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

We give a Matsumoto-type presentation of \(K_2\) $K_2$ -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 induced by U. Rehmann’s approach in the case of division rings.