Weak polynomial identities and their applications

Let R be an associative algebra over a field K generated by a vector subspace V. The polynomial f(x 1, . . . , xn ) of the free associative algebra K〈x 1, x 2, . . .〉 is a weak polynomial identity for the pair (R, V) if it vanishes in R when evaluated on V. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.

CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

The problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

Lie commutators in a free diassociative algebra

We give a criterion for Leibniz elements in a free diassociative algebra. In the diassociative case one can consider two versions of Lie commutators. We give criterions for elements of diassociative algebras to be Lie under these commutators. One of them corresponds to Leibniz elements. It generalizes the Dynkin-Specht-Wever criterion for Lie elements in a free associative algebra.

Odd automorphisms of two generated braided free associative algebras

It is proved that an endomorphism $\varphi$ of an braided free associative algebra in two generators over an arbitrary field $k$ with an involutive diagonal braiding $\tau = (- 1, -1, -1, -1)$ given by the rule $\varphi (x_1) = x_1, \, \varphi (x_2) = \alpha x_2 + \beta x^m_1,$ where $\alpha, \, \beta \in k, \, m $ is an odd number, is an odd automorphism. It is also proved that the linear endomorphism $\psi$ of this algebra is an automorphism if and only if $\psi$ is affine. It is shown that the group of all automorphisms of braided free associative algebra in two variables over an arbitrary field $ k $ with an involutive diagonal braiding $ \tau = (- 1, -1, -1, -1) $ coincides with the group of odd automorphisms of this algebra.

On the augmentation topology of automorphism groups of affine spaces and algebras

We study topological properties of Ind-groups [Formula: see text] and [Formula: see text] of automorphisms of polynomial and free associative algebras via Ind-schemes, toric varieties, approximations, and singularities. We obtain a number of properties of [Formula: see text], where [Formula: see text] is the polynomial or free associative algebra over the base field [Formula: see text]. We prove that all Ind-scheme automorphisms of [Formula: see text] are inner for [Formula: see text], and all Ind-scheme automorphisms of [Formula: see text] are semi-inner. As an application, we prove that [Formula: see text] cannot be embedded into [Formula: see text] by the natural abelianization. In other words, the Automorphism Group Lifting Problem has a negative solution. We explore close connection between the above results and the Jacobian conjecture, as well as the Kanel-Belov–Kontsevich conjecture, and formulate the Jacobian conjecture for fields of any characteristic. We make use of results of Bodnarchuk and Rips, and we also consider automorphisms of tame groups preserving the origin and obtain a modification of said results in the tame setting.

FIXED ELEMENTS OF NONINJECTIVE ENDOMORPHISMS OF POLYNOMIAL ALGEBRAS IN TWO VARIABLES

Let $A_{2}$ be a free associative algebra or polynomial algebra of rank two over a field of characteristic zero. The main results of this paper are the classification of noninjective endomorphisms of $A_{2}$ and an algorithm to determine whether a given noninjective endomorphism of $A_{2}$ has a nontrivial fixed element for a polynomial algebra. The algorithm for a free associative algebra of rank two is valid whenever an element is given and the subalgebra generated by this element contains the image of the given noninjective endomorphism.

A Deformation of the Orlik-Solomon Algebra

A deformation of the Orlik-Solomon algebra of a matroid $\mathfrak{M}$ is defined as a quotient of the free associative algebra over a commutative ring $R$ with $1$. It is shown that the given generators form a Gröbner basis and that after suitable homogenization the deformation and the Orlik-Solomon have the same Hilbert series as $R$-algebras. For supersolvable matroids, equivalently fiber type arrangements, there is a quadratic Gröbner basis and hence the algebra is Koszul.