Infinite Hopf Families of Algebras and Yang-Baxter Relations

TL;DR

This paper introduces a formalism for generalized quantum affine algebras using Yang-Baxter relations, establishing their structure as an infinite Hopf family and exploring their algebraic properties and examples.

Contribution

It develops a new Yang-Baxter relation-based framework for infinite Hopf families of (super-) algebras and connects it with Drinfeld currents and algebra comorphisms.

Findings

Explicit structure of infinite Hopf family on $L$ matrices

Relation with Drinfeld current realization for 4x4 R-matrices

Construction method for commuting operators from generalized RLL algebras

Abstract

A Yang-Baxter relation-based formalism for generalized quantum affine algebras with the structure of an infinite Hopf family of (super-) algebras is proposed. The structure of the infinite Hopf family is given explicitly on the level of $L$ matrices. The relation with the Drinfeld current realization is established in the case of $4\times4$ $R$-matrices by studying the analogue of the Ding-Frenkel theorem. By use of the concept of algebra ``comorphisms'' (which generalize the notion of algebra comodules for standard Hopf algebras), a possible way of constructing infinitely many commuting operators out of the generalized $RLL$ algebras is given. Finally some examples of the generalized $RLL$ algebras are briefly discussed.