On the centre of two-parameter quantum groups

Abstract

We describe Poincaré-Birkhoff-Witt bases for the two-parameter quantum groups U = Ur,s(sln) following Kharchenko and show that the positive part of U has the structure of an iterated skew polynomial ring. We define an ad-invariant bilinear form on U , which plays an important role in the construction of central elements. We introduce an analogue of the Harish-Chandra homomorphism and use it to determine the centre of U .

Whittaker modules for Uq(sl2)

“…It is also worth noting that some of the results of this chapter generalize to the two-parameter quantum group U r,s (sl 2 ) studied in [2]. However, the center of U r,s (sl 2 ) is somewhat more complicated than that of U q (sl 2 ) (see [1]), and thus the results from this chapter regarding the decomposition of a Whittaker module into simple submodules do not apply directly to U r,s (sl 2 ).…”

“…This gives a filtration M [1] ⊆ M [2] ⊆ · · · ⊆ M. Proof. Since Y ⊆ X, Dedekind's lemma (see, for example, [6, p. 27…”

“…The generator h, when regarded as a polynomial, is the minimal polynomial of the linear transformation given by the action of C on V . 1 1 · · · h a k k is the factorization of h into powers of irreducible (linear) polynomials h i . Let…”

Yetter–Drinfeld modules under cocycle twists

“…as a coalgebra, and the algebra structure is given by (3) ) ⊗ a (2) b, (1) for all m ∈ M, such that the following compatibility condition is satisfied: …”

“…For a wide class of such examples, Radford's construction lends itself to computations using Gröbner basis techniques. We further illustrate the construction by briefly explaining how to use Singular::Plural to compute bases and dimensions of all simple modules for some of the one-and two-parameter quantum groups u q (sl 3 ), u r,s (sl 3 ). Many such computations appeared in the second author's PhD thesis [19].…”