Simple algebras of Weyl type, II

Over a field of any characteristic, for a commutative associative algebra , and for a commutative subalgebra of , the vector space which consists of polynomials of elements in with coefficients in and which is regarded as operators on forms naturally an associative algebra. It is proved that, as an associative algebra, is simple if and only if is -simple. Suppose is -simple. Then, (a) is a free left -module; (b) as a Lie algebra, the subquotient is simple (except for one case), where is the center of . The structure of this subquotient is explicitly described. This extends the results obtained by Su and Zhao.

     

On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

We prove a part of the Cachazo-Douglas-Seiberg-Witten conjecture uniformly for any simple Lie algebra . The main ingredients in the proof are: Garland's result on the Lie algebra cohomology of ; Kostant's result on the `diagonal' cohomolgy of and its connection with abelian ideals in a Borel subalgebra of ; and a certain deformation of the singular cohomology of the infinite Grassmannian introduced by Belkale-Kumar.

     

Novikov algebras and Novikov structures on Lie algebras

We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras of triangular matrices. Finally we show that there are families of Lie algebras of arbitrary high solvability class which admit Novikov structures.     

Tanaka prolongation of free Lie algebras

With the exception of the three step real free Lie algebra on two generators, all real free Lie algebras of step at least three are shown to have trivial Tanaka prolongation. This result, together with the known results concerning the step two real free Lie algebras and the step three real free Lie algebra on two generators, gives a complete list of Tanaka prolongations for real free Lie algebras.      

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.     

Simple sagle algebras

It is proved that ifS is a simple finite-dimensional anticommutative algebra over a field ϕ of characteristic zero satisfying the identityJ(x, y, z)t=J(t, z, xy)+J(t, y, zx)+J(t, x,yz), whereJ(x, y, z)=(xy)z+(zx)y+(yz)x, thenS is a Lie algebra. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 607–611, April, 1999.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

Leibniz algebras constructed by Witt algebras

ABSTRACT

We describe infinite-dimensional Leibniz algebras whose associated Lie algebra is the Witt algebra and we prove the triviality of low-dimensional Leibniz cohomology groups of the Witt algebra with the coefficients in itself.     

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.     

Novikov-Poisson algebras and associative commutative derivation algebras

We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras. Supported by RFBR (grant No. 05-01-00230), by SB RAS (Integration project No. 1.9), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 186–202, March–April, 2008.     

Local automorphisms of standard operator algebras

Let X be an infinite-dimensional separable real or complex Banach space and A a closed standard operator algebra on X. Then every local automorphism of A is an automorphism. The assumptions of infinite-dimensionality, separability, and closeness are all indispensable.     

Classification of Lie algebras of specific type in complexified Clifford algebras

We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These 16 Lie algebras are direct sums of subspaces of quaternion types. We obtain isomorphisms between these Lie algebras and classical matrix Lie algebras in the cases of arbitrary dimension and signature. We present 16 Lie groups: one Lie group for each Lie algebra associated with this Lie group. We study connection between these groups and spin groups.     

Local automorphisms of operator algebras on Banach spaces

In this paper we extend a result of Semrl stating that every 2-local automorphism of the full operator algebra on a separable infinite dimensional Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we obtain the same conclusion for the much larger class of Banach spaces with Schauder bases. The proof rests on an analogous statement concerning the 2-local automorphisms of matrix algebras for which we present a short proof. The need to get such a proof was formulated in Semrl's paper.

     

Extremal Bases for the Adjoint Representations of the Simple Lie Algebras

We construct n distinct weight bases, which we call extremal bases, for the adjoint representation of each simple Lie algebra 𝔤 of rank n: One construction for each simple root. We explicitly describe actions of the Chevalley generators on the basis elements. We show that these extremal bases are distinguished by their “supporting graphs” in three ways. (In general, the supporting graph of a weight basis for a representation of a semisimple Lie algebra is a directed graph with colored edges that describe the supports of the actions of the Chevalley generators on the elements of the basis.) We show that each extremal basis constructed is essentially the only basis with its supporting graph (i.e., each extremal basis is solitary), and that each supporting graph is a modular lattice. Each extremal basis is shown to be edge-minimizing: Its supporting graph has the minimum number of edges. The extremal bases are shown to be the only edge-minimizing as well as the only modular lattice weight bases (up to scalar multiples) for the adjoint representation of 𝔤. The supporting graph for an extremal basis is shown to be a distributive lattice if and only if the associated simple root corresponds to an end node for a “branchless” simple Lie algebra, i.e., type A, B, C, F, or G. For each extremal basis, basis elements for the Cartan subalgebra are explicitly expressed in terms of the h _i Chevalley generators.     

Lie algebras with few centralizers

We will characterize all finite dimensional Lie algebras with at most |F|²+|F|+2 centralizers, where F is the underlying field of Lie algebras under consideration.     

Linearly compact algebraic Lie algebras and coalgebraic Lie coalgebras

It is proved that if the dual Lie algebra of a Lie coalgebra is algebraic, then it is algebraic of bounded degree. This result is an analog of the D.Radford's result for associative coalgebras.

     

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.