Quantum enveloping algebras with von Neumann regular Cartan-like generators and the Pierce decomposition

TL;DR

This paper introduces quantum bialgebras with von Neumann regular Cartan-like generators, exploring their antipodes, R-matrices, and structural decompositions, thereby extending the theory of quantum groups with new algebraic features.

Contribution

It presents a novel class of quantum bialgebras with idempotents and von Neumann regular generators, including explicit R-matrices and structural decompositions like Pierce decomposition.

Findings

Constructed invertible and von Neumann regular antipodes

Established Hopf and von Neumann-Hopf algebra structures

Provided explicit forms of R-matrices respecting Pierce decomposition

Abstract

Quantum bialgebras derivable from Uq(sl2) which contain idempotents and von Neumann regular Cartan-like generators are introduced and investigated. Various types of antipodes (invertible and von Neumann regular) on these bialgebras are constructed, which leads to a Hopf algebra structure and a von Neumann-Hopf algebra structure, respectively. For them, explicit forms of some particular R-matrices (also, invertible and von Neumann regular) are presented, and the latter respects the Pierce decomposition.