Binary Lie Algebras Research Articles

Malcev-like binary Lie algebras of dimension 5

We study anti-commutative algebras, which are extensions of a one-dimensional algebra by a four-dimensional nilpotent Lie algebra and at the same time extensions of the two-dimensional non-Abelian Lie algebra by an abelian algebra. These algebras, just like the five-dimensional solvable Malcev algebras, have a flag of subalgebras and can therefore be considered their closest relatives. We characterize binary Lie algebras in this class, called Malcev-like algebras, playing interesting role in non-associative Lie theory. We find normal forms of their multiplications, and determine their isomorphism classes.

3-Lie algebras, ternary Nambu-Lie algebras and the Yang-Baxter equation

We construct ternary self-distributive (TSD) objects from compositions of binary Lie algebras, 3-Lie algebras and, in particular, ternary Nambu-Lie algebras. We show that the structures obtained satisfy an invertibility property resembling that of racks. We prove that these structures give rise to Yang-Baxter operators in the tensor product of the base vector space and, upon defining suitable twisting isomorphisms, we obtain representations of the infinite (framed) braid group. We consider examples for low-dimensional Lie algebras, where the ternary bracket is defined by composition of the binary ones, along with simple 3-Lie algebras. We show that the Yang-Baxter operators obtained are not gauge equivalent to the transposition operator, and we consider the problem of deforming the operators to obtain new solutions to the Yang-Baxter equation. We discuss the applications of this deformation procedure to the construction of (framed) link invariants.

Classification of a family of 4-dimensional anti-commutative algebras and their automorphisms

We determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field K of characteristic zero having an analogous family of flags of subalgebras as the four-dimensional non-Lie binary Lie algebras, and hence can be considered as the closest relatives of binary Lie algebras. These algebras are extensions of K by the 3-dimensional nilpotent Lie algebra and at the same time extensions of a two-dimensional Lie algebra by a two-dimensional abelian algebra. We describe their groups of automorphisms as extensions of a subgroup of the group of automorphisms of the three-dimensional nilpotent Lie algebra by K.

New Examples of Binary Lie Superalgebras and Algebras

New Examples of Binary Lie Superalgebras and Algebras

Levi and Malcev Theorems for Finite-Dimensional Algebras from the Variety Defined by the Identities x2 = J( x,y, zu) =0

We study the variety of binary Lie algebras defined by the identities [Formula: see text], where [Formula: see text] denotes the Jacobian of [Formula: see text], [Formula: see text], [Formula: see text]. Building on previous work by Carrillo, Rasskazova, Sabinina and Grishkov, in the present article it is shown that the Levi and Malcev theorems hold for this variety of algebras.

Central extensions of 4-dimensional binary Lie algebras

The class of binary Lie algebras contains the one of Malcev algebras (and so the one of Lie algebras). We provide a classification of central extensions of complex non-Malcev binary Lie algebras of dimensions either 3 or 4. From here, a classification of central extensions of complex non-Lie anticommutative ℭ𝔇-algebras of dimensions either 3 or 4 is also given.

The algebraic and geometric classification of nilpotent binary Lie algebras

We give a complete algebraic classification of nilpotent binary Lie algebras of dimension at most 6 over an arbitrary field of characteristic not 2 and a complete geometric classification of nilpotent binary Lie algebras of dimension 6 over [Formula: see text]. As an application, we give an algebraic and geometric classification of nilpotent anticommutative [Formula: see text]-algebras of dimension at most 6.

Degenerations of binary Lie and nilpotent Malcev algebras

ABSTRACTWe describe degenerations of four-dimensional binary Lie algebras, and five- and six-dimensional nilpotent Malcev algebras over ℂ. In particular, we describe all irreducible components of these varieties.

Universal Poisson Envelope for Binary-Lie Algebras

In this article the universal Poisson enveloping algebra for a binary-Lie algebra is constructed. Taking a basis 𝔹 of a binary-Lie algebra B, we consider the symmetric algebra S(B) of polynomials in the elements of 𝔹. We consider two products in S(B), the usual product of polynomials fg and the braces {f, g}, defined by the product in B and the Leibniz rule. This algebra is a general Poisson algebra. We find an ideal I of S(B) such that the factor algebra S(B)/I is the universal Poisson envelope of B. We provide some examples of this construction for known binary-Lie algebras.

Binary Lie algebras satisfying the third Engel condition

Let Φ be a unital associative commutative ring with 1/2. The local nilpotency is proved of binary Lie Φ-algebras satisfying the third Engel condition. Moreover, it is proved that this class of algebras does not contain semiprime algebras.

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.

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.

Lattice definability of a semisimple finite-dimensional binary-Lie algebra over an algebraically closed field of characteristic 0

Lattice definability of a semisimple finite-dimensional binary-Lie algebra over an algebraically closed field of characteristic 0

ON THE CONJUGACY OF LEVI FACTORS IN FINITE-DIMENSIONAL BINARY-LIE ALGEBRAS

The main result is the determination of necessary and sufficient conditions for conjugacy of Levi factors of a finite-dimensional binary-Lie algebra over a field of characteristic 0 (Theorem 5). In the course of the proof a detailed study of binary-Lie algebras coinciding with their squares is carried out. In particular, it is proved that the quotient of such an algebra modulo its center is a Mal'tsev algebra. Bibliography: 26 titles.

The Lie theorem for solvable binary-Lie algebras

The Lie theorem for solvable binary-Lie algebras

STRUCTURE AND REPRESENTATION OF BINARY-LIE ALGEBRAS

In this paper semisimple finite-dimensional binary-Lie algebras over a field of characteristic 0 are described. It is proved that a finite-dimensional binary-Lie semisimple algebra over a field of characteristic 0 is a Mal'cev algebra, and an arbitrary finite-dimensional irreducible binary-Lie module over a semisimple Mal'cev algebra is Mal'cev. Bibliography: 10 titles.

On the exceptional central simple non-Lie Mal’cev algebras

Malcev algebras belong to the class of binary Lie algebras. Any Lie algebra is a Malcev algebra. In this paper we show that for each seven-dimensional central simple non-Lie Malcev algebra any finite dimensional Malcev module is completely reducible also for positive characteristics. This contrasts with each modular semisimple Lie algebra. As a consequence we get that the classical structure theory for characteristic zero is valid also in the modular case if semisimplicity is replaced by G 1 {G_1} -separability. The Wedderburn principal theorem is proved for Malcev algebras.

Binary Lie algebras of characteristic two

Binary Lie algebras of characteristic two