Related papers: Triality, Structurable and Pre-structurable Algebr…
The normal symmetric triality algebras (STA's) and the normal Lie related triple algebras (LRTA's) have been recently introduced by the second author, in connection with the principle of triality. It turns out that the unital normal LRTA's…
We give a review of recent works for non-associative algebras, especially Lie algebras satisfying the triality relation. They are also intimately related to S_4 (symmetric group of 4-objects) symmetry of the Lie algebras.
This paper is devoted to the classification and studying properties of complex unital $3$-dimensional structurable algebras. We provide a complete list of non-isomorphic classes, identifying five algebras for type $(2, 1)$ and two algebras…
By analogy with the definition of group with triality we introduce Lie algebra with triality as Lie algebra L wich admits the group of automorphisms S_3={s,r | s^2=r^3=1, srs=r^2} such that for any x\in L we have…
We study and describe possibilities for arities of elementary theories and of their expansions. Links for arities with respect to Boolean algebras, to disjoint unions and to compositions of structures are shown. The dynamics for arities of…
In this paper we study the categories of braided categorical associative algebras and braided crossed modules of associative algebras and we relate these structures with the categories of braided categorical Lie algebras and braided crossed…
We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.
In this paper, we introduce the notion of compatible anti-pre-Lie algebras and study relationship between them and the related structures such as anti-$\mathcal{O}$-operators, commutative $2$-cocycles on compatible Lie algebras. Moreover,…
We develop a theory of toroidal vertex algebras and their modules, and we give a conceptual construction of toroidal vertex algebras and their modules. As an application, we associate toroidal vertex algebras and their modules to toroidal…
The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and induced $3$-Lie algebras.
A Lie superalgebra is attached to any finite-dimensional J-ternary algebra over an algebraically closed field of characteristic 3, using a process of semisimplification via tensor categories. Some of the exceptional simple Lie algebras,…
Using the notion of existentially closed structures, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of some groups and Lie algebras with prescribed properties.
The aim of this paper is to introduce the notion of (noncommutative) transposed Poisson conformal algebras, which serve as the conformal analogues of transposed Poisson algebras and admit a rich class of identities. We show that the tensor…
Some cohomology classes associated with an ideal in a Lie algebra, a Poisson structure on the basic functions algebra of contact structure, its Poisson cohomology and geometric (pre)quantization are considered from the algebraic point of…
Trialitarian automorphisms are related to automorphisms of order 3 of the Dynkin diagram of type D4. Octic etale algebras with trivial discriminant, containing quartic subalgebras, are classified by Galois cohomology with value in the Weyl…
We study post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$, motivated by nil-affine actions of Lie groups. We prove existence results for such structures depending on the interplay of the algebraic…
This short paper presents a generalisation of Tressl's structure theorem for differentially finitely generated algebras over differential rings of characteristic 0 to the case of separable algebras over differential rings of arbitrary…
A converse to Lie's theorem for Leibniz algebras is found and generalized. The result is used to find cases in which the generalized property, called triangulable, is 2-recognizeable; that is, if all 2-generated subalgebras are…
The PostLie algebra is an enriched structure of the Lie algebra that has recently arisen from operadic study. It is closely related to pre-Lie algebra, Rota-Baxter algebra, dendriform trialgebra, modified classical Yang-Baxter equations and…
We characterize finite-dimensional Lie algebras over an arbitrary field of characteristic zero which admit a non-trivial (quasi-) triangular Lie bialgebra structure.