FRT Construction for Dynamical Yang-Baxter Maps

TL;DR

This paper introduces (H, X)-bialgebroids and their dynamical representations, establishing a framework connecting dynamical Yang-Baxter maps with algebraic structures and tensor categories.

Contribution

It proposes the notions of (H, X)-bialgebroids and shows how dynamical Yang-Baxter maps generate these structures, linking them to tensor categories of representations.

Findings

Dynamical representations form a tensor category.

Construction of (H, X)-bialgebroid from R(lambda).

Isomorphism between categories of L-operators and dynamical representations.

Abstract

Notions of an (H, X)-bialgebroid and of its dynamical representation are proposed. The dynamical representations of each (H, X)-bialgebroid form a tensor category. Every dynamical Yang-Baxter map R(lambda) satisfying suitable conditions, a generalization of the set-theoretical solution to the quantum Yang-Baxter equation, gives birth to an (H, X)-bialgebroid A_R. The categories of L-operators for R(lambda) and of dynamical representations of A_R are isomorphic as tensor categories.