arXiv subjects
Publications and source records attributed to Alexei Kanel-Belov.
We show that there is no iterated identity satisfied by all finite groups. For $w$ being a non-trivial word of length $l$, we show that there exists a finite group $G$ of cardinality at most $\exp(l^C)$ which does not satisfy the iterated identity $w$. The proof uses the approach of Borisov and Sapir, who used dynamics of polynomial mappings for the proof of non residual finiteness of some groups.
We study the Zariski topology of the ind-groups of polynomial and free associative algebras $\Aut(K[x_1,...,x_n])$ (which is equivalent to the automorphism group of the affine space $\Aut(K^n))$) and $\Aut(K< x_1,..., x_n>$ via $\Ind$-schemes, toric varieties, approximations and singularities. We obtain some nice properties of $\Aut(\Aut(A))$, where $A$ is polynomial or free associative algebra over a field $K$. We prove that all $\Ind$-scheme automorphisms of $\Aut(K[x_1,...,x_n])$ are inner for $n\ge 3$, and all $\Ind$-scheme automorphisms of $\Aut(K< x_1,..., x_n>)$ are semi-inner. We also establish that any effective action of torus $T^n$ on $\Aut(K< x_1,..., x_n>)$ is linearizable provided $K$ is infinity. That is, it is conjugated to a standard one. As an application, we prove that $\Aut(K[x_1,...,x_n])$ cannot be embedded into $\Aut(K< x_1,...,x_n>)$ induced by the natural abelianization. In other words, the {\it Automorphism Group Lifting Problem} has a negative solution. We explore the close connection between the above results and the Jacobian conjecture, and Kontsevich-Belov conjecture, and formulate the Jacobian conjecture for fields of any characteristic.
Kaplansky asked about the possible images of a polynomial $f$ in several noncommuting variables. In this paper we consider the case of $f$ a Lie polynomial. We describe all the possible images of $f$ in $M_2(K)$ and provide an example of $f$ whose image is the set of non-nilpotent trace zero matrices, together with 0. We provide an arithmetic criterion for this case. We also show that the standard polynomial $s_k$ is not a Lie polynomial, for $k>2.$
In this paper we study a correspondence between cyclic modules over the first Weyl algebra and planar algebraic curves in positive characteristic. In particular, we show that any such curve has a preimage under a morphism of certain ind-schemes. This property might pave the way for an indirect proof of existence of a canonical isomorphism between the group of algebra automorphisms of the first Weyl algebra over the field complex numbers and the group of polynomial symplectomorphisms of $\mathbb{C}^2$.
The situation surrounding the Olympiads is paradoxical. On the one hand, considerable resources are spent on the Olympiads. On the other hand, there are widespread arguments about the harm of the Olympiads, often very strange ones. For example, there is a perception that success in the Olympiads has little to do with a scientific career. Sometimes it comes to curious assertions, such as that "there are no Olympiad winners among the eminent mathematical researchers," even though in truth the percentage of famous mathematicians among former Olympiad winners is much greater than among those who did not participate in such competitions
We present a proof of Kemer's representability theorem for affine PI algebras over a field of characteristic zero.
The paper is devoted to technology of math. education in Russia in special mathematical schools ("listkovaya systema"). This system has positive sides and most classes in Russian mathematical schools or circles almost always uses it. However, this system is not good to teach students, in particular to theoretical thinking. We discourse positive and negative sides of it and explain why it should be complimented. The second part of paper devoted to some specific items - information theory, algebra and non-Euclidean geometry. We show, where is dialog is necessary.
The paper deals with popular mathematics and mathematical education of talented students. The paper is devoted to high school students mathematical conferences, problem contest for them, academical policy. The paper was supported by grant RFFI No 14-01-00548.
We define a generalization $\mathfrak{G}$ of the Grassmann algebra $G$ which is well-behaved over arbitrary commutative rings $C$, even when $2$ is not invertible. In particular, this enables us to define a notion of superalgebras that does not become degenerate in such a setting. Using this construction we are able to provide a basis of the non-graded multilinear identities of the free superalgebra with supertrace, valid over any ring. We also show that all identities of $\mathfrak{G}$ follow from the Grassmann identity, and explicitly give its co-modules, which turn out to be generalizations of the sign representation. In particular, we show that the co-module is a free $C$-module of rank $2^{n-1}$.
From Rauzy graph Rauzy Scheme can be obtaining by uniting sequence of vertices of ingoing and outgoing degree 1 by arches. This notion is a tool to describe Rauzy graph behavior. For morphic superword we prove periodicity of Rauzy schemes. This fact has consequence in discrete dynamic systems and logic. This fact is also generalization of fact that quadratic irrationals have periodic chain fractions.
Let $p$ be a multilinear polynomial in several noncommuting variables, with coefficients in a algebraically closed field $K$ of arbitrary characteristic. In this paper we classify the possible images of $p$ evaluated on $3\times 3$ matrices. The image is one of the following: \begin{itemize} \item \{0\}, \item the set of scalar matrices, \item a (Zariski) dense subset of $\sl_3(K)$, the matrices of trace 0, \item a dense subset of $M_3(K)$, \item the set of $3-$scalar matrices (i.e., matrices having eigenvalues $(β, β\varepsilon, β\varepsilon^2)$ where $\varepsilon$ is a cube root of 1), or \item the set of scalars plus $3-$scalar matrices.
Any multilinear non-central polynomial $p$ (in several noncommuting variables) takes on values of degree $n$ in the matrix algebra $M_n(F)$ over an infinite field $F$. The polynomial $p$ is called {\it $ν$-central} for $M_n(F)$ if $p^ν$ takes on only scalar values, with $k$ minimal such. Multilinear $ν$-central polynomials do not exist for any $ν$ with $n>3$, thereby answering a question of Drensky. Saltman proved that an arbitrary polynomial $p$ cannot be $ν$-central for $M_n(F)$ for $n$ odd unless $n$ is prime; we show for $n$ even, that $ν$ must be 2.
In the paper the notion of {\em Rauzy scheme} is introduced. From Rauzy graph Rauzy Scheme can be obtaining by uniting sequence of vertices of ingoing and outgoing degree 1 by arches. This notion is a tool to describe Rauzy graph behavior. For morphic superword we prove periodicity of Rauzy schemes. This is generalization of fact that quadratic irrationals have periodic chain fractions.
Let G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)} and x_{i,g_i}'s variables that correspond to the g_i's, i=1,...,r. Let F be the corresponding free G-graded algebra where F is a field of zero characteristic. Here the degree of a monomial is determined by the product of the indices in G. Let I be a G-graded T-ideal of F which is PI (e.g. any ideal of identities of a G-graded finite dimensional algebra is of this type). We prove that the Hilbert series of F /I is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F /I is a rational function.
Let W be an associative PI algebra over a field F of characteristic zero, graded by a finite group G. Let id_{G}(W) denote the T-ideal of G-graded identities of W. We prove: 1. {[G-graded PI equivalence]} There exists a field extension K of F and a finite dimensional Z/2ZxG-graded algebra A over K such that id_{G}(W)=id_{G}(A^{*}) where A^{*} is the Grassmann envelope of A. 2. {[G-graded Specht problem]} The T-ideal id_{G}(W) is finitely generated as a T-ideal. 3. {[G-graded PI-equivalence for affine algebras]} Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite dimensional algebra A over K such that id_{G}(W)=id_{G}(A).