Related papers: Malcev dialgebras
We implement two algorithms in MATHEMATICA for classifying automorphisms of lower-dimensional non-commutative Lie algebras. The first algorithm is a brute-force approach whereas the second is an evolutionary strategy. These algorithms are…
We introduce degree n Sabinin algebras, which are defined by the polynomial identities up to degree n in a Sabinin algebra. Degree 4 Sabinin algebras can be characterized by the polynomial identities satisfied by the commutator, associator…
Perez-Izquierdo and Shestakov recently extended the PBW theorem to Malcev algebras. It follows from their construction that for any Malcev algebra $M$ over a field of characteristic $\ne 2, 3$ there is a representation of the universal…
Let $k$ be an arbitrary field and $d$ a positive integer. For each degenerate symmetric or antisymmetric bilinear form $M$ on $k^{d}$ we determine the structure of the Lie algebra of matrices that preserve $M$, and of the Lie algebra of…
Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities $a(bc) = a(cb)$, $(ba)a^2 = (ba^2)a$, and $(b,a^2,c) = 2(b,a,c)a$. These identities are satisfied by the product $ab = a \dashv b + b…
This work completes the study of the solvable Leibniz algebras, more precisely, it completes the classification of the $3$-filiform Leibniz algebras of maximum length \cite{3-filiform}. Moreover, due to the good structure of the algebras of…
New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the…
We give a relational description of higher commutator operators, which were introduced by Bulatov, in varieties with a Mal'cev term. Furthermore, we use this result to prove that for every algebra with a Mal'cev term there exists a largest…
We study real and complex Manin triples for a complex reductive Lie algebra, $\g$. The first part includes, and extends to complex Manin triples, our earlier work [De]. First, we generalize results of E. Karolinsky, on the classification of…
It is known that any finite idempotent algebra that satisfies a nontrivial Maltsev condition must satisfy the linear one-equality Maltsev condition (a variant of the term discovered by M. Siggers and refined by K. Kearnes, P. Markovi\'c,…
We investigate the rectifiable spaces, the Mal'cev algebras, the almost quasivarieties of topological algebraic systems and their free systems and others. It specifies and corrects the roughest mistakes, incorrect statements and nonsense of…
In the note, all indecomposable canonical forms of linear systems with dimension less than or equal to $4$ are determined based on Belitskii's algorithm. As an application, an effective way to calculate dimensions of equivalence classes of…
We determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field $\mathbb K$ of characteristic zero having an analogous family of flags of subalgebras as the four-dimensional non-Lie binary Lie…
We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems respectively with a third and a fourth order ladder operators satisfying…
In two seminal papers M. Kontsevich introduced graph homology as a tool to compute the homology of three infinite dimensional Lie algebras, associated to the three operads `commutative,' `associative' and `Lie.' We generalize his theorem to…
Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by…
Convergence is a crucial issue in iterative algorithms. Damping is commonly employed to ensure the convergence of iterative algorithms. The conventional ways of damping are scalar-wise, and either heuristic or empirical. Recently, an…
We first give a sufficient condition for a Mal'cev-Neumann ring of formal series to be a noncrossed product division algebra. This result is then used to give an elementary proof of the existence of noncrossed product division algebras (of…
We classify all two-dimensional simple algebras (which may be non-associative) over an algebraically closed field. For each two-dimensional algebra $\mathcal{A}$, we describe a minimal (with respect to inclusion) generating set for the…
We present coalgebras of some classes of nonassociative algebras whose associator satisfies invariance conditions given by the action of the 3-order symmetric group. Amongst these algebras we find the well-known Vinberg algebras, the…