) = B (u, q_{i}^{- 1} K_{i} E_{i} v),$

for all $u, v \in M$ , $h \in P^{\lor}$ and $i \in I$ (Verification, step 2.1).

For $λ$ dominant and $L (λ)$ the normalized simple highest weight module, such a form exists, is unique, and is nondegenerate on $L (λ)$ ; moreover distinct label-weight spaces with distinct characters are orthogonal by (ii). Uniqueness is proved here (Verification, step 2.2), as is the orthogonality; consequently, when the pairing separates $P$ , the distinct label-weight spaces of $L (λ)$ are orthogonal and the restriction of $B$ to each finite-dimensional label-weight space is nondegenerate (Verification, step 4.1). The existence, the identification of the radical on the Verma module and the descent to $L (λ)$ are proved in the local Cartan-projection lemma lem-cartan-projection-constructs-the-quantum-verma-contravariant-form, which is the declared justification of this definition. That lemma is proved on this page, and its statement is the supplier of the existence and nondegeneracy clauses above; establishing those clauses is its content rather than a claim of this definition.

Facts & Assumptions

Given: A symmetrizable Cartan datum with $a_{i i} = 2$ , its Drinfeld–Jimbo algebra $U_{q} (g)$ with defining ideal $I_{D J}$ on the free algebra $T$ on the generators $E_{i}, F_{i}, K_{h}$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations), and a normalized highest weight module $(M, v_{λ})$ as above.

[F1]

$U_{q} (g) = T / I_{D J}$ where $I_{D J}$ is generated by the toral relations $K_{0} - 1$ , $K_{h} K_{h^{'}} - K_{h + h^{'}}$ ; the toral-action relations $K_{h} E_{i} - q^{⟨ α_{i}, h ⟩} E_{i} K_{h}$ and $K_{h} F_{i} - q^{- ⟨ α_{i}, h ⟩} F_{i} K_{h}$ ; the mixed relations $E_{i} F_{j} - F_{j} E_{i} - δ_{i j} (K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1})$ ; and the two Serre sums ${S e r r e}_{i j}^{\pm}$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F2]

The toral relations give $K_{h} K_{i}^{- 1} = K_{i}^{- 1} K_{h}$ and $K_{h} E_{i} K_{h}^{- 1} = q^{⟨ α_{i}, h ⟩} E_{i}$ , $K_{h} F_{i} K_{h}^{- 1} = q^{- ⟨ α_{i}, h ⟩} F_{i}$ ; in particular $K_{j} F_{i} K_{j}^{- 1} = q_{j}^{- a_{j i}} F_{i}$ and $K_{j} E_{j} K_{j}^{- 1} = q_{j}^{2} E_{j}$ , because $⟨ α_{i}, d_{j} h_{j} ⟩ = d_{j} a_{j i}$ (Symmetrizable Cartan data for quantum groups, The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F3]

The datum has $⟨ α_{j}, h_{i} ⟩ = a_{i j}$ and $d_{i} a_{i j} = d_{j} a_{j i}$ , so $d_{j} ⟨ α_{i}, h_{j} ⟩ = d_{j} a_{j i} = d_{i} a_{i j}$ (Symmetrizable Cartan data for quantum groups).

[F4]

The triangular decomposition: $U_{q} (g) = U_{q}^{-} U_{q}^{0} U_{q}^{+}$ as a vector space; $U_{q}^{+} v_{λ} \subseteq Q (q) v_{λ}$ and $U_{q}^{0} v_{λ} = Q (q) v_{λ}$ , so $M = U_{q}^{-} v_{λ}$ ; the label spaces satisfy $M_{μ} = 0$ unless $μ \leq λ$ , and $M_{λ} = Q (q) v_{λ}$ (Triangular decomposition of a quantized enveloping algebra, Positive, negative and toral quantum subalgebras and their root gradings).

[F5]

Semilinearity of $τ$ and the linearity of $σ$ : $σ$ fixes $q \in Q (q)$ and is $Q (q)$ -linear, while $τ$ applies $q \mapsto q^{- 1}$ and inverts the torus (The Chevalley involution, bar involution, and contravariant anti-involution preserve the Drinfeld–Jimbo ideal).

Verification

technique · descend the anti-map $\sigma$ through the defining ideal by checking the relation families, rewrite the invariance condition as the generator adjunctions, derive the orthogonality from the toral invariance, and prove uniqueness by reduction to $M=U_q^-v_\lambda$

1.1givenF1F2F3construct

Define $σ$ on the free algebra $T$ by the stated values on generators, extended as a $Q (q)$ -linear anti-algebra map. Every image of a defining relation lies in $I_{D J}$ : the toral relations are fixed or reversed; for the toral-action relation $K_{h} E_{i} - q^{⟨ α_{i}, h ⟩} E_{i} K_{h}$ one gets $σ (E_{i}) σ (K_{h}) - q^{⟨ α_{i}, h ⟩} σ (K_{h}) σ (E_{i}) = q_{i} F_{i} K_{i}^{- 1} K_{h} - q^{⟨ α_{i}, h ⟩} K_{h} q_{i} F_{i} K_{i}^{- 1} = q_{i} F_{i} K_{h} K_{i}^{- 1} - q_{i} q^{⟨ α_{i}, h ⟩} q^{- ⟨ α_{i}, h ⟩} F_{i} K_{h} K_{i}^{- 1} = 0$ by [F2], and the $F$ -variant is analogous; for the mixed relation with $i = j$ one gets $K_{i} E_{i} F_{i} K_{i}^{- 1} - F_{i} E_{i} = E_{i} F_{i} - F_{i} E_{i} = (K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1})$ by [F2], and for $i \neq j$ the image is $q_{i} q_{j}^{- 1} (K_{j} E_{j} F_{i} K_{i}^{- 1} - F_{i} K_{i}^{- 1} K_{j} E_{j})$ ; since $E_{j} F_{i} = F_{i} E_{j}$ and $K_{j} K_{i}^{- 1} = K_{i}^{- 1} K_{j}$ , the first bracket equals $q^{2 d_{j} - d_{i} a_{i j}} F_{i} E_{j} K_{j} K_{i}^{- 1}$ and the second equals $q^{2 d_{j} - d_{i} a_{i j}} F_{i} E_{j} K_{i}^{- 1} K_{j}$ (using $K_{j} F_{i} K_{j}^{- 1} = q^{- d_{j} a_{j i}} F_{i} = q^{- d_{i} a_{i j}} F_{i}$ and $K_{j} E_{j} K_{j}^{- 1} = q^{2 d_{j}} E_{j}$ from [F2], [F3]), so their difference vanishes; finally $σ ({S e r r e}_{i j}^{+}) = q_{i}^{- m} q_{j}^{- 1} K_{i}^{- m} K_{j}^{- 1} \sum_{r = 0}^{m} (- 1)^{r} {(\binom{m}{r})}_{i} F_{i}^{r} F_{j} F_{i}^{m - r} = (- 1)^{m} q_{i}^{- m} q_{j}^{- 1} K_{i}^{- m} K_{j}^{- 1} {S e r r e}_{i j}^{-}$ with $m = 1 - a_{i j}$ , a multiple of a defining relation, and $σ ({S e r r e}_{i j}^{-}) = (- 1)^{m} q_{i}^{- m} q_{j}^{- 1} K_{i}^{m} K_{j} {S e r r e}_{i j}^{+}$ . To check the common scalar, write $σ (E_{i}) = q_{i}^{- 1} K_{i}^{- 1} F_{i}$ and move all toral factors left. The crossing exponent is $- d_{i} m (m - 1) - d_{i} m a_{i j} = 0$ , since $m = 1 - a_{i j}$ ; the negative case uses $σ (F_{i}) = q_{i}^{- 1} K_{i} E_{i}$ with the same exponent. Hence $σ$ descends to an anti-algebra endomorphism of $U_{q} (g)$ ; since $σ^{2}$ fixes every generator and is an algebra endomorphism ( $σ^{2} (x y) = σ (σ (y) σ (x)) = σ^{2} (x) σ^{2} (y)$ ), $σ^{2} = i d$ , so $σ$ is an involutive anti-automorphism, $Q (q)$ -linear, with $σ (K_{h}) = K_{h}$ and distinct from the semilinear $τ$ of [F5].

2.1step 1.1algebra

If $B$ satisfies condition (iv), then for $x = K_{h}, E_{i}, F_{i}$ one gets the three displays by substituting $σ (K_{h}) = K_{h}$ , $σ (E_{i}) = q_{i} F_{i} K_{i}^{- 1}$ , $σ (F_{i}) = q_{i}^{- 1} K_{i} E_{i}$ . Conversely, assume the three displays hold. Let $S \subseteq U_{q} (g)$ be the set of $x$ with $B (x u, v) = B (u, σ (x) v)$ for all $u, v \in M$ . It contains $1$ (since $σ (1) = 1$ and $B (u, v) = B (u, v)$ ) and the generators; if $x, y \in S$ , then $B ((x y) u, v) = B (x (y u), v) = B (y u, σ (x) v) = B (u, σ (y) σ (x) v) = B (u, σ (x y) v)$ by step 1.1, so $x y \in S$ . Hence $S = U_{q} (g)$ , which is condition (iv). In particular $σ$ fixes the torus, so $B (K_{h} u, v) = B (u, K_{h} v)$ : the form is toral-invariant and not toral-inverting.

2.2step 1.1F4algebra

Uniqueness holds even without assuming symmetry or homogeneity. For two normalized $σ$ -contravariant bilinear forms put $D = B - B^{'}$ . For every nonempty negative word $w$ , $σ (w) v_{λ} = 0$ , and contravariance applied to $σ (w)$ gives $D (v_{λ}, w v_{λ}) = D (σ (w) v_{λ}, v_{λ}) = 0$ ; for the empty word this is normalization. Since $M = U_{q}^{-} v_{λ}$ , $D (v_{λ}, M) = 0$ . For $u = a v_{λ}$ and arbitrary $v \in M$ , contravariance now gives $D (u, v) = D (v_{λ}, σ (a) v) = 0$ . Thus $B = B^{'}$ , and this uniqueness may be applied to a form and its transpose before symmetry is established.

3.1step 2.1algebra

Homogeneity and orthogonality of different characters: let $u \in M_{μ}$ , $v \in M_{τ}$ and $h \in P^{\lor}$ . By step 2.1, $q^{⟨ μ, h ⟩} B (u, v) = B (K_{h} u, v) = B (u, K_{h} v) = q^{⟨ τ, h ⟩} B (u, v)$ ; since $q^{⟨ μ, h ⟩} \neq q^{⟨ τ, h ⟩}$ unless the characters agree, $B (u, v) = 0$ whenever $χ_{μ} \neq χ_{τ}$ . This is condition (ii), restated as a proved orthogonality: label-weight spaces with different characters are orthogonal, so under a separating pairing distinct label-weight spaces are orthogonal.

4.1step 3.1step 2.2given∎

In the separating case, nondegeneracy of the global form passes to each finite-dimensional label-weight space: if the pairing separates $P$ and $b \in M_{μ}$ satisfies $B (b, M_{μ}) = 0$ , then $B (b, M_{τ}) = 0$ for every $τ \neq μ$ by step 3.1 and $B (b, M_{μ}) = 0$ by assumption, so $b$ lies in the radical of $B$ on $M$ ; hence $b = 0$ whenever $B$ is nondegenerate on $M$ . The remaining clauses are the content of the Cartan-projection lemma: for $λ$ dominant and $L (λ)$ the normalized simple highest weight module, lem-cartan-projection-constructs-the-quantum-verma-contravariant-form constructs the normalized homogeneous $σ$ -invariant form on the Verma module $M (λ)$ (existence), identifies its radical with the maximal proper graded submodule, descends it to $L (λ)$ and proves nondegeneracy of the descended form. By step 2.2 that descended form is the unique one satisfying (i)-(iv). The existence and nondegeneracy clauses of this definition are thus delegated to that declared supplier, which is proved on this page; every other clause of the definition is proved above.

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Sources

Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (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)
George Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110, Birkhauser, 1993; second printing 1994; author-hosted scan) (standard reference, not scraped)