ABSTRACT. We prove Bergman's theorem [1] on centralizers by using generic matrices and Kontsevich's quantization method. For any field k of positive characteristics, set A = k x 1 , . . . , x s be a free associative algebra, then any centralizer C(f ) of nontrivial element f ∈ A\k is a ring of polynomials on a single variable. We also prove that there is no commutative subalgebra with transcendent degree ≥ 2 of A. INTRODUTIONQuantization ideas provides a new vision in some classical areas in mathematics including polynomial automorphisms and Jacobian conjecture. In the papers [2, 3] , Alexei Kanel Belov and Maxim Kontsevich used antiquantizations for to obtain a proof for to show the equivalence of the Jacobian conjecture and the Dixmier conjecture. And P. S. Kolesnikov [4,5] reproved the famous Makar-Limanov theorem on the construction of algebraically closed skew fields by using formal quantization by using the formal quantization.In the series of papers [6,7,8,9, 10], V. A. Artamonov discovered relations of metabelian algebras and the Serre conjecture with quantization. Some of Artamonov's ideas had been used by U. Umirbaev [11] for Jacobian conjecture in the case of metabelian algebras. And in 2007, Umirbaev in his classical paper [12], used metabelian algebras in order to prove the famous Anick conjecture (it was second ingredient, the main ingredient was the famous theory of U.Umirbaev and I.Shestakov).As to review some classical algebra and algebraic geometry, especially polynomial automorphisms from quantization point of view *

This paper consist of 3 sections. In the first section, we will give a brief introduction to the "Feigin's homomorphisms" and will see how they will help us to prove our main and fundamental theorems related to quantum Serre relations and screening operators. In the second section, we will introduce Local integral of motions as the space of invariants of nilpotent part of quantum affine Lie algebras and will find two and three point invariants in the case of Uq( ŝ l 2 ) by using Volkov's scheme. In the third section, we will introduce lattice Virasoro algebras as the space of invariants of Borel part Uq(B + ) of Uq(g) for simple Lie algebra g and will find the set of generators of Lattice Virasoro algebra connected to sl 2 and Uq(sl 2 ) And as a new result, we found the set of some generators of lattice Virasoro algebra. Contents1. Feigin's homomorphisms and quantum groups Feigin's homomorphisms on U q (n) 1.1.the contribution between Quantum Serre relations and screening operators 1.2. sl(3) case 1.3. affinized Lie algebra ŝl(2) 2. Local integral of motions; Volkov's scheme Example U q ( ŝl 2 ); two point invariants Example U q ( ŝl 2 ); three point invariants 3. Lattice Virasoro algebra Lattice Virasoro algebra connected to sl 2 Lattice Virasoro algebra connected to U q (sl 2 ) Formulation for to extend to four and more invariant points Generators of lattice Virasoro algebra coming from 2-dimensional representation of sl 2 Results; Generators of lattice Virasoro algebra coming from 3 and 4-dimensional representation of sl 2 ConclusionDate: June 30, 2015.

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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