Quantum Group Actions, Twisting Elements, and Deformations of Algebras

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras

Twisted Graded Hecke Algebras

We generalize graded Hecke algebras to include a twisting twococycle for the associated finite group. We give examples where the parameter spaces of the resulting twisted graded Hecke algebras are larger than that of the graded Hecke algebras. We prove that twisted graded Hecke algebras are particular types of deformations of a crossed product.

Twisting cocycles in fundamental representaion and triangular bicrossproduct Hopf algebras

We find the general solution to the twisting equation in the tensor bialgebra T (R) of an associative unital ring R viewed as that of fundamental representation for a universal enveloping Lie algebra and its quantum deformations. We suggest a procedure of constructing twisting cocycles belonging to a given quasitriangular subbialgebra H ⊂ T (R). This algorithm generalizes Reshetikhin’s approach...

6 Simple Hopf Algebras and Deformations of Finite Groups

We show that certain twisting deformations of a family of supersolvable groups are simple as Hopf algebras. These groups are direct products of two generalized dihedral groups. Examples of this construction arise in dimensions 60 and pq, for prime numbers p, q with q|p − 1. We also show that certain twisting deformation of the symmetric group is simple as a Hopf algebra. On the other hand, we p...

Hochschild Cohomology and Quantum Drinfeld Hecke Algebras

Abstract. Quantum Drinfeld Hecke algebras are generalizations of Drinfeld Hecke algebras in which polynomial rings are replaced by quantum polynomial rings. We identify these algebras as deformations of skew group algebras, giving an explicit connection to Hochschild cohomology. We compute the relevant part of Hochschild cohomology for actions of many reflection groups and we exploit computatio...