Ivan Shestakov

The tame and the wild automorphisms of polynomial rings in three variables

Let C = F [x1, x2, . . . , xn] be the polynomial ring in the variables x1, x2, . . . , xn over a field F , and let AutC be the group of automorphisms of C as an algebra over F . An automorphism τ ∈ AutC is called elementary if it has a form τ : (x1, . . . , xi−1, xi, xi+1, . . . , xn) 7→ (x1, . . . , xi−1, αxi + f, xi+1, . . . , xn), where 0 6= α ∈ F, f ∈ F [x1, . . . , xi−1, xi+1, . . . , xn]. The subgroup of AutC generated by all the elementary automorphisms is called the tame subgroup, and the elements from this subgroup are called tame automorphisms of C. Non-tame automorphisms of the algebra C are called wild. It is well known [6], [9], [10], [11] that the automorphisms of polynomial rings and free associative algebras in two variables are tame. At present, a few new proofs of these results have been found (see [5], [8]). However, in the case of three or more variables the similar question was open and known as “The generation gap problem” [2], [3] or “Tame generators problem” [8]. The general belief was that the answer is negative, and there were several candidate counterexamples (see [5], [8], [12], [7], [19]). The best known of them is the following automorphism σ ∈ Aut(F [x, y, z]), constructed by Nagata in 1972 (see [12]): σ(x) = x+ (x − yz)z, σ(y) = y + 2(x − yz)x+ (x − yz)z, σ(z) = z.

Poisson brackets and two-generated subalgebras of rings of polynomials

Let A = F [x1, x2, . . . , xn] be a ring of polynomials over a field F on the variables x1, x2, . . . , xn. It is well known (see, for example, [11]) that the study of automorphisms of the algebra A is closely related with the description of its subalgebras. By the theorem of P. M. Cohn [4], a subalgebra of the algebra F [x] is free if and only if it is integrally closed. The theorem of A. Zaks [13] says that the Dedekind subalgebras of the algebra A are rings of polynomials in a single variable. A. Nowicki and M. Nagata [8] proved that the kernel of any nontrivial derivation of the algebra F [x, y], char(F ) = 0, is also a ring of polynomials in a single generator. An original solution of the occurrence problem for the algebra A, using the Groebner basis, was given by D. Shannon and M. Sweedler [9]. However, the method of the Groebner basis does not give any information about the structure of concrete subalgebras. Recall that the solubility of the occurrence problem for rings of polynomials over fields of characteristic 0 was proved earlier by G. Noskov [7]. The present paper is devoted to the investigation of the structure of twogenerated subalgebras of A. In the sequel, we always assume that F is an arbitrary field of characteristic 0. Let us denote by f the highest homogeneous part of an element f ∈ A, and by 〈f1, f2, . . . , fk〉 the subalgebra of A generated by the elements f1, f2, . . . , fk ∈ A. Definition 1. A pair of polynomials f1, f2 ∈ A is called ∗-reduced if they satisfy the following conditions: 1) f1, f2 are algebraically dependent; 2) f1, f2 are algebraically independent; 3) f1 / ∈ 〈f2〉, f2 / ∈ 〈f1〉. Recall that a pair f1, f2 with condition 3) is usually called reduced. Condition 1) means that we exclude the trivial case when f1, f2 are algebraically independent. We do not consider the case when f1, f2 are algebraically dependent. Concerning this case, recall the well-known theorem of S. S. Abhyankar and T. -T. Moh [1], which says that if f, g ∈ F [x] and 〈f, g〉 = F [x], then f ∈ 〈ḡ〉 or ḡ ∈ 〈f〉.

An envelope for Malcev algebras

We prove that for every Malcev algebra M there exist an algebra U(M) and a monomorphism ι:M→U(M)− of M into the commutator algebra U(M)− such that the image of M lies into the alternative center of U(M), and U(M) is a universal object with respect to such homomorphisms. The algebra U(M), in general, is not alternative, but it has a basis of Poincare–Birkhoff–Witt type over M and inherits some good properties of universal enveloping algebras of Lie algebras. In particular, the elements of M can be characterized as the primitive elements of the algebra U(M) with respect to the diagonal homomorphism Δ:U(M)→U(M)⊗U(M). An extension of Ado–Iwasawa theorem to Malcev algebras is also proved.

Prime alternative superalgebras of arbitrary characteristic

Simple nonassociative alternative superalgebras are classified. Any such superalgebra either is trivial (i.e., has zero odd part) or has characteristic 2 or 3 and is isomorphic over its center to a superalgebra of one of the following five types: in characteristic 3, these are two superalgebras of dimensions 3 and 6 and a “twisted superalgebra of vector type,” which either is infinite-dimensional or has dimension 2·3n; in characteristic 2, those are either a Cayley-Dixon algebra with a grading induced by the Cayley-Dixon process or a “double Cayley-Dixon algebra.” Under certain constraints on the structure of even parts, we also give a description of prime nonassociative alternative nontrivial superalgebras in terms of central orders of simple superalgebras. The simple superalgebras of dimensions 3 and 6 are then used to construct simple Jordan superalgebras of characteristic 3 and of dimensions 12 and 21, respectively.

Non-associative algebra and its applications

On Non-unitary Representations of the Rotation Group and Magnetic Monopoles. Generalized Derivations of Quantum Polynomials. Abelian Group Gradings on Simple Algebras. Groebner-Shirshov Basis for Lie Algebras. Classification of Solvable Three-Dimensional Lip-Triple Systems. Nonassociative Algebra Structures on Irreducible Representations of the Simple Lie Algebra. On Locally Finite Split Lie Triple Systems. On a Special Kind of Malcev Algebras. New Realizations of Hadronic Supersymmetry Based on Octonions. Application of Octonionic Algebras in Hadronic Physics. Application of Octonionic Algebras in Hardonic Physics. On Flexible Right-Nilalgebras Satisfying x(zy) = y(zx). Lie Algebras: Applications to the Classical Electromagnetic Fields. Helicity Basis and Parity. A New Look at the Freudenthal. On the Theory of Left Loops. Approximation of Locally Compact Groups by Finite Quasigroups. Some Classes of Nonassociative Algebras. Some Results on the Theory of Smooth Bol-Bruck Loops. A Nonzero Element of Degree 7 in the Center of the Free Alternative Algebra. The Identities of the Simple Non-Special Jordan Algebra. Ternary Derivations of Finite Dimensional Real Division Algebras. The Transformation Algebras of Bernstein Graph Algebras. On Composition, Quadratic, and some Triple Systems. Combinatorial Rank of a Frobenius-Lusztig Kernel. On Representations on Right Nilalgebras of Right Nilindex Four and Dimension Four. An Introduction to Associator Quantization. Prosymmetric Spaces. The Exponential Function and the Fundamental Theorem of Algebra for the Cayley Dickson Algebras. Dimension Filtration on Binary Systems. Right Alternative Bimodules. Some Applications of Quasigroups and Loops in Physics. Gravity within the Framework of Nonassociative Geometry. One-to-One Correspondence between Bi-Linear and Triple Product Algebras. Algebras satisfying Local Symmetric Triality Principle. Operads and Nonassociative Deformations. Representations of Quantum Algebras at Roots of 1. Algebras, Hyperalgebras, Nonassociative Bialgebras, and Loops. Algebraic Structures on Lie Algebras, Vinberg Algebras. Algebraic and Differential Structures in Renormalized Perturbation Quantum Field Theory. Survey on Smooth Quasigroups Development. On Kikkawa Spaces. Bol and Bruck Identities in Recent Research. New Example of a Simple Jordan Superalgebra with Associative Even Part. Unital Irreducible Representations of Small Simple Jordan Superalgebras. The Lie Product on the Lie Bialgebra Duals of the Witt and Virasoro Algebras. Derivations and Automorphisms of Free Algebras. On Derivations and Automorphisms of a Lie Algebra. Subrings of Finite Division Rings. Realizaiton of Finite Groups by Nets in Complex Projective Plane.

The Nagata automorphism is wild

It is proved that the well known Nagata automorphism of the polynomial ring in three variables over a field of characteristic zero is wild, that is, it can not be decomposed into a product of elementary automorphisms.

Every Akivis Algebra is Linear

AbstractAn Akivis algebra is a vector space V endowed with a skew-symmetric bilinear product [x,y] and a trilinear product A(x,y,z) that satisfy the identity

FREE MALCEV SUPERALGEBRA ON ONE ODD GENERATOR

\begin{gathered} [[x,y],z] + [[y,z],x] + [[z,x],y] \hfill \\ = {\mathcal{A}}(x,y,z) + {\mathcal{A}}(y,z,x) + {\mathcal{A}}(z,x,y) - {\mathcal{A}}(y,x,z) - {\mathcal{A}}(x,z,y) - {\mathcal{A}}(z,y,x). \hfill \\ \end{gathered}