A Matsumoto type theorem for GL_n over rings of non-commutative Laurent polynomials

TL;DR

This paper extends Matsumoto's theorem to the $K_2$-groups over rings of non-commutative Laurent polynomials, generalizing results for loop groups to a non-commutative setting.

Contribution

It provides a Matsumoto-type presentation for $K_2$-groups over non-commutative Laurent polynomial rings, a novel extension of classical results.

Findings

Established a non-commutative version of Matsumoto's theorem.

Connected $K_2$-groups with non-commutative Laurent polynomial rings.

Extended Rehmann's approach to this new setting.

Abstract

We give a Matsumoto-type presentation of $K_2$-groups over rings of non-commutative Laurent polynomials, which is a non-commutative version of M. Tomie's result for loop groups. Our main idea is due to U. Rehmann's approach in case of division rings.