Bergman’s centralizer theorem and quantization

What is it about?

Bergman's centralizer theorem states that the maximum commutative subalgebra of a free associative algebra A is isomorphic to the ring of polynomials in one variable. Quantization ideas provide a new vision in some classical areas in mathematics including polynomial automorphisms and the Jacobian conjecture. The Kontsevich quantization method describes how to construct a generalized star product operator algebra from a given arbitrary Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. The main theorem in the paper under review directly implies Bergman's theorem on centralizers. The goal in this paper is to demonstrate the direct relation between Bergman's theorem and Kontsevich quantization in a classic way. The proof of the theorem has been broken into multiple steps. A generic matrix is a matrix whose entries are distinct commutative indeterminates, and the so-called algebra of generic matrices of order m is generated by associative generic m×m matrices. The paper uses the fact that when we come to the quantization of generic matrices, those matrices are allowed to commute but to have no other relations. The authors prove that any two commuting elements in the free associative algebra also commute in some algebra of generic matrices. They also prove that if A is a free associative algebra, then there is no commutative subalgebra with transcendent degree greater than or equal to 2 of A. It is seen that two commuting generic matrices f, g with tr.deg(f,g)=2 do not commute after quantization. Reviewed by Vida Milani

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