We introduce the notion of 'bar category' by which we mean a monoidal category equipped with additional structure induced by complex conjugation. Examples of our theory include bimodules over a *-algebra, modules over a conventional Hopf *-algebra and modules over a more general object which call a 'quasi-*-Hopf algebra' and for which examples include the standard quantum groups u q (g) at q a ...

The aim of this paper is to explain the bialgebra structure of the positive part of the quantized universal enveloping algebra (Drinfeld-Jimbo quantum group) of type Anusing the Lie algebra theory concepts. Recently has been introduced a generalization of Lie algebras, the basic T -Lie algebras [1]. Using the T -Lie algebra concept some new (we think) quantum groups of type Ancan be constructed...

We introduce an algebra which describes the multiplication structure of a family of q-series containing a q-analogue of multiple zeta values. The double shuffle relations are formulated in our framework. They contain a q-analogue of Hoffman’s identity for multiple zeta values. We also discuss the dimension of the space spanned by the linear relations realized in our algebra.

We show that, over an arbitrary field, q-rook monoid algebras are iterated inflations of Iwahori-Hecke algebras, and, in particular, are cellular. Furthermore we give an algebra decomposition which shows a q-rook monoid algebra is Morita equivalent to a direct sum of Iwahori-Hecke algebras. We state some of the consequences for the representation theory of q-rook monoid algebras.

We obtain an explicit expression for the defining relation of the deformed WN algebra, DWA(ŝlN)q,t. Using this expression we can show that, in the q → 1 limit, DWA(ŝlN)q,t with t = e 2πi N q k+N N reduces to the slN -version of the Lepowsky-Wilson’s Z-algebra of level k, ZA(ŝlN)k . In other words DWA(ŝlN)q,t with t = e − 2πi N q k+N N can be considered as a q-deformation of ZA(ŝlN)k . In the ap...

Let H be a finite-dimensional quasi-Hopf algebra. We show for each quotient quasibialgebra Q of H that Q is a quasi-Hopf algebra whose dimension divides the dimension of H.

It will be shown that the defining relations for fuzzy torus and deformed (squashed) sphere proposed by J. Arnlind, et al (hep-th/0602290) can be rewriten as a new algebra which contains q-deformed commutators. The quantum parameter q (|q| = 1) is a function of ~. It is shown that the q → 1 limit of the algebra with the parameter μ < 0 describes fuzzy S and that the squashed S with q 6= 1 and μ...

A classification of finite dimensional irreducible representations of the nonstandard q-deformation U ′ q (so n) of the universal enveloping algebra U (so(n, C)) of the Lie algebra so(n, C) (which does not coincides with the Drinfeld–Jimbo quantized universal enveloping algebra U q (so n)) is given for the case when q is not a root of unity. It is shown that such representations are exhausted b...