Non-extremal weight modules for quantized universal enveloping algebras

Cohomological construction of quantized universal enveloping algebras

Given an associative algebra A, and the category, C, of its finite dimensional modules, additional structures on the algebra A induce corresponding ones on the category C. Thus, the structure of a rigid quasi-tensor (braided monoidal) category on RepA is induced by an algebra homomorphism A → A ⊗ A (comultiplication), coassociative up to conjugation by Φ ∈ A (associativity constraint) and cocom...

Computing with Quantized Enveloping Algebras: PBW-Type Bases, Highest-Weight Modules and R-Matrices

Quantized enveloping algebras have been widely studied, almost exclusively by theoretical means (see, for example, De Concini and Procesi, 1993; Jantzen, 1996; Lusztig, 1993). In this paper we consider the problem of computing with a quantized enveloping algebra. For this we need a basis of it, along with a method for computing the product of two basis elements. To this end we will use so-calle...

Extremal weight modules of quantum affine algebras

Let ĝ be an affine Lie algebra, and let Uq(ĝ) be the quantum affine algebra introduced by Drinfeld and Jimbo. In [11] Kashiwara introduced a Uq(ĝ)-module V (λ), having a global crystal base for an integrable weight λ of level 0. We call it an extremal weight module. It is isomorphic to the Weyl module introduced by Chari-Pressley [6]. In [12, §13] Kashiwara gave a conjecture on the structure of...

More Tight Monomials in Quantized Enveloping Algebras

Constructing Canonical Bases of Quantized Enveloping Algebras

Since the invention of canonical bases of quantized enveloping algebras, one of the main problems has been to establish what they look like. Explicit formulas are only known in a few cases corresponding to root systems of low rank, namely A1 (trivial), A2 ([Lusztig 90]), A3 ([Xi 99a]), and B2 ([Xi 99b]). Furthermore, there is evidence suggesting that for higher ranks the formulas become so comp...