A new one-parameter family of $2\times 2$ quantum matrices

Abstract

In this note the authors use the so called FRT approach [\Cite{Reshetikhin89:Quantization:178--206}[90j:17039]N. Yu.Reshetikhin, L. A. Takhtadzhyan and L. D. Faddeev, Algebra iAnaliz 1 (1989), no. 1, 178 206; MR 90j:17039 ]R\sb q(T\otimes T)=(T\otimes T)R\sb q to define a newbialgebra, whose generators are the elements of T, from aspecial R matrix R\sb q for the Yang Baxter equation. It isby now a standard process to find a Hopf algebra or bialgebra solong as one knows an R matrix (for a similar discussion one cansee a paper by 1102129N. H. Jing, M. L. Ge and Y. S. Wu [Lett.Math. Phys. 21 (1991), no. 3, 193 203; MR\Cite{Jing91:new:193--203}[92k:17023]]).\par The authors study indetail the representation theory of the quantum matrix algebraB\sb q(2) obtained in this way. In particular, building certaincorrespondences among B\sb q(2), its dual B\sb q(2)\sp 0 andM\sb 2(C)\sp *, they construct a co representation forB\sb q(2) and obtain the corresponding exact Koszul complex.Proofs of the results announced in this note are contained inanother paper by the authors [``A new one parameter family of2\times 2 matrix bi algebras'', Preprint; per bibl.]