Related papers: `Mixed' Jordan-Lie Superalgebra

We classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras $Q(n)$, $n \geq 2$, over an algebraically closed field of characteristic different from $2$ (and not dividing $n+1$ in the Lie case): fine…

We classify two classes of B_2-graded Lie algebras which have a second compatible grading by an abelian group A: (a) graded-simple Lie algebras for A torsion-free and (b) division-A-graded Lie algebras. Our results describe the centreless…

We compare a number of different definitions of structure algebras and TKK constructions for Jordan (super)algebras appearing in the literature. We demonstrate that, for unital superalgebras, all the definitions of the structure algebra and…

Graded bundles are a particularly nice class of graded manifolds and represent a natural generalisation of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids we define the notion of a weighted…

We extend the concepts of the associator and commutator from algebras with a binary multiplication law to algebras with a ternary multiplication law using cube roots of unity. By analogy with the Jacobi identity for the binary commutator,…

Each choice of a K\"ahler class on a compact complex manifold defines an action of the Lie algebra $\slt$ on its total complex cohomology. If a nonempty set of such K\"ahler classes is given, then we prove that the corresponding…

Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear…

Using elementary linear algebra, this paper clarifies and proves some concepts about a recently introduced octonion-like associative division algebra over R. This octonion-like algebra is actually the same as the split-biquaternion algebra,…

A notion of n-Lie algebra introduced by V.T. Filippov can be viewed as a generalization of a concept of binary Lie algebra to the algebras with n-ary multiplication law. A notion of Lie algebra can be extended to Z_2-graded structures…

A three-dimensional polynomial algebra of order $m$ is defined by the commutation relations $[P_0, P_\pm]$ $=$ $\pm P_\pm$, $[P_+, P_-]$ $=$ $\phi^{(m)}(P_0)$ where $\phi^{(m)}(P_0)$ is an $m$-th order polynomial in $P_0$ with the…

This paper concerns an associative graded algebra A that is the enveloping algebra of a Lie algebra with exponential growth. The algebra A is 3-Calabi-Yau. There is a Z-form of A so for every field k there is an algebra A_k. An algebraic…

Nonassociative commutative algebras $A$ generated by idempotents $e$ whose adjoint operators ${\rm ad}_e\colon A \rightarrow A$, given by $x \mapsto xe$, are diagonalizable and have few eigenvalues are of recent interest. When certain…

Let $M_n$ be the algebra of $n \times n$ complex matrices. We consider arbitrary subalgebras $\mathcal{A}$ of $M_n$ which contain the algebra of all upper-triangular matrices (i.e.\ block upper-triangular subalgebras), and their Jordan…

This paper is based on a talk given at the 14-th International Workshop on Differential Geometry and Its Applications, hosted by the Petroleum Gas University from Ploiesti, between July 9-th and July 11-th, 2019. After presenting some…

There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics. From a purely mathematical point of view, the point of view of Jordan algebra theory might give new strength to such…

We consider $\G$-graded commutative algebras, where $\G$ is an abelian group. Starting from a remarkable example of the classical algebra of quaternions and, more generally, an arbitrary Clifford algebra, we develop a general viewpoint on…

A square matrix $A$ has the usual Jordan canonical form that describes the structure of $A$ via eigenvalues and the corresponding Jordan blocks. If $A$ is a linear relation in a finite-dimensional linear space ${\mathfrak H}$ (i.e., $A$ is…

A new kind of graded Lie algebra (we call it $Z_{2,2}$ graded Lie algebra) is introduced as a framework for formulating parasupersymmetric theories. By choosing suitable bose subspace of the $Z_{2,2}$ graded Lie algebra and using relevant…

A unified treatment of both superconformal and quasisuperconformal algebras with quadratic non-linearity is given. General formulas describing their structure are found by solving the Jacobi identities. A complete classification of…

Non-associtive algebras is a research direction gaining much attention these days. New developments show that associative algebras and some not-associative structures can be unified at the level of Yang-Baxter structures. In this paper, we…