On Hopf algebroid structure of kappa-deformed Heisenberg algebra

TL;DR

This paper explores the Hopf algebroid structure of the $ppa$-deformed Heisenberg algebra, detailing its algebraic properties and coproduct gauge freedom within the framework of quantum phase spaces.

Contribution

It provides a detailed algebraic description of the Hopf algebroid structure of the $ppa$-deformed Heisenberg algebra, including the right bialgebroid and antipode map.

Findings

Explicit structure of the $ppa$-deformed quantum phase space as a Heisenberg double.

Derivation of the target map using J-H. Lu's formula.

Identification of coproduct gauge freedom in the algebraic framework.

Abstract

The $(4+4)$-dimensional $\kappa$-deformed quantum phase space as well as its $(10+10)$-dimensional covariant extension by the Lorentz sector can be described as Heisenberg doubles: the $(10+10)$-dimensional quantum phase space is the double of $D=4$ $\kappa$-deformed Poincar\'e Hopf algebra $\mathbb{H}$ and the standard $(4+4)$-dimensional space is its subalgebra generated by $\kappa$-Minkowski coordinates $\hat{x}_\mu$ and corresponding commuting momenta $\hat{p}_\mu$. Every Heisenberg double appears as the total algebra of a Hopf algebroid over a base algebra which is in our case the coordinate sector. We exhibit the details of this structure, namely the corresponding right bialgebroid and the antipode map. We rely on algebraic methods of calculation in Majid-Ruegg bicrossproduct basis. The target map is derived from a formula by J-H. Lu. The coproduct takes values in the bimodule tensor product over a base, what is expressed as the presence of coproduct gauge freedom.