Quantum deformations for the diagonal R-matrices

TL;DR

This paper explores two types of quantum deformations of the linear group using diagonal R-matrices, analyzing their algebraic relations, coactions, and differential calculi on quantum spaces.

Contribution

It introduces and compares two deformation methods for GL(n) with diagonal R-matrices, and constructs differential calculi on associated quantum structures.

Findings

Relations between braided and quantum deformed algebras established

Differential calculi compatible with tensor grading constructed

Coactions on quantum planes analyzed

Abstract

We consider two different types of deformations for the linear group $ GL(n)$ which correspond to using of a general diagonal R-matrix. Relations between braided and quantum deformed algebras and their coactions on a quantum plane are discussed. We show that tensor-grading-preserving differential calculi can be constructed on braided groups , quantum groups and quantum planes for the case of the diagonal R-matrix.