Centralizers in Free Associative Algebras and Generic Matrices

TL;DR

The paper studies centralizers in free associative algebras through a deformation-quantization-inspired approach based on generic matrices and PI-theory. It establishes that the centralizer $C(f;K\langle X\rangle)$ of any non-scalar element is integrally closed and has transcendence degree $1$, enabling a proof that such centralizers are isomorphic to the polynomial ring $K[t]$ (the Bergman centralizer theorem). It also proves that the algebra of generic matrices with traces (or forms in positive characteristic) is integrally closed and develops a framework connecting generic matrices to a skew field via a center, with generic elements corresponding to elements of this skew field. Finally, the work discusses the rationality of degree-one subfields in the skew-field of fractions of generic matrices, presenting open questions and partial rationality results grounded in local isomorphisms and reduction arguments.

Abstract

This paper is concerned with the completion of the proof of the Bergman centralizer theorem by using generic matrices based on our previous quantization proof \cite{KBRZh}. Additionally, we establish that the algebra of generic matrices with characteristic coefficients is integrally closed.

Theorems & Definitions (57)

• Theorem 1
• Theorem 2
• Definition 1.1
• proof : Proof of Theorem \ref{['ThmGenMatrixdomain']}
• Remark 1.2
• Proposition 1.3
• proof
• Corollary 1.5
• proof
• Proposition 1.6