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Positive, negative and toral quantum subalgebras and their root gradings

Statement

Let $U_{q} (g)$ be the Drinfeld–Jimbo algebra of The Drinfeld-Jimbo quantized enveloping algebra by generators and relations with its root-lattice grading $U_{q} (g) = ⨁_{β \in Q} U_{q} (g) [β]$ . Define

$U_{q}^{+} : = ⟨ E_{i} : i \in I ⟩, U_{q}^{-} : = ⟨ F_{i} : i \in I ⟩, U_{q}^{0} : = ⟨ K_{h} : h \in P^{\lor} ⟩,$

the $Q (q)$ -subalgebras generated by the indicated images, and let $Q_{+} = ⨁_{i} Z_{\geq 0} α_{i}$ . Then

$U_{q}^{\pm} = ⨁_{β \in Q_{+}} U_{q}^{\pm} [\pm β], U_{q}^{+} [β] : = U_{q} (g) [β] \cap U_{q}^{+}, U_{q}^{-} [- β] : = U_{q} (g) [- β] \cap U_{q}^{-} .$

The degree-zero component $U_{q} (g) [0]$ contains $U_{q}^{0}$ , and for every $x \in U_{q} (g) [β]$ and $h \in P^{\lor}$ ,

$K_{h} x K_{h}^{- 1} = q^{β (h)} x .$

Thus each $K_{h} \in U_{q}^{0}$ acts by conjugation on the degree- $β$ component as the scalar $q^{β (h)}$ . The elements $K_{h}$ span $U_{q}^{0}$ and satisfy $K_{h} K_{h^{'}} = K_{h + h^{'}}$ with $K_{0} = 1$ ; hence $U_{q}^{0}$ is a quotient of the group algebra $Q (q) [P^{\lor}]$ . The subalgebras $U_{q}^{\pm}$ are spanned by finite monomials in the $E_{i}$ and $F_{i}$ , respectively. Define the nonnegative and nonpositive product spans by

$U_{q}^{\geq 0} : = {span}_{Q (q)} (U_{q}^{0} U_{q}^{+}), U_{q}^{\leq 0} : = {span}_{Q (q)} (U_{q}^{-} U_{q}^{0}) .$

This definition asserts no linear independence of these spanning families and no injectivity of the natural maps; those claims belong to Triangular decomposition of a quantized enveloping algebra.

Facts & Assumptions

Given: The Drinfeld–Jimbo presentation is homogeneous in the root lattice, with the degrees of $E_{i}, F_{i}, K_{h}$ specified in The Drinfeld-Jimbo quantized enveloping algebra by generators and relations. The datum provides the pairing of roots and coroots in Symmetrizable Cartan data for quantum groups.

[F1]

$U_{q} (g) = ⨁_{β \in Q} U_{q} (g) [β]$ with $\deg E_{i} = α_{i}$ , $\deg F_{i} = - α_{i}$ , and $\deg K_{h} = 0$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F2]

$K_{h} E_{i} K_{h}^{- 1} = q^{⟨ α_{i}, h ⟩} E_{i}$ and $K_{h} F_{i} K_{h}^{- 1} = q^{- ⟨ α_{i}, h ⟩} F_{i}$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F3]

$K_{0} = 1$ and $K_{h} K_{h^{'}} = K_{h + h^{'}}$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F4]

For a ring homomorphism $f : R \to S$ , the first isomorphism theorem gives $R / \ker f ≅ im f$ (First isomorphism theorem for rings: $R / \ker f ≅ im f$ ).

[F5]

The root lattice is $Q = ⨁_{i} Z α_{i}$ , and its pairing with $P^{\lor}$ is integer-valued and additive (Symmetrizable Cartan data for quantum groups).

Proof

technique · Use the root grading on finite generator words, and map the group algebra of the torus lattice onto the generated toral subalgebra

1.1F1givenalgebra

Every word in the $E_{i}$ is homogeneous of degree in $Q_{+}$ ; the empty word has degree $0$ . By generation, every element of $U_{q}^{+}$ is a finite sum of such words. Grouping those terms by degree and using the direct grading of $U_{q} (g)$ gives $U_{q}^{+} = ⨁_{β \in Q_{+}} (U_{q} (g) [β] \cap U_{q}^{+})$ .

1.2F1givenalgebra

Every word in the $F_{i}$ is homogeneous of degree in $- Q_{+}$ , including the degree-zero empty word. Grouping finite sums by their degrees in the direct grading gives $U_{q}^{-} = ⨁_{β \in Q_{+}} (U_{q} (g) [- β] \cap U_{q}^{-})$ .

1.3F3F4construct

Define $Q (q) [P^{\lor}]$ to be the vector space of finite sums $\sum_{h} c_{h} X_{h}$ with multiplication $X_{h} X_{h^{'}} = X_{h + h^{'}}$ . By [F3], $ϕ (X_{h}) = K_{h}$ extends linearly and multiplicatively to a unital algebra homomorphism into $U_{q}^{0}$ . Its image is all of $U_{q}^{0}$ , since the image contains every generator $K_{h}$ and consists of finite linear combinations of them. Thus [F4] identifies $U_{q}^{0}$ with $Q (q) [P^{\lor}] / \ker ϕ$ , and the $K_{h}$ span it.

1.4F1F2F3F5algebra

The relations in [F3] give $K_{h}^{- 1} = K_{- h}$ . By [F2], conjugation by $K_{h}$ acts on a word of degree $β$ by multiplying each $E_{i}$ factor by $q^{α_{i} (h)}$ , each $F_{i}$ factor by $q^{- α_{i} (h)}$ , and each $K_{h^{'}}$ factor by $1$ . Additivity of the pairing in [F5] makes the product scalar $q^{β (h)}$ . Every element is a finite sum of generator-word images; grouping those words by degree and using the direct grading [F1] shows that an element of $U_{q} (g) [β]$ is a finite linear combination of degree- $β$ word images. Thus the formula holds for every such $x$ . Since the $K_{h}$ have degree zero, their generated subalgebra $U_{q}^{0}$ lies in $U_{q} (g) [0]$ .

2.1givenalgebra∎

By definition, finite monomials in the $E_{i}$ and $F_{i}$ span $U_{q}^{+}$ and $U_{q}^{-}$ . Finite sums of products $u v$ with $u \in U_{q}^{0}, v \in U_{q}^{+}$ and with $u \in U_{q}^{-}, v \in U_{q}^{0}$ span the two stated product subspaces. No basis or independence assertion is involved.

Depends on

Used by

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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 (standard reference, not scraped)