FRT-duals as Quantum Enveloping Algebras

The Hopf algebra generated by the l-functionals on the quantum double C_q[G] \bowtie C_q[G] is considered, where C_q[G] is the coordinate algebra of a standard quantum group and q is not a root of unity. It is shown to be isomorphic to C_q[G]^op \bowtie U_q(g). This was conjectured by T. Hodges in [Ho]. As an algebra it can be embedded into U_q(g) \otimes U_q(g). Here it is proven that there is no bialgebra structure on U_q(g) \otimes U_q(g), for which this embedding becomes a homomorphism of bialgebras. In particular, it is not an isomorphism. As a preliminary a lemma of [Ho] concerning the structure of l-functionals on C_q[G] is generalized. For the classical groups a certain choice of root vectors is expressed in terms of l-functionals. A formula for their coproduct is derived.Comment: Revised version of math.QA/0109157, 12 page

Representation of quantum algebras arising from non-compact quantum groups : quantum orbit method and super-tensor products

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996.Includes bibliographical references (leaves 85-87).by Leonid I. Korogodsky.Ph.D

The colored Jones function is q-holonomic

A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions. Using a general state sum definition of the colored Jones function of a link in 3-space, we prove from first principles that the colored Jones function is a multisum of a q-proper-hypergeometric function, and thus it is q-holonomic. We demonstrate our results by computer calculations.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper29.abs.htm

Simple Compact Quantum Groups I

The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups $B_u(Q)$ for $Q \in GL(n, {\mathbb C})$ satisfying $Q \bar{Q} = \pm I_n$, $n \geq 2$; (b) The quantum automorphism groups $A_{aut}(B, \tau)$ of finite dimensional $C^*$-algebras $B$ endowed with the canonical trace $\tau$ %endowed with a tracial functional $tr$ when $\dim(B) \geq 4$, including the quantum permutation groups $A_{aut}(X_n)$ on $n$ points ($n \geq 4$); (c) The standard deformations $K_q$ of simple compact Lie groups $K$ and their twists $K_q^u$, as well as Rieffel's deformation $K_J$.Comment: AMS-LATEX file, 49 page

Braided homology for quantum groups

We study braided Hochschild and cyclic homology of ribbon algebras in braided monoidal categories, as introduced by Baez and by Akrami and Majid. We compute this invariant for several examples coming from quantum groups and braided groups