Graded twisting of categories and quantum groups by group actions

Quantum Group Actions, Twisting Elements, and Deformations of Algebras

We construct twisting elements for module algebras of restricted two-parameter quantum groups from factors of their R-matrices. We generalize the theory of Giaquinto and Zhang to universal deformation formulas for categories of module algebras and give examples arising from R-matrices of two-parameter quantum groups.

Graded Quantum Groups

Starting from a Hopf algebra endowed with an action of a group π by Hopf automorphisms, we construct (by a “twisted” double method) a quasitriangular Hopf π-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular Hopf π-coalgebras for any finite group π and for infinite groups π such as GLn(k). In particular, we define the graded quantum groups, which are Hopf π-coalg...

q-EUCLIDEAN SPACE AND QUANTUM GROUP WICK ROTATION BY TWISTING

We study the quantum matrix algebra R21x1x2 = x2x1R and for the standard 2×2 case propose it for the co-ordinates of q-deformed Euclidean space. The algebra in this simplest case is isomorphic to the usual quantum matrices Mq(2) but in a form which is naturally covariant under the Euclidean rotations SUq(2)⊗SUq(2). We also introduce a quantum Wick rotation that twists this system precisely into...

Internal Categories and Quantum Groups

Group Actions on Algebras and Module Categories

Let k be a field and A a finite dimensional (associative with 1) k-algebra. By modA we denote the category of finite dimensional left A-modules. In many important situations we may suppose that A is presented as a quiver with relations (Q, I) (e.g. if k is algebraically closed, then A is Morita equivalent to kQ/I). We recall that if A is presented by (Q, I), then Q is a finite quiver and I is a...