Covariant connections on bicovariant differential calculus

TL;DR

This paper investigates bicovariant differential calculus, demonstrating the decomposition of two-tensors, the existence of torsionless connections, and conditions for unique Levi-Civita connections, especially in deformations of classical Lie groups.

Contribution

It establishes the existence of bicovariant torsionless and Levi-Civita connections under specific conditions, extending the understanding of differential calculus on quantum groups.

Findings

Decomposition of two-tensors into symmetric and anti-symmetric parts.

Existence of bicovariant torsionless connections.

Conditions for unique Levi-Civita connections in quantum group deformations.

Abstract

Given a bicovariant differential calculus $(\mathcal{E}, d)$ such that the braiding map is diagonalisable in a certain sense, the bimodule of two-tensors admits a direct sum decomposition into symmetric and anti-symmetric tensors. This is used to prove the existence of a bicovariant torsionless connection on $\mathcal{E}$. Following Heckenberger and Schm{\"u}dgen, we study invariant metrics and the compatibility of covariant connections with such metrics. A sufficient condition for the existence and uniqueness of bicovariant Levi-Civita connections is derived. This condition is shown to hold for cocycle deformations of classical Lie groups.