Duality and representations for new exotic bialgebras. (English) Zbl 1060.81041
Summary: We find the exotic matrix bialgebras which correspond to the two nontriangular nonsingular \(4\times 4\) \(R\)-matrices in the classification of Hietarinta, namely, \(R_{S0,3}\) and \(R_{S1,4}\). We find two new exotic bialgebras \(S03\) and \(S14\) which are not deformations of the classical algebras of functions on \(\text{GL}(2)\) or \(\text{GL}(1|1)\). 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 \(R_{S1,4}\) in which the corresponding algebra turns out to be a special case of the two-parameter quantum group deformation GL\(_{p,q}(2)\).
MSC:
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
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