The quantum group SLq⋆(2) and quantum algebra Uq(sl2⋆) based on a new associative multiplication on 2 × 2 matrices

Abstract

We present classical groups SL⋆(2) and SU⋆(2) as well as classical Lie algebra sl2⋆(C) associated with a new associative multiplication on 2 × 2 matrices. The idea of the new multiplication is generalized to the action of a 2 × 2 square matrix on a 2 × 1 column one. The coordinate Hopf algebra O(SLq⋆(2)) is introduced as a q-generalization of Hopf algebra O(SL⋆(2)), and it is shown that the coordinate algebra corresponding to the quantum plane Cq2 is a SLq⋆(2)-left-covariant algebra. Furthermore, the quantized universal enveloping algebra Uq(sl2⋆) with the Hopf structure as a dual of O(SLq⋆(2)) is introduced. For each of the Hopf algebras O(SLq⋆(2)) and Uq(sl2⋆), we associate two different real forms with two inequivalent families of *-involutions, with O(SUq⋆(2)) and Uq(su2⋆) as one of the real forms. It is shown that the Hopf algebra pairing is a dual pairing of two Hopf *-algebras O(SUq⋆(2)) and Uq(su2⋆).