Quantum group covariant noncommutative geometry

TL;DR

This paper develops an algebraic framework for quantum group covariant noncommutative geometry, introducing noncommutative connections and curvatures that generate covariant quantum algebras, with potential applications to topological and gravitational theories.

Contribution

It provides a novel algebraic formulation of quantum group covariant noncommutative geometry using the $R$-matrix approach, including explicit realizations and invariants related to physical models.

Findings

Defined noncommutative connections and curvatures as comodules under quantum group coaction.

Constructed covariant quantum algebras and identified invariants analogous to physical Lagrangians.

Presented explicit realization via coset construction $GL_{q}(N+1)/(GL_{q}(N)\otimes GL(1))$.

Abstract

The algebraic formulation of the quantum group covariant noncommutative geometry in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider structure groups taking values in the quantum groups and introduce the notion of the noncommutative connections and curvatures transformed as comodules under the "local" coaction of the structure group which is exterior extension of $GL_{q}(N)$. These noncommutative connections and curvatures generate $ GL_{q}(N)$-covariant quantum algebras. For such algebras we find combinations of the generators which are invariants under the coaction of the "local" quantum group and one can formally consider these invariants as the noncommutative images of the Lagrangians for the topological Chern-Simons models, non-abelian gauge theories and the Einstein gravity. We present also an explicit realization of such covariant quantum algebras via the investigation of the coset construction $GL_{q}(N+1)/(GL_{q}(N)\otimes GL(1))$.