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.     

Algebraic geometry over algebraic structures. II. Foundations

In this paper, we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove so-called unification theorems that describe coordinate algebras of algebraic sets in several different ways.     

Infinite-dimensional lie algebras. II

In a previous paper, systems of vectors of the kind that occur in Lie algebras of types A and D were classified in the infinite-dimensional case. The present paper carries out the analogous project for types B and C. The starting point is a set of six axioms for a set of vectors in an inner product space. The fact that F₄ is the only exceptional system and that its dimension is only 4 makes it possible to keep the discussion elementary and self-contained.     

Wild automorphisms of free metabelian algebras

We establish a necessary condition for the invertibility of an endomorphism of a free associative algebra. As an application, we offer examples of wild automorphisms of certain free metabelian algebras.     

On ideals of free lie algebras

The only possible ideals which are finitely generated as subalgebras of a free Lie algebra are the zero ideal or the whole algebra. This should not be understood as “all nontrivial ideals which have infinite rank are proper ideals”.In this paper we prove that a nontrivial ideal which has infinite rank may be the whole algebra under certain conditions.     

Rigidity of free algebras in varieties of lie algebras

In this paper the main result is the rigidity of varietally free Lie algebras within the varieties where they are free. In fact, these algebras are usually free in some larger varieties, such as various types of the commutator varieties, and this is demonstrated in this paper as well. A related paper is [2] where, among some other results, we claimed the rigidity of free metabelian algebras within the respective variety of Lie algebras.     

Weitzenböck derivations of free metabelian Lie algebras

A nonzero locally nilpotent linear derivation δ   of the polynomial algebra K[_Xd]=K[_x1,…,_xd]$K[X_d]=K[x_1\text{,}\dots \text{,}{x}_d]$ in several variables over a field K   of characteristic 0 is called a Weitzenböck derivation. The classical theorem of Weitzenböck states that the algebra of constants K[_Xd]^δ$K{[X_d]}^{\delta }$ (which coincides with the algebra of invariants of a single unipotent transformation) is finitely generated. Similarly one may consider the algebra of constants of a locally nilpotent linear derivation δ of a finitely generated (not necessarily commutative or associative) algebra which is relatively free in a variety of algebras over K  . Now the algebra of constants is usually not finitely generated. Except for some trivial cases this holds for the algebra of constants (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$ of the free metabelian Lie algebra _Ld/_Ld″$L_d/L_d″$ with d   generators. We show that the vector space of the constants (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$ in the commutator ideal _Ld′/_Ld″$L_d\prime /L_d″$ is a finitely generated K[_Xd]^δ$K{[X_d]}^{\delta }$-module. For small d  , we calculate the Hilbert series of (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$ and find the generators of the K[_Xd]^δ$K{[X_d]}^{\delta }$-module (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$. This gives also an (infinite) set of generators of the algebra (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$.     

Pointed torsion free modules of affine lie algebras

It is shown that all pointed torsion free modules for affine Lie algebras belong to C⁽¹⁾ _n and A⁽¹⁾ _n-1 and are the result of the natural construction of tensoring the Laurent polynomials with a torsion free module of the “underlying” simple finite dimensional Lie Algebra. These latter modules have been completely determined by Britten and Lemire [1].     

Test elements for monomorphisms of free lie algebras and lie superalgebras †

A test criterion for an endomorphism φ of the free Lie (super) algebra L of finite rank to be an automorphism is obtained: φ is an automorphism of L if and only if for an element u ∈ L with the maximal rank the element φ(u) belongs to the orbit of u with respect to the automorphism group of L. In particular, test elements for monomorphisms of L are exactly the elements of the maximal rank     

Directed pseudo-graphs and lie algebras over finite fields

The main goal of this paper is to show an application of Graph Theory to classifying Lie algebras over finite fields. It is rooted in the representation of each Lie algebra by a certain pseudo-graph. As partial results, it is deduced that there exist, up to isomorphism, four, six, fourteen and thirty-four 2-, 3-, 4-, and 5-dimensional algebras of the studied family, respectively, over the field ?/2?. Over ?/3?, eight and twenty-two 2- and 3-dimensional Lie algebras, respectively, are also found. Finally, some ideas for future research are presented.     

Induced modules over group algebras of metabelian groups of finite rank

ABSTRACT

Investigation of multiplace functions by algebraic methods plays an important role in modern mathematics were we consider various operations on sets of functions, which are naturally defined. The basic operation for functions is superposition (composition), but there are some other naturally defined operations, which are also worth of consideration. For example, the operation of set-theoretic intersection and the operation of projections. In this paper we find an abstract characterization of the set of multiplace functions which are closely related to these three operations.     

Enumeration of maximal subalgebras in free restricted lie algebras

Given a finitely generated restricted Lie algebra L over the finite field \(\mathbb{F}_q \), and n ≥ 0, denote by a _n (L) the number of restricted subalgebras H ? L with \(\dim _{\mathbb{F} _q} \) L/H = n. Denote by ã _n (L) the number of the subalgebras satisfying the maximality condition as well. Considering the free restricted Lie algebra L = F _d of rank d ≥ 2, we find the asymptotics of ã _n (F _d ) and show that it coincides with the asymptotics of a _n (F _d ) which was found previously by the first author. Our approach is based on studying the actions of restricted algebras by derivations on the truncated polynomial rings. We establish that the maximal subalgebras correspond to the so-called primitive actions. This means that “almost all” restricted subalgebras H ? F _d of finite codimension are maximal, which is analogous to the corresponding results for free groups and free associative algebras.