Representation theory of the Drinfeld doubles of a family of Hopf algebras

Abstract

We investigate the Drinfeld doubles D( n,d ) of a certain family of Hopf algebras. We determine their simple modules and their indecomposable projective modules, and we obtain a presentation by quiver and relations of these Drinfeld doubles, from which we deduce properties of their representations, including the Auslander-Reiten quivers of the D( n,d ). We then determine decompositions of the tensor products of most of the representations described, and in particular give a complete description of the tensor product of two simple modules. This study also leads to explicit examples of Hopf bimodules over the original Hopf algebras.

Factorizable ribbon quantum groups in logarithmic conformal field theories

“…Irreducible quantum-group modules can be "glued" together to produce indecomposable representations. Already for U q sℓ(2), its indecomposable representations (which have been classified, rather directly, in [10] or can be easily deduced from a more general analysis in [35]) are rather numerous. Apart from the projective modules, to be considered separately in 3.3, indecomposable representations are given by families of modules W ± r (n), M ± r (n), and O ± r (n, z) that can be respectively represented as…”

“…It follows from [35] that the U q sℓ(2) Grothendieck ring (3.1) is in fact the result of "forceful semisimplification" of the following tensor product algebra of irreducible representations. First, if r + s − p 1, then, obviously, only irreducible representations occur in the decomposition: Finally, if r + s − p 3 and is odd, r + s − p = 2n + 1 with n 1, then…”

“…We note that in each of the last two formulas, the first sum in the right-hand side contains p − max(r, s) terms, and therefore disappears whenever max(r, s) = p (see [35] for the tensor products of other modules in 3.2.1).…”

Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral su(2) WZNW Model

“…5 Note that, to comply with our previous conventions for the zero modes, we have chosen here the "dual" Drinfeld double with respect to the one in [8], keeping the same Hopf structure for U q . In effect, our universal R-matrix (3.4) coincides with R −1 21 of [8] and hence, the M -matrix (3.9) is the inverse of the one given by Eq.…”

“…It has been argued at an early stage that the quantum group counterpart of an integer level su(n) k WZNW model is the restricted quantum universal enveloping algebra (QUEA) U q sℓ(n) at q an even root of unity that is factored by the relations E h α = 0 = F h α , K 2h i = 1 I for h = k + n (q h = −1) (1.1) (see [12], Chapter 4; after intermediate sporadic applications, see e.g. [15], it was studied more systematically in [8,5]). It is a finite dimensional QUEA that has a finite number of irreducible representations but a rather complicated tensor product decomposition, partly characterized by its Grothendieck ring (GR) which "forgets" the indecomposable structure of the resulting representations (see Section 2.3 below for a precise definition, and Section 3.4 for a description of the GR in the present context for n = 2 ).…”