Related papers: Semi-Associative $3$-Algebras
The Frank Lie algebras are simple Lie algebras that only occur over fields of characteristic 3. These come equipped with distinguished inner derivations that make them algebras in the category $\textbf{Rep}(\alpha_3)$. We apply the…
In this paper, we present some basic properties concerning the quasi-derivation algebra $QDer(\mathcal{A})$ and the quasi-centroid algebra $QC(\mathcal{A})$ of associative algebra $\mathcal{A}$. Furthermore, using the result on…
As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators…
We prove a triangulation theorem for semi-algebraic sets over a p-adically closed field, quite similar to its real counterpart. We derive from it several applications like the existence of flexible retractions and splitting for…
There is a notion of non-commutative Lie algebra called "Leibniz algebra", which is characterized by the condition: left bracketing is a derivation. The purpose of this article is to introduce and study a new notion of algebra, called…
We introduce a notion of pre-alternative algebra which may be seen as an alternative algebra whose product can be decomposed into two pieces which are compatible in a certain way. It is also the "alternative" analogue of a dendriform…
The notion of associativity (which differs from the straightforward generalization of the usual associativity given by the move of parentheses in the relevant expression) for operations of high arity is introduced. It is proved that the…
The Abelian semigroup expansion is a powerful and simple method to derive new Lie algebras from a given one. Recently it was shown that the $S$-expansion of $\mathfrak{so}\left( 3,2\right) $ leads us to the Maxwell algebra $\mathcal{M}$. In…
We generalize some of the standard homological techniques to $\cW$-algebras, and compute the semi-infinite cohomology of the $\cW_3$ algebra on a variety of modules. These computations provide physical states in $\cW_3$ gravity coupled to…
Every metric symplectic Lie algebra has the structure of a quadratic extension. We give a standard model and describe the equivalence classes on the level of corresponding quadratic cohomology sets. Finally, we give a scheme to classify the…
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…
On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear forms satisfying the usual cocycle equation. We note their relationship with antiderivations and compute them for some classes of Lie algebras, including…
We study moduli spaces of mirror non-compact Calabi-Yau threefolds enhanced with choices of differential forms. The differential forms are elements of the middle dimensional cohomology whose variation is described by a variation of mixed…
We prolonge the list of C*-algebras for which all extensions by any stable separable C*-algebra are semi-invertible. In particular, we handle certain amalgamations, both of C*-algebras and of groups. Concerning groups we consider both…
In this paper, we give a study of the $\mathbb{C}[\partial]$-split extending structures problem for associative conformal algebras. Using the unified product as a tool, which includes interesting products such as bicrossed product, cocycle…
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,…
A self-dual algebras is one isomorphic as a module to the opposite of its dual; a quasi self-dual algebra is one whose cohomology with coefficients in itself is isomorphic to that with coefficients in the opposite of its dual. For these…
It is shown that the non-trivial cocycles on simple Lie algebras may be used to introduce antisymmetric multibrackets which lead to higher-order Lie algebras, the definition of which is given. Their generalised Jacobi identities turn out to…
A nonassociative generalization of supersymmetry is studied, where supersymmetry generators are considered to be the nonassociative ones. Associators for the product of three and four multipliers are defined. Using a special choice of the…
The $n$-Lie bialgebras are studied. In Section 2, the $n$-Lie coalgebra with rank $r$ is defined, and the structure of it is discussed. In Section 3, the $n$-Lie bialgebra is introduced. A triple $(L, \mu, \Delta)$ is an $n$-Lie bialgebra…