The Socle of a Metric n-Lie Algebra

We define the socle of an n-Lie algebra as the sum of all the minimal ideals. An n-Lie algebra is called metric if it is endowed with an invariant nondegenerate symmetric bilinear form. We characterize the socle of a metric n-Lie algebra, which is closely related to the radical and the center of the metric n-Lie algebra. In particular, the socle of a metric n-Lie algebra is reductive, and a metric n-Lie algebra is solvable if and only if the socle coincides with its center. We also calculate the metric dimensions of simple and reductive n-Lie algebras and give a lower bound in the nonreductive case.     

Metric n-Lie Algebras

An n-Lie algebra is said to be metric if it is endowed with an invariant, non-degenerate, symmetric bilinear form. We prove that any simple n-Lie algebra over an algebraically closed field of characteristic zero admits a unique metric structure and vice versa. Further, we present two metric n-Lie algebras, which are indecomposable but admit many more metric structures.     

Nilpotent n-Lie Algebras

We find examples of nilpotent n-Lie algebras and prove n-Lie analogs of classical group theory and Lie algebra results. As an example we show that a nilpotent ideal I of class c in a n-Lie algebra A with A/I ² nilpotent of class d is nilpotent and find a bound on the class of A. We also find that some classical group theory and Lie algebra results do not hold in n-Lie algebras. In particular, non-nilpotent n-Lie algebras can admit a regular automorphism of order p, and the sum of nilpotent ideals need not be nilpotent.     

One-relator quotients of free products of cyclic groups

Abstract

The question of existence of n-ary systems playing the role of enveloping algebras for Filippov (n-Lie) algebras is investigated. We consider the ternary algebras whose reduced binary algebras are associative, the associative algebras, and two other classes of ternary algebras as possible enveloping algebras.     

Some Properties On Isoclinism in n-Lie Algebras

The concept of n-Lie algebra were introduced by Filippov in 1987. One notes that not all properties of Lie algebras can be carried over to n-Lie algebras, as Williams showed in 2009. In the present article, among other results it is shown that the notions of isoclinism and isomorphism for two finite dimensional n-Lie algebras of the same dimension are equivalent. This was already done for ordinary Lie algebras by Moneyhun in 1994.     

On Schur’s theorem and its converses for n-Lie algebras

An n-Lie algebra analogue of Schur’s theorem and its converse as well as a Lie algebra analogue of Baer’s theorem and its converse are presented. Also, it is shown that, an n-Lie algebra with finite dimensional derived subalgebra and finitely generated central factor is isoclinic to some finite dimensional n-Lie algebra.     

Isoclinism between pairs of n-Lie algebras and their properties

In this paper, we investigate the notion of isoclinism on a pair of n-Lie algebras, which forms an equivalence relation. In addition, we prove that each equivalence class contains a stem pair of n-Lie algebras, which has minimal dimension amongst the finite dimensional pairs of n-Lie algebras. Finally, some more results are obtained when two isoclinic pairs of n-Lie algebras are given.     

Binary lie algebras of small dimensions

We give a simple and shorter proof of the Gainov theorem in [1], which dealt with classifying non-Lie binary Lie algebras of dimension ≤4 over a field of characteristic ≠2. Concurrently, the case of characteristic 2 is treated, and we find out an exotic 4-dimensional non-Lie Mal'tsev algebra, which is a split extension of an irreducible 1-dimensional Mal'tsev module over a simple 3-dimensional Lie algebra. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 320–328, May–June, 1998.     

A New Approach to Representations of 3-Lie Algebras and Abelian Extensions

In this paper, we introduce the notion of generalized representation of a 3-Lie algebra, by which we obtain a generalized semidirect product 3-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of generalized representations of 3-Lie algebras and computation of 2-cocycles of the new cohomology are provided. Also, we show that a split abelian extension of a 3-Lie algebra is isomorphic to a generalized semidirect product 3-Lie algebra. Furthermore, we describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements.     

Universal Associative Envelopes of (n+ 1)-Dimensional n-Lie Algebras

For n even, we prove Pozhidaev's conjecture on the existence of associative enveloping algebras for simple n-Lie (Filippov) algebras. More generally, for n even and any (n + 1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(L) and show that the natural map L → U(L) is injective. We use noncommutative Gröbner bases to present U(L) as a quotient of the free associative algebra on a basis of L and to obtain a monomial basis of U(L). In the last section, we provide computational evidence that the construction of U(L) is much more difficult for n odd.     

Ternary Numbers, Algebras of Reflexive Numbers and Berger Graphs

The Calabi-Yau spaces with SU(n) holonomy can be studied by the algebraic way through the integer lattice where one can construct the Newton reflexive polyhedra or the Berger graphs. Our conjecture is that the Berger graphs can be directly related with the n-ary algebras. To find such algebras we study the n-ary generalization of the well-known binary norm division algebras, , which helped to discover the most important “minimal” binary simple Lie groups, U(1), SU(2) and G(2). As the most important example, we consider the case n = 3, which gives the ternary generalization of quaternions (octonions), 3 ⁿ , n = 2, 3, respectively. The ternary generalization of quaternions is directly related to the new ternary algebra (group) which are related to the natural extensions of the binary su(3) algebra (SU(3) group). Using this ternary algebra we found the solution for the Berger graph: a tetrahedron.

“Why geniosis live so short? They wanna stay kids.”

Alexey Dubrovski: On leave from JINR, Russia. Guennadi Volkov: On leave from PNPI, Russia.     

Algebras and differential equations with additive symmetry

The set of all m-ary algebra structures on a given vector space affords, by the change of basis action, a representation of the general linear group. The invariants of a given subgroup are identified with those algebras admitting that subgroup as algebra automorphisms. Any finite dimensional representation of the additive group as automorphisms is obtained as the exponential of a nilpotent derivation. The latter can be embedded in the Lie algebra sl(2) so that the maximal vectors in an irreducible decomposition of the set of algebras as an sl(2) module are the invariants of the given action of the additive group. Dimension formulas and explicit bases are computed for the space of algebras with certain additive group actions. Employing the equivalence of the categories of m-ary algebras and systems of autonomous mth order homogeneous differential equations, the algebraic results are connected to the construction of first integrals and semi-invariants.     

3-李代数的辛结构

研究了3-李代数和度量3-李代数的辛结构.对任意3-李代数L,构造了无限多个度量辛3-李代数.证明了度量3-李代数(A,B)是度量辛3-李代数的充要条件,即存在可逆导子D,使得D∈Der_B(A).同时证明了每一个度量辛3-李代数(A,B,ω)是度量辛3-李代数(A,B,ω)的T_θ~*-扩张.最后,利用度量辛3-李代数经过特殊导子的双扩张得到了新的度量辛3-李代数.     

The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras

Let H be a Hopf algebra over the field k which is a finite module over a central affine sub-Hopf algebra R. Examples include enveloping algebras of finite dimensional k-Lie algebras in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity. The ramification behaviour of the maximal ideals of Z(H) with respect to the subalgebra R is studied, and the conclusions are then applied to the cases of classical and quantised enveloping algebras. In the case of for semisimple a conjecture of Humphreys [28] on the block structure of is confirmed. In the case of for semisimple and an odd root of unity we obtain a quantum analogue of a result of Mirković and Rumynin, [35], and we fully describe the factor algebras lying over the regular sheet, [9]. The blocks of are determined, and a necessary condition (which may also be sufficient) for a baby Verma -module to be simple is obtained. Received: 24 June 1999; in final form: 30 March 2000 / Published online: 17 May 2001     

(n + 1)-ary derivations of semisimple Filippov algebras

The structure of generalized and (n+1)-ary derivations of simple and semisimple finite-dimensional Filippov algebras over an algebraically closed field of characteristic zero is described. An example of a semisimple ternary Mal’tsev algebra is given which is not a Filippov algebra and admits a nontrivial 4-ary derivation.     

Matrix Characterization of 4-Ary Algebraic Operations of Idempotent Algebras

Let (U; F) be an idempotent algebra. There is an r-ary essentially algebraic operation in F where there is not any (r + 3)-ary algebraic operation depending on at least r + 1 variables. In this paper, we prove that the set of all 4-ary algebraic operations of this algebras forms a finite De Morgan algebra, and then we characterize this De Morgan algebra.     

On classification of n-Lie algebras

In this paper, we prove the isomorphic criterion theorem for (n+2)-dimensional n-Lie algebras, and give a complete classification of (n + 2)-dimensional n-Lie algebras over an algebraically closed field of characteristic zero.     

The structure and representation of <Emphasis Type="Italic">n</Emphasis>-ary algebras of DNA recombination

In this paper we investigate the structure and representation of n-ary algebras arising from DNA recombination, where n is a number of DNA segments participating in recombination. Our methods involve a generalization of the Jordan formalization of observables in quantum mechanics in n-ary splicing algebras. It is proved that every identity satisfied by n-ary DNA recombination, with no restriction on the degree, is a consequence of n-ary commutativity and a single n-ary identity of the degree 3n-2. It solves the well-known open problem in the theory of n-ary intermolecular recombination.     

Endoprimal implication algebras

An algebra A is endoprimal if, for all , the only maps which preserve the endomorphisms of A are the n-ary term functions of A. The theory of natural dualities has been a very effective tool for finding finite endoprimal algebras. We study endoprimality within the variety of implication algebras, which does not contain any non-trivial dualisable algebras. We show that there are no non-trivial finite endoprimal implication algebras. We also give some examples of infinite implication algebras which are endoprimal. Received July 28, 1998; accepted in final form January 18, 1999.     

Annihilators of Mal'tsev algebras

Let be an associative ring with 1, containing 1/6,and let A be an arbitrary Mal'tsev -algebra. For some fully invariant ideals of an algebra A, we prove that their products lie in the annihilator Annof this algebra. As a consequence, it is inferred that for every algebra A over a field of characteristic 0, satisfying the nth Engel condition, there exists an N such that .Also, some identities in Mal'tsev algebras having faithful representations are proved to hold in separated Mal'tsev algebras; in the class of alternative algebras, new central and kernel functions are constructed. Translated fromAlgebra i Logika, Vol. 33, No. 5, pp. 576–595, September–October, 1994.