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The article is devoted to some ``strange'' phenomena of representation theory and their interrelations. Cross-projective representations of pairs of anticommutative algebras, alloys, their universal envelopping Lie algebras and their…
The purpose of this paper is to investigate ternary multiplications constructed from a binary multiplication, linear twisting maps and a trace function. We provide a construction of ternary Hom-Nambu and Hom-Nambu-Lie algebras starting from…
We offer some partition functions related to ternary quadratic forms, and note on their upper bounds and related properties. We offer these results as an application of a simple method related to conjugate Bailey pairs presented in a prior…
This manuscript introduces $J_3$-numbers, a seemingly missing three-dimensional intermediate between complex numbers related to points in the Cartesian coordinate plane and Hamilton's quaternions in the 4D space. The current development is…
To a regular projection of a knot we associate a finite dimensional non-commutative associative algebra which is self-injective and special biserial.
We classify the $4$-dimensional nilpotent bicommutative algebras over $\mathbb C$ from both algebraic and geometric approaches.
The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of…
The positive part $U^+_q$ of $U_q(\hat{\mathfrak{sl}}_2)$ has a presentation with two generators $W_0$, $W_1$ and two relations called the $q$-Serre relations. The algebra $U^+_q$ contains some elements, said to be alternating. There are…
The characterization of commutators in associative algebras is a classical problem in ring theory. In this paper, we address this problem for the natural class of generalized block-triangular algebras. To this end, we introduce a new…
We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized…
We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized…
This paper is a sequel to [3]. We formulate a natural algebraic geometry conjecture, give some of its number theoretic and analytical consequences, and show that those can be used to get further advances in wave turbulence theory.
A Lie algebra is said to be quadratic if it admits a symmetric invariant and non-degenerated bilinear form. Semisimple algebras with the Killing form are examples of these algebras, while orthogonal subspaces provide abelian quadatric…
For $\mathfrak{g}$ a simple Lie algebra and $G$ its adjoint group, the Chevalley map and work of Coxeter gives a concrete description of the algebra of $G$-invariant polynomials on $\mathfrak{g}$ in terms of traces over various…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…
A formal description of quaternions by means of exterior calculus is presented. Considering a three-dimensional space-time characterized by three time-like coordinates, we have been able to consistently recover a suitable formulation of…
A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different…
The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general…
In this paper, we develop the theory of symmetric triads with multiplicities. First, we classify abstract symmetric triads with multiplicities. Second, we determine the symmetric triads with multiplicities corresponding to commutative…
Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among…