Varieties of affine Kac-Moody algebras

In this paper the identities of the complex affine Kac-Moody algebras are studied. It is proved that the identities of twisted affine algebras coincide with those of the corresponding nontwisted algebras. Moreover, in the class of nontwisted affine Kac-Moody algebras, each of these algebras is uniquely defined by its identities. It is shown that the varieties of affine algebras, as well as the varieties defined by finitely generated three-step solvable Lie algebras, have exponential growth. Translated fromMatematicheskie Zametki, Vol. 62 No. 1, pp. 95–102, July 1997. Translated by A. I. Shtern     

Two remarks on varieties of Lie algebras

The varieties of Lie algebras in which every subalgebra of the free algebra is also free are completely described, and a proof is given for a theorem on the verbal ideals of the ideals of an absolutely free Lie algebra.Translated from Matematicheskie Zametki, Vol. 4, No. 4, pp. 389–398, October, 1968.The author takes this opportunity to thank A. L. Shmel'kin for suggesting the proof of the analog of the Neumann-Weigold theorem for varieties of Lie algebras.     

Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

The centralizer algebras of lie color algebras

In his 1977 paper, V.G. Kac classified the finite-dimensional simple complex Lie superalgebras. After Kac’s paper, M. Scheunert initiated the study of a generalization of Lie superalgebras - the Lie color algebras. We construct some new families of simple Lie color algebras. Following the work of A. Berele and A. Regev and A.N. Sergeev, who studied the general linear and sq(n)-series superalgebra cases, and the work of G. Benkart, C. Lee Shader, and A. Ram, who studied the orthosymplectic cases, we examine the centralizer algebras of some classical Lie superalgebras and their Lie color algebra counterparts acting on tensor space and derive Schur-Weyl duality results for these algebras and their centralizers.     

Left-symmetric algebras of derivations of free algebras

A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie algebra of derivations. Left and right nilpotent elements of left-symmetric algebras of derivations are studied. Simple left-symmetric algebras of derivations and Novikov algebras of derivations are described. It is also proved that the positive part of the left-symmetric algebra of derivations of a free nonassociative symmetric m-ary algebra in one free variable is generated by one derivation and some right nilpotent derivations are described.     

The polynomial growth of colength of varieties of Lie algebras

It is proved that the colength of every API-variety of Lie algebras grows polynomially, and we give a number of examples in which the colength grows more rapidly than any polynomial function does. These indicate that for many of the important varieties of Lie algebras, such as varieties of solvable algebras of derived length 3, varieties generated by some infinite-dimensional simple algebras of Cartan type, or by certain Katz-Mudi algebras, the growth of colength will be superpolynomial. Supported by RFFR grants No. 96-01-00146 and No. 96-15-96050. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 161–175, March–April, 1999.     

Exponentiation of infinite dimensional Z-graded lie algebras

In this article we give a new technique for exponentiating infinite dimensional graded representations of graded Lie algebras that allows for the exponentiation of some non-locally nilpotent elements. Our technique is to naturally extend the representation of the Lie algebra g on the space V naturally to a representation on a subspace £ of the dual space V ^*. After introducing the technique, we prove that it enables the exponentiation of all elements of free Lie Algebras and afhne Kac-Moody Lie algebras.     

c-Graded filiform Lie algebras

In this paper we generalize naturally graded filiform Lie algebras as well as filiform Lie algebras admitting a connected gradation of maximal length, by introducing the concept of c-graded complex filiform Lie algebras. We deal with the particular case of 3-graded filiform Lie algebras and we obtain their classification in arbitrary dimension. We finally show a link among derived algebras, graded filiform and rigid solvable Lie algebras.     

Correlation of Poisson algebras and Lie algebras in the language of identities

In the paper, the varieties of Poisson algebras whose ideals of identities contain the identity {x, y}· {z, t} = 0 are studied, and the correlation of these varieties with varieties of Lie algebras is investigated. A variety of Poisson algebras of almost exponential growth is presented. An example of a variety of Poisson algebras with fractional exponent is also given.     

Tubular algebras and affine Kac-Moody algebras

The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.     

On the Hilbert series op relatively free algebras

Subvarieties of the variety of algebras genelaetu by the algebra of the upper triangular matrices are studied in this paper. The main result is that the relatively free algebras have rational Hilbert (or Poincarê) series when the base field is infinite. As a corollary, in characteristic 0 the rationality is obtained for varieties not containing the algebra of all 2 × 2 matrices. Some of the results are transfered to Lie and Jordan algebras.     

Bases, filtrations and module decompositions of free Lie algebras

We use the technique known as elimination to devise some new bases of the free Lie algebra which (like classical Hall bases) consist of Lie products of left normed basic Lie monomials. Our bases yield direct decompositions of the homogeneous components of the free Lie algebra with direct summands that are particularly easy to describe: they are tensor products of metabelian Lie powers. They also give rise to new filtrations and decompositions of free Lie algebras as modules for groups of graded algebra automorphisms. In particular, we obtain some new decompositions for free Lie algebras and free restricted Lie algebras over fields of positive characteristic.     

Rota-Baxter algebras and dendriform algebras

In this paper we study the adjoint functors between the category of Rota-Baxter algebras and the categories of dendriform dialgebras and trialgebras. In analogy to the well-known theory of the adjoint functor between the category of associative algebras and Lie algebras, we first give an explicit construction of free Rota-Baxter algebras and then apply it to obtain universal enveloping Rota-Baxter algebras of dendriform dialgebras and trialgebras. We further show that free dendriform dialgebras and trialgebras, as represented by binary planar trees and planar trees, are canonical subalgebras of free Rota-Baxter algebras.     

Prime and semiprime lattice ordered lie algebras

The concepts of prime Lie algebras and semiprime Lie algebras are important in the study of Lie algebras. The purpose of this paper is to investigate generalizations of these concepts to lattice ordered Lie algebras over partially ordered fields. Some results concerning the properties of l-prime and l-semiprime lattice ordered Lie algebras are obtained. A necessary and sufficient condition for a lattice ordered Lie algebra to be an l-prime Lie l-algebra is presented.     

Solvable Leibniz algebras with quasi-filiform Lie algebras of maximum length nilradicals

Abstract

In this article, solvable Leibniz algebras, whose nilradical is quasi-filiform Lie algebra of maximum length, are classified. The rigidity of such Leibniz algebras with two-dimensional complemented space to the nilradical is proved.

Communicated by K. C. Misra     

Restricted Lie algebras all whose elements are semisimple

People studied the properties and structures of restricted Lie algebras all whose elements are semisimple. It is the main objective of this paper to continue the investigation in order to obtain deeper structure theorems. We obtain some sufficient conditions for the commutativity of restricted Lie algebras, generalize some results of R. Farnsteiner and characterize some properties of a finite-dimensional semisimple restricted Lie algebra all whose elements are semisimple. Moreover, we show that a centralsimple restricted Lie algebra all whose elements are semisimple over a field of characteristic p > 7 is a form of a classical Lie algebra.     

Varieties of linear algebras with colength one

In the case of characteristic zero it is proved that there exist exactly three varieties of linear algebras with the colength equal to one for all degrees. Those are the variety of all associative-commutative algebras, the variety of all metabelian Lie algebras, and the variety of soluble Jordan algebras of the step 2 with the identity x ² x ≡ 0.     

Classification of Lie bialgebras over current algebras

In this paper, we give a classification of Lie bialgebra structures on Lie algebras of type \mathfrak g{\mathfrak {g}} [[x]] and \mathfrak g[x]{\mathfrak g[x]}, where \mathfrak g{\mathfrak g} is a simple complex finite dimensional Lie algebra.