Related papers: Malcev dialgebras
We describe degenerations of four-dimensional binary Lie algebras, and five- and six-dimensional nilpotent Malcev algebras over \mathbb{C}. In particular, we describe all irreducible components of these varieties.
We address the question of the dualizability of nilpotent Mal'cev algebras, showing that nilpotent finite Mal'cev algebras with a non-abelian supernilpotent congruence are inherently non-dualizable. In particular, finite nilpotent…
Comtrans algebras, arising in web geometry, have two trilinear operations, commutator and translator. We determine a Gr\"obner basis for the comtrans operad, and state a conjecture on its dimension formula. We study multilinear polynomial…
Given a positive integer d, the Kaplansky-Lvov conjecture states that the set of values of a multilinear noncommutative polynomial f on the matrix algebra M_d(C) is a vector subspace. In this article the technique of using one-wiggle…
The paper is devoted to classification problem of finite dimensional complex none Lie filiform Leibniz algebras. The motivation to write this paper is an unpublished yet result of J.R.Gomez, B.A.Omirov on necessary and sufficient conditions…
The indecomposable solvable Lie algebras with graded nilradical of maximal nilindex and a Heisenberg subalgebra of codimension one are analyzed, and their generalized Casimir invariants calculated. It is shown that rank one solvable…
Leibniz algebras are certain generalization of Lie algebras. In this paper we give the classification of four dimensional non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of bilinear…
For each 3-dimensional non-Lie Leibniz algebra over the complex numbers, we describe the algebra of polynomial invariants and determine its group of automorphisms. As a consequence, we establish that any two non-nilpotent 3-dimensional…
We first discuss the construction by Perez-Izquierdo and Shestakov of universal nonassociative enveloping algebras of Malcev algebras. We then describe recent results on explicit structure constants for the universal enveloping algebras…
Moens proved that a finite-dimensional Lie algebra over field of characteristic zero is nilpotent if and only if it has an invertible Leibniz-derivation. In this article we prove the analogous results for finite-dimensional Malcev, Jordan,…
For finitely generated nilpotent groups, we employ Mal'cev coordinates to solve several classical algorithmic problems efficiently. Computation of normal forms, the membership problem, the conjugacy problem, and computation of presentations…
An algebraic algorithm is developed for computation of invariants ('generalized Casimir operators') of general Lie algebras over the real or complex number field. Its main tools are the Cartan's method of moving frames and the knowledge of…
I present here some new results which make explicit the role of the division algebras R,C,H,O in the construction and classification of, respectively, N=1,2,4,8 global supersymmetric quantum mechanical and classical dynamical systems. In…
The commutator $[a,b] = ab - ba$ in a free Zinbiel algebra (dual Leibniz algebra) is an anticommutative operation which satisfies no new relations in arity 3. Dzhumadildaev discovered a relation $T(a,b,c,d)$ which he called the tortkara…
In this paper we define the basic concepts for left or right Leibniz algebras and prove some of the main results. Our proofs are often variations of the known proofs and several results seem to be new.
In this paper we consider images of (ordinary) noncommutative polynomials on matrix algebras endowed with a graded structure. We give necessary and sufficient conditions to verify that some multilinear polynomial is a central polynomial, or…
We use computer algebra to determine all the multilinear polynomial identities of degree $\le 7$ satisfied by the trilinear operations $(a \cdot b) \cdot c$ and $a \cdot (b \cdot c)$ in the free dendriform dialgebra, where $a \cdot b$ is…
We study the universal enveloping algebras of the one-parameter family of solvable 5-dimensional non-Lie Malcev algebras. We explicitly determine the universal nonassociative enveloping algebras (in the sense of Perez-Izquierdo and…
A new purely algebraic algorithm is presented for computation of invariants (generalized Casimir operators) of Lie algebras. It uses the Cartan's method of moving frames and the knowledge of the group of inner automorphisms of each Lie…
The alternating ternary sum in an associative algebra, $abc - acb - bac + bca + cab - cba$, gives rise to the partially alternating ternary sum in an associative dialgebra with products $\dashv$ and $\vdash$ by making the argument $a$ the…