Non-commutative Euclidean structures in compact spaces

TL;DR

This paper introduces a new compact non-commutative structure covariant under quantum group SOq(3) at roots of unity, exploring algebraic properties and representations of q-deformed operators.

Contribution

It develops a covariant compact non-commutative structure for quantum groups at roots of unity, including algebraic and representation-theoretic insights.

Findings

Constructed a covariant non-commutative algebra under SOq(3)

Derived methods to compute operators in specific representations

Compared properties of q-deformed Heisenberg algebra with classical case

Abstract

Based on results for real deformation parameter q we introduce a compact non- commutative structure covariant under the quantum group SOq(3) for q being a root of unity. To match the algebra of the q-deformed operators with necesarry conjugation properties it is helpful to define a module over the algebra genera- ted by the powers of q. In a representation where X is diagonal we show how P can be calculated. To manifest some typical properties an example of a one-di- mensional q-deformed Heisenberg algebra is also considered and compared with non-compact case.