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Triangular decomposition of a quantized enveloping algebra

Statement

Let $U_{q} (g)$ be the Drinfeld–Jimbo algebra of a finite symmetrizable Cartan datum, over $k = Q (q)$ , and let $U_{q}^{-}, U_{q}^{0}, U_{q}^{+}$ be its generated subalgebras of Positive, negative and toral quantum subalgebras and their root gradings.

(i) Multiplication is a vector-space isomorphism $m : U_{q}^{-} \otimes_{k} U_{q}^{0} \otimes_{k} U_{q}^{+} ⟶ U_{q} (g), x_{-} \otimes x_{0} \otimes x_{+} ⟼ x_{-} x_{0} x_{+} .$ Products $b_{-} K_{h} b_{+}$ , with $b_{\pm}$ ranging over supplied bases of the two halves and $h \in P^{\lor}$ , form a basis of $U_{q} (g)$ . Arbitrary simple-generator words span the halves; their Serre relations preclude claiming their independence.

(ii) The halves are exactly the algebras presented by their separate Serre relations, and $U_{q}^{0} ≅ k [P^{\lor}]$ . Thus all three abstract factor maps are injective and the $K_{h}$ are linearly independent.

(iii) For every $β \in Q$ , the total root grading satisfies $U_{q} (g) [β] ≅ ⨁_{α, γ \in Q_{+}, α - γ = β} U_{q}^{-} [- γ] \otimes_{k} U_{q}^{0} \otimes_{k} U_{q}^{+} [α] .$ In particular degree zero contains all equal-positive/negative-degree summands, including $F_{i} E_{i}$ ; it is not just the toral subalgebra. As a left $U_{q}^{0}$ -module this component is free, with rank equal to the possibly infinite sum of the products of the two half-component dimensions. Under AC (The Axiom of Choice), the half-component dimensions are their classical PBW ranks, and products of the generic PBW bases of Generic quantum Serre halves have classical PBW ranks and a nondegenerate Hopf pairing with the toral basis give a PBW basis of the whole algebra. AC enters only through that supplier's formal-embedding proof; (i), (ii) and the graded tensor decomposition are choice-free.

Facts & Assumptions

Given: The datum, its Drinfeld–Jimbo algebra and its generated subalgebras.

[F1]

The independently presented halves $A^{\pm}$ and free toral algebra $C = k [P^{\lor}]$ form an associative algebra on $A^{-} \otimes C \otimes A^{+}$ ; its normal-factor multiplication to $U_{q} (g)$ is an isomorphism and identifies the factor images with the generated subalgebras. This follows from the locally proved Serre-free normal forms, opposite Serre commutators and factor-ideal identity (The generic quantum halves form a Drinfeld–Jimbo crossed double).

[F2]

The generated subalgebras are root-graded with $\deg F_{i} = - α_{i}$ , $\deg E_{i} = α_{i}$ , $\deg K_{h} = 0$ , and toral conjugation on a homogeneous element is given by its additive weight (Positive, negative and toral quantum subalgebras and their root gradings).

[F3]

Under AC, both separately presented generic halves have classical PBW ranks and their regular lifts of ordered classical PBW monomials are generic bases, including over $Q (q)$ (Generic quantum Serre halves have classical PBW ranks and a nondegenerate Hopf pairing, The Axiom of Choice).

Proof

1.1F1algebra

By [F1], the algebra $U_{q} (g)$ is identified with the tensor space of the independently presented factors, and their injective images are $U_{q}^{-}, U_{q}^{0}, U_{q}^{+}$ . Under these identifications the multiplication map $m$ is exactly the normal-factor isomorphism in [F1]. This proves (i) and (ii) without any additional assumption about a quotient presentation's freeness. For supplied factor bases, finite bilinear expansions show that their pure tensors span the tensor product; coordinate functionals on the supplied bases separate the coefficients of every finite sum of those tensors. Thus their pure tensors, and hence their images $b_{-} K_{h} b_{+}$ , are a basis.

2.1F1F2step 1.1algebra

The normal-factor isomorphism preserves total root degree by [F2], so the degree- $β$ tensor subspace is precisely the direct sum over $α - γ = β$ displayed in (iii). Every algebra element uses finitely many such summands. The equal-degree summands at $β = 0$ include $F_{i} \otimes 1 \otimes E_{i}$ ; both factors are nonzero because their simple-root component presentations have no Serre relations. Their independence from $1 \otimes C \otimes 1$ follows from the tensor decomposition. Hence $U_{q} (g) [0]$ contains these additional summands and is not just $U_{q}^{0}$ .

3.1F1F2F3step 1.1step 2.1algebra∎

Homogeneous bases of the halves can be obtained by enumerating their words and retaining the first vectors outside the previous span, without choice. For homogeneous half basis vectors $b_{-}$ of degree $- γ$ and $b_{+}$ of degree $α$ , the map $C \to U_{q} (g)$ sending $c$ to $c b_{-} b_{+}$ identifies a free left toral copy: moving $K_{h}$ past $b_{-}$ multiplies $b_{-} K_{h} b_{+}$ by the nonzero scalar $q^{- γ (h)}$ . Step 1.1 therefore identifies the fixed total-degree component with the direct sum of these free left toral copies. Its rank is the sum of the products of the two component dimensions, which may be infinite. With the stated AC hypothesis, [F3] identifies those finite half dimensions with the classical PBW ranks and supplies the generic ordered-monomial bases; inserting their products into step 1.1 gives the asserted full PBW basis. This completes all claims with the exact total grading.

Depends on

Used by

Divided-power commutation and the simple U_qᵢ(sl₂) string modules Theorem

Dependency tree · two levels

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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 (book-length lecture notes, last updated 18 January 2024) (standard reference, not scraped)
Benjamin Enriquez, PBW and Duality Theorems for Quantum Groups and Quantum Current Algebras, Journal of Lie Theory 13 (2003), 21-64 (standard reference, not scraped)