Quantum Enveloping Algebras with von Neumann Regular Cartan-like Generators and the Pierce Decomposition

Abstract

Quantum bialgebras derivable from U q (sl 2) 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.