Deformed Covariant Quantum Phase Spaces as Hopf Algebroids

TL;DR

This paper explores the structure of kappa-deformed quantum phase spaces in four dimensions, demonstrating their realization as Hopf algebroids and analyzing their algebraic properties and coproduct gauge freedoms.

Contribution

It provides an explicit algebraic construction of the Hopf algebroid structure for standard kappa-deformed quantum covariant phase space in four dimensions.

Findings

Explicit Hopf algebroid structure derived algebraically

Coproducts are not unique, exhibiting gauge freedom

Interpretation of quantum phase spaces as Hopf algebroids provided

Abstract

We consider the general D=4 (10+10)-dimensional kappa-deformed quantum phase space as given by Heisenberg double \mathcal{H} of D=4 kappa-deformed Poincare-Hopf algebra H. The standard (4+4) -dimensional kappa - deformed covariant quantum phase space spanned by kappa - deformed Minkowski coordinates and commuting momenta generators ({x}_{\mu },{p}_{\mu }) is obtained as the subalgebra of \mathcal{H}. We study further the property that Heisenberg double defines particular quantum spaces with Hopf algebroid structure. We calculate by using purely algebraic methods the explicite Hopf algebroid structure of standard kappa - deformed quantum covariant phase space in Majid-Ruegg bicrossproduct basis. The coproducts for Hopf algebroids are not unique, determined modulo the coproduct gauge freedom. Finally we consider the interpretation of the algebraic description of quantum phase spaces as Hopf algebroids.