Noncommutative geometry on central extension of U(u(2))

TL;DR

This paper develops a noncommutative differential calculus on a central extension of the algebra U(u(2)), comparing methods via quantum doubles and coalgebraic structures, with potential applications in noncommutative geometry.

Contribution

It introduces a new approach to extend quantum partial derivatives on U(u(2)) using central extensions and Cayley-Hamilton identities, advancing noncommutative differential calculus.

Findings

Extended quantum derivatives to a larger algebra via central extension.

Prolongated derivatives to skew-field elements using Cayley-Hamilton identities.

Discussed potential applications in noncommutative geometry.

Abstract

In our previous publications we have introduced analogs of partial derivatives on the algebras U(gl(N)). In the present paper we compare two methods of introducing these analogs: via the so-called quantum doubles and by means of a coalgebraic structure. In the case N=2 we extend the quantum partial derivatives from U(u(2)) (the compact form of the algebra U(gl(2))) on a bigger algebra, constructed in two steps. First, we define the derivatives on a central extension of this algebra, then we prolongate them on some elements of the corresponding skew-field by using the Cayley-Hamilton identities for certain matrices with noncommutative entries. Eventual applications of this differential calculus are discussed.