Colored Yang-Baxter operators and representations of the braid and symmetric groups

Let Delta be an abelian group, and let Delta be a Delta-graded algebra which is commutative with respect to a symmetric bicharacter epsilon: on Delta. Associated to any Delta-graded Delta-module M there is a tensor Delta-algebra colored by Delta with epsilon-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 epsilon-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 epsilon. Algebra endomorphisms induced in a functorial fashion from A-module endomorphisms of the original M are also obtained.