Centralizers in free group algebras and nonsingular curves

This paper surveys results concerning the quantization approach to the Jacobian Conjecture and related topics on noncommutative algebras. We start with a brief review of the paper and its motivations. The first section deals with the approximation by tame automorphisms and the Belov–Kontsevich Conjecture. The second section provides quantization proof of Bergman’s centralizer theorem which has not been revisited for almost 50 years and formulates several related centralizer problems. In the third section, we investigate a free algebra analogue of a classical theorem of Białynicki-Birula’s theorem and give a noncommutative version of this famous theorem. Additionally, we consider positive-root torus actions and obtain the linearity property analogous to the Białynicki-Birula theorem. In the last sections, we introduce Feigin’s homomorphisms and we see how they help us in proving our main and fundamental theorems on screening operators and in the construction of our lattice Wn-algebras associated with sln, which is by far the simplest known approach concerning constructing such algebras until now.

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Centralizers in Free Associative Algebras and Generic Matrices

Kanel-Belov

Razavinia

Zhang

2018

Preprint

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In this short note, we complete a proof of Bergman's centralizer theorem for the free associative algebra using generic matrices approach based on our previous work [1]. IntroductionLet k be a field and k Z := k z 1 , . . . , z s be the free associative algebra over a field k with non-commuting variables in Z. G. Bergman proves in [2] that the centralizer C of a non-scalar element f in k Z is isomorphic to k[t], the ring of polynomials in one indeterminate over k.Up to our knowledge, there, no new proofs has been appeared after Bergman [2] for almost fifty years. We are using a method of deformation quantization presented by M. Kontsevich to give an alternative proof of Bergman's centralizer theorem. In our previous works [1,4], we got that the centralizer is a commutative domain of transcendence degree one.The claim in the abstract of our previous paper [1] was quite imprecise, because it just reproves and claims a part of Bergman's centralizer theorem that, the centralizer C is a commutative domain of transcendence degree one. For us, it was the most interesting part of the proof of the Bergman's centralizer theorem.However, L.Makar-Limanov notified us (and we are gratefull to him) few weeks ago that by opinion of specialists the other part is of much more importance and

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Centralizers in free group algebras and nonsingular curves

Abstract: The centralizer of any non-scalar element of a free group algebra over a field is the coordinate ring of a nonsingular curve.

Cited by 3 publications

References 6 publications

Centralizers in Free Associative Algebras and Generic Matrices

Centralizers in Free Associative Algebras and Generic Matrices

Polynomial Automorphisms, Deformation Quantization and Some Applications on Noncommutative Algebras

Centralizers in Free Associative Algebras and Generic Matrices

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