Adjamagbo Determinant and Serre conjecture for linear groups over Weyl algebras

TL;DR

This paper extends Suslin's theorem to Weyl algebras over fields of characteristic zero using the Adjamagbo determinant, proving that matrices with determinant one are products of elementary matrices.

Contribution

It generalizes Suslin's $K_1$-analogue of the Serre Conjecture to Weyl algebras via noncommutative determinant theory.

Findings

Proves matrices with determinant one over Weyl algebras are elementary products.

Extends classical results from polynomial rings to noncommutative Weyl algebras.

Utilizes the theory of determinants over Ore domains for the proof.

Abstract

Thanks to the theory of determinants over an Ore domain, also called Adjamagbo determinant by the Russian school of non commutative algebra, we extend to any Weyl algebra over a field of characteristic zero Suslin theorem solving what Suslin himself called the $K_1$-analogue of the well-known Serre Conjecture and asserting that for any integer $n$ greater than 2, any $n$ by $n$ matrix with coefficients in any algebra of polynomials over a field and with determinant one is the product of elementary matrices with coefficients in this algebra