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The algebras for all possible Lorentzian and Euclidean kinematics with $\frak{so}(3)$ isotropy except static ones are re-classified. The geometries for algebras are presented by contraction approach. The relations among the geometries are…
We use elementary algebraic properties of left, right multiplication operators to prove some deep structural properties of left $m$-invertible, $m$-isometric, $m$-selfadjoint and other related classes of Banach space operators, often adding…
Let $A_n$ be an $n$-dimensional algebra with zero multiplication over a field $K$ of characteristic $0$. Then its universal (multiplicative) enveloping algebra $U_n$ in the variety of left-symmetric algebras is a homogeneous quadratic…
In the paper, I consider properties and mappings of free algebra with unit. I consider also conjugation of free algebra with unit.
Two-dimensional real hyporeductive triple algebras (h.t.a.) are investigated. A classification of such algebras is presented. As a consequence, a classification of two-dimensional real Lie triple algebras (i.e. generalized Lie triple…
The paper has a form of a talk on the given topic. It consists of three parts. The first part of the paper contains main notions, the second one is devoted to logical geometry, the third part describes types and isotypeness. The problems…
We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some…
The purpose of this paper is twofold. First, we review applications of the bar duality of operads to the construction of explicit cofibrant replacements in categories of algebras over an operad. In view toward applications, we check that…
The irreducible alternative superbimodules are studied. The complete classification is obtained for even bimodules of arbitrary dimension and for finite-dimensional irreducible superbimodules over an algebraically closed field.
The main aim of this paper is to classify the distinct multiplicative Lie algebra structures (up to isomorphism) on a given group. We also see that for a given group $G$, every homomorphism from the non-abelian exterior square $G \wedge G$…
We discuss certain ternary algebraic structures appearing more or less naturally in various domains of theoretical and mathematical physics. Far from being exhaustive, this article is intended above all to draw attention to these algebras,…
Let $L$ be a Lie algebra with its enveloping algebra $U(L)$ over a field. In this paper we survey results concerning the isomorphism problem for enveloping algebras: given another Lie algebra $H$ for which $U(L)$ and $U(H)$ are isomorphic…
We investigate the existence of left-invariant closed G$_2$-structures on seven-dimensional non-solvable Lie groups, providing the first examples of this type. When the Lie algebra has trivial Levi decomposition, we show that such a…
Over an algebraically closed field we classify all minimal representation-infinite algebras where the lattice of two-sided ideals is not distributive. As a consequence there are only finitely many isomorphism classes of minimal…
We study the problem of classifying local projective structures in dimension two having non trivial Lie symmetries. In particular we obtain a classification of flat projective structures having positive dimensional Lie algebra of projective…
A brief survey of real algebraic structures on topological spaces is given. This article is written for the Gokova Gemetry/Topology Conference proceedings.
This paper focuses on the derivations and automorphism groups of certain finite-dimensional associative algebras over the field of complex numbers. Using classification results for algebras of dimensions two, three, and four, along with…
We describe the graded isomorphisms of rings of endomorphisms of graded flags over graded division algebras. As a consequence describe the isomorphism classes of upper block triangular matrix algebras (over an algebraically closed field of…
We characterize vertex algebras (in a suitable sense) as algebras over a certain graded co-operad. We also discuss some examples and categorical implications of this characterization.
Structures of dual Lie bialgebras on the one sided Witt algebra, the Witt algebra and the Virasoro algebra are investigated. As a result, we obtain some infinite dimensional Lie algebras.