Identities of metabelian alternative algebras

We study metabelian alternative (in particular, associative) algebras over a field of characteristic 0. We construct additive bases of the free algebras of mentioned varieties, describe some centers of these algebras, compute the values of the sequence of codimensions of corresponding T-ideals, and find unitarily irreducible components of the decomposition of mentioned varieties into a union and their bases of identities. In particular, we find a basis of identities for the metabelian alternative Grassmann algebra. We prove that the free algebra of a variety that is generated by the metabelian alternative Grassmann algebra possesses the zero associative center.     

Nonfinitely Based Varieties of Right Alternative Metabelian Algebras

Since 1976, it is known from the paper by V. P. Belkin that the variety RA₂ of right alternative metabelian (solvable of index 2) algebras over an arbitrary field is not Spechtian (contains nonfinitely based subvarieties). In 2005, S. V. Pchelintsev proved that the variety generated by the Grassmann RA₂-algebra of finite rank r over a field ?, for char(?) ≠ 2, is Spechtian iff r = 1. We construct a nonfinitely based variety 𝔐 generated by the Grassmann 𝒱-algebra of rank 2 of certain finitely based subvariety 𝒱 ? RA₂ over a field ?, for char(?) ≠ 2, 3, such that 𝔐 can also be generated by the Grassmann envelope of a five-dimensional superalgebra with one-dimensional even part.     

On Spechtian varieties of right alternative algebras

A sufficient condition is proved for the Specht property of varieties of right alternative metabelian algebras over a field of characteristic distinct from 2. As a consequence, the Specht property of some varieties generated by right alternative metabelian algebras A satisfying a commutator identity is stated. In particular, it is proved that if A ⁽⁻⁾ is a binary Lie algebra, then var(A) is Spechtian. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 89–100, 2006.     

Über endliche galoissche schiefkörpererweiterungen ohne endliche galoisgruppen

We define a generalized Brauer-Severi variety to be the projective Variety whose closed points correspond to rank m right ideals in a central simple algebra. We show that these varieties are forms of the Grassmann variety and their function fields are generic partial splitting fields for the associated algebras. The subgroup of the Brauer group split by the function field is calculated. Following Schofield and van den Bergh, we calculate the index of a simple ailgebra extended by the function field of a generalized Brauer-Severi variety associated with any other central simple algebra.     

Test Sets and Test Rank of a Free Metabelian Lie Algebra

Abstract

We prove that the test rank of a free metabelian Lie algebra L of rank n equals n-1 and a test set of L belongs to the derived subalgebra of L.     

Speciality of Metabelian Mal'tsev Algebras

It is proved that, for any metabelian Mal'tsev algebra M over a field of characteristic 2,3, there is an alternative algebra A such that the algebra M can be embedded in the commutator algebra A^(-). Moreover, the enveloping alternative algebra A can be found in the variety of algebras with the identity [x,y][z,t] = 0. The proof of this result is based on the construction of additive bases of the free metabelian Mal'tsev algebra and the free alternative algebra with the identity [x,y][z,t] = 0.     

Endomorphisms of Free Metabelian Lie Algebras Which Preserve Orbits

For the free rank 2 metabelian Lie algebra over an infinite field we prove that an endomorphism of the algebra which preserves the automorphic orbit of a nonzero element is an automorphism. We construct some counterexamples over finite fields.     

Ideals of Free Metabelian Lie Algebras and Primitive Elements

Necessary and sufficient conditions are established for elements of a finite rank free metabelian Lie algebra in order that every endomorphism of this algebra be uniquely determined by its values at these elements.     

The hypercentral structure of the group of unitriangular automorphisms of a free metabelian Lie algebra

We describe the hypercentral structure of the group of unitriangular automorphisms of a free metabelian Lie algebra over an arbitrary field. Using it, we prove that this group admits no faithful representation by matrices over a field provided that the algebra rank is at least four.     

POLYNOMIAL IDENTITIES ON SUPERALGEBRAS AND ALMOST POLYNOMIAL GROWTH

Let A be a superalgebra over a field of characteristic zero. In this paper we investigate the graded polynomial identities of A through the asymptotic behavior of a numerical sequence called the sequence of graded codimensions of A. Our main result says that such sequence is polynomially bounded if and only if the variety of superalgebras generated by A does not contain a list of five superalgebras consisting of a 2-dimensional algebra, the infinite dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with trivial and nontrivial gradings. Our main tool is the representation theory of the symmetric group.     

Test Sets in Free Metabelian Lie Algebras

We calculate the test rank of a finite rank free metabelian Lie algebra over an arbitrary field and characterize the test sets for these algebras. We prove that each automorphism that is the identity modulo the derived subalgebra and that acts as the identity on some test set is an inner automorphism.     

Strictly nontame primitive elements of the free metabelian Lie algebra of rank 3

We prove that the free metabelian Lie algebra M ₃ of rank 3 over an arbitrary field K admits strictly nontame primitive elements.     

Primitive and measure-preserving systems of elements on the varieties of metabelian and metabelian profinite groups

Given a free metabelian group S of finite rank r, r ≥ 2, we prove that a system of elements g ₁, ..., g _n ∈ S for n = 1 or n = r preserves measure on the variety of all metabelian groups if and only if the system is primitive. Similar results hold for a free profinite group $\hat S$ and the variety of finite metabelian groups for each n, 1 ≤ n ≤ r. Some corollaries to these theorems are derived.     

Growth of codimensions of metabelian algebras

Numerical invariants of identities of nonassociative algebras are considered. It is proved that the codimension sequence of any finitely generated metabelian algebra has an exponentially bounded codimension growth. It is shown that the upper PI-exponent increases at most by 1 after adjoining an external unit. It is proved that for two-step left-nilpotent algebras the lower PI-exponent increases at least by 1.     

Algebraic geometry over free metabelian lie algebras. I. U-algebras and universal classes

This paper is the first in a series of three, the object of which is to lay the foundations of algebraic geometry over the free metabelian Lie algebra F. In the current paper, we introduce the notion of a metabelian U-Lie algebra and establish connections between metabelian U-Lie algebras and special matrix Lie algebras. We define the Δ-localization of a metabelian U-Lie algebra A and the direct module extension of the Fitting radical of A and show that these algebras lie in the universal closure of A. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 37–63, 2003.     

Approximations of Lie algebras

It is proved that a finitely generated metabelian Lie algebra over an arbitrary field can be approximated by finite-dimensional algebras and a stronger result is also obtained over fields of nonzero characteristics.Translated from Matematicheskie Zametki, Vol. 12, No. 6, pp. 713–716, December, 1972.     

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ?-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.     

Translation-invariant integrals,and Fourier analysis on Clifford and Grassmann algebras

The generalization of Berezin's Grassmann algebra integral to a Clifford algebra is shown to be translation-invariant in a certain sense. This enables the construction of analogs of twisted convolutions of Grassmann algebra elements and of the Fourier-Weyl transformation, which is an isomorphism from a Clifford algebra to the Grassmann algebra over the dual space, equipped with a twisted convolution product. As an application a noncommutative central limit theorem for states of a Clifford algebra is proved.