We present an integral formula for the universal R -matrix of quantum affine algebra Uq(ĝ) with ’Drinfeld comultiplication’. We show that the properties of the universal R -matrix follow from the factorization properties of the cycles in proper configuration spaces. For general g we conjecture that such cycles exist and unique. For Uq(ŝl2) we describe precisely the cycles and present a new simp...

In this paper we defined the concept of module amenability of Banach algebras and module connes amenability of module dual Banach algebras.Also we assert the concept of module Arens regularity that is different with [1] and investigate the relation between module amenability of Banach algebras and connes module amenability of module second dual Banach algebras.In the following we studythe...

S.L. Woronowicz proved in 1991 that quantum SU(1,1) does not exist as a locally compact quantum group. Results by L.I. Korogodsky in 1994 and more recently by Woronowicz gave strong indications that the normalizer S̃U(1, 1) of SU(1, 1) in SL(2,C) is a much better quantization candidate than SU(1, 1) itself. In this paper we show that this is indeed the case by constructing S̃Uq(1, 1), a new examp...

We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and comultiplication laws, thereby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the conc...

The dynamical algebra of the q-deformed harmonic oscillator is constructed. As a result, we find the free deformed Hamiltonian as well as the Hamiltonian of the deformed oscillator as a complicated, momentum dependent interaction Hamiltonian in terms of the usual canonical variables. Furthermore we construct a well-defined algebra SUq(1,1) with consistent conjugation properties and comultiplica...

We consider the quantum double D(G) of a compact group G, following an earlier paper. We use the explicit comultiplication on D(G) in order to build tensor products of irreducible ∗-representations. Then we study their behaviour under the action of the R-matrix, and their decomposition into irreducible ∗-representations. The example of D(SU(2)) is treated in detail, with explicit formulas for d...

We introduce a twisted quantum affine algebra associated to each simply laced finite dimensional simple Lie algebra. This new algebra is a Hopf algebra with a Drinfeld-type comultiplication. We obtain this algebra by considering its vertex representation. The vertex representation quantizes the twisted vertex operators of Lepowsky-Wilson and Frenkel-LepowskyMeurman. We also introduce a twisted ...

We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra non-commutative symmetric functions described as subalgebra rooted ordered trees algebra. Its natural basis can be identified with set sentences over alphabet $A$ (the colours). present also its graded dual $\mathrm{QSym}_A$ quasi-symmetric together realization in terms power series partially commutative variables. pro...

We introduced a braided Sweedler cohomology, which is adequate to work with the H-braided cleft extensions studied in [G-G1]. Introduction In [Sw] a cohomology theory H(H,A) for a commutative module algebra A over a cocommutative Hopf algebra H was introduced. This cohomology is related to those of groups an Lie algebras in the following sense. When H is a group algebra k[G], then H(H,A) is can...

A strict 2-group is a 2-category with one object in which all morphisms and all 2morphisms have inverses. 2-Groups have been studied in the context of homotopy theory, higher gauge theory and Topological Quantum Field Theory (TQFT). In the present article, we develop the notions of trialgebra and cotrialgebra, generalizations of Hopf algebras with two multiplications and one comultiplication or...