Hom-quantum groups II: cobraided Hom-bialgebras and Hom-quantum geometry

TL;DR

This paper introduces cobraided Hom-bialgebras, non-associative generalizations of quantum groups, providing new constructions, solutions to Hom-Yang-Baxter equations, and non-associative quantum geometric frameworks.

Contribution

It develops the theory of cobraided Hom-bialgebras, including construction methods and their applications to Hom-quantum groups and non-associative quantum geometry.

Findings

Constructed Hom-type quantum groups and matrices.

Derived solutions to operator quantum Hom-Yang-Baxter equations.

Established non-associative quantum geometric structures.

Abstract

A class of non-associative and non-coassociative generalizations of cobraided bialgebras, called cobraided Hom-bialgebras, is introduced. The non-(co)associativity in a cobraided Hom-bialgebra is controlled by a twisting map. Several methods for constructing cobraided Hom-bialgebras are given. In particular, Hom-type generalizations of FRT quantum groups, including quantum matrices and related quantum groups, are obtained. Each cobraided Hom-bialgebra comes with solutions of the operator quantum Hom-Yang-Baxter equations, which are twisted analogues of the operator form of the quantum Yang-Baxter equation. Solutions of the Hom-Yang-Baxter equation can be obtained from comodules of suitable cobraided Hom-bialgebras. Hom-type generalizations of the usual quantum matrices coactions on the quantum planes give rise to non-associative and non-coassociative analogues of quantum geometry.