Finite-dimensional pointed Hopf algebras of type $A_n$ related to the Faddeev-Reshetikhin-Takhtajan $U(R)$ construction

Abstract

Two "quantum enveloping algebras", here denoted by $U(R)$ and $U^{\sim }(R)$, are associated in [FRTa] and [FRTb] to any Yang-Baxter operator R. The latter is only a bialgebra, in general; the former is a Hopf algebra. In this paper, we study the pointed Hopf algebras $U(R_Q)$, where $R_Q$ is the Yang-Baxter operator associated with the multi-parameter deformation of $G{L}_n$ supplied in [AST]; cf also [S,Re]. Some earlier results concerning these Hopf algebras $U(R_Q)$ were obtained in [To,CLMT,CM]; a related (but different) Hopf algebra was studied in [DP]. The main new results obtained here concerning these quantum enveloping algebras are: 1)We list, in an extremely explicit form, those quantum enveloping algebras $U(R_Q)$ which are finite-dimensional--let $\mathcal{U}$ denote the collection of these. 2)We verify that the pointed Hopf algebras in $\mathcal{U}$ are quasitriangular and of Cartan type $A_n$ in the sense of Andruskiewich-Schneider. 3)We show that every $U(R_Q)$ is a Hopf quotient of a double cross-product (hence, as asserted in 2), is quasitriangular if finite-dimensional.) 4) CAUTION: These Hopf algebras are NOT always cocycle twists of the standard 1-parameter deformation. This somewhat surprising fact is an immediate consequence of the data furnished here-- clearly a cocycle twist will not convert an infinite-dimensional Hopf algebra to a finite-dimensional one! Furthermore, these Hopf algebras in $\mathcal{U}$ are (it is proved) not all cocycle twists of each other. 5)We discuss also the case when the quantum determinant is central in $A(R_Q)$, so it makes sense to speak of a $Q$-deformation of the special linear group.

20 pages