Duality and representations for new exotic bialgebras

Abstract

We find the exotic matrix bialgebras which correspond to the two nontriangular nonsingular 4×4 R-matrices in the classification of Hietarinta, namely, RS0,3 and RS1,4. We find two new exotic bialgebras S03 and S14 which are not deformations of the classical algebras of functions on GL(2) or GL(1|1). With this we finalize the classification of the matrix bialgebras which unital associative algebras generated by four elements. We also find the corresponding dual bialgebras of these new exotic bialgebras and study their representation theory in detail. We also discuss in detail a special case of RS1,4 in which the corresponding algebra turns out to be a special case of the two-parameter quantum group deformation GLp,q(2).