On The non-commutative Riemannian geometry of GL_q(n)

TL;DR

This paper develops a non-commutative Riemannian geometry framework for GL_q(n) by constructing unique linear connections based on generalized permutations, analyzing their properties and symmetries.

Contribution

It introduces a method to define linear connections on GL_q(n) using generalized permutations, including conditions for their stability and uniqueness.

Findings

Existence of a unique linear connection for each generalized permutation

Analysis of bicovariance, torsion, and commutative limit properties

Identification of suitable generalized permutations for GL_q(n)

Abstract

A recently proposed definition of a linear connection in non-commutative geometry, based on a generalized permutation, is used to construct linear connections on GL_q(n). Restrictions on the generalized permutation arising from the stability of linear connections under involution are discussed. Candidates for generalized permutation on GL_q(n) are found. It is shown that, for a given generalized permutation, there exists one and only one associated linear connection. Properties of the linear connection are discussed, in particular its bicovariance, torsion and commutative limit.