COLORED YANG-BAXTER OPERATORS AND REPRESENTATIONS OF THE BRAID AND SYMMETRIC GROUPS

Abstract

Let be an abelian group, and let A be a -graded algebra which is commutative with respect to a symmetric bicharacter on . Associated to any -graded A-module M there is a tensor A-algebra colored by with -compatible left and right A-module structures. It is proved that this tensor algebra comes equipped with a set of—up to a scalar—unique Yang-Baxter operators satisfying a specific set of natural conditions, by means of which nontrivial representations of the braid and symmetric groups are obtained. It is shown that, when M is freely generated by homogeneous elements, the submodule of invariant elements under the corresponding representation is also freely generated, and has a canonical -commutative algebra structure. Several symmetric-like and exterior-like algebras in the literature can be obtained as examples of the so constructed algebras of invariant elements for particular choices of . Algebra endomorphisms induced in a functorial fashion from A-module endormorphisms of the original M are also obtained.

Acknowledgments

OASV has been partially supported by CONACyT grants 28491-E, and E130.1880, and MBRLM grant 1411-99. CVM has been partially supported by DGICyT grant PB94–1196 and Acció Integrada 1998-99 Comissionat per a Universitats i Recerca, Generalitat de Catalunya. Both authors would like to thank the referee for his/her enormously valuable comments, as they led us to a clearer and more simplified exposition.