MULTIPARAMETER QUANTUM FORMS OF THE ENVELOPING ALGEBRA $U_{{\mathfrak g}{\mathfrak l}_N}$ RELATED TO THE FADDEEV-RESHETIKHIN-TAKHTAJAN U(R) CONSTRUCTIONS

Abstract

Two quantum enveloping algebras UR and ÛR are associated in [RTF] to any Yang-Baxter operator R. These are constructed as subalgebras of A(R)* with specific generating sets. Also, [RTF] construct specific relations on the generators for UR, leaving open the question whether these generate all relations on these generators—let us say R is “perfect” when this is the case. Given an [Formula: see text] -tuple [Formula: see text] of nonzero elements qij,r in the groundfield, ([AST], [R], [S]) construct a multiparameter deformation [Formula: see text] of GLN associated with a Yang-Baxter operator [Formula: see text]. The method of ‘braiding maps’, introduced in [LT], is applied, in order to derive a PBW basis and a generators-and-relations presentation for a suitable generalization of [Formula: see text]. These results imply that [Formula: see text] is perfect, for generic [Formula: see text]. The construction [Formula: see text] is in some ways unsatisfactory if r is a root of 1. A construction [Formula: see text] is proposed, which is in some ways better behaved, coincides with [Formula: see text] if r is not a root of 1, and also makes sense over arbitrary commutative rings.