Related papers: Some remarks on evolution algebras corresponding t…
In this paper, we give a purely cohomological interpretation of the extension problem for associative algebras; that is the problem of extending an associative algebra by another associative algebra. We then give a similar interpretation of…
We present new characterizations of the rings in which every element is the sum of two idempotents and a nilpotent that commute, and the rings in which every element is the sum of two tripotents and a nilpotent that commute. We prove that…
We determine the complete degeneration picture inside the variety of nilpotent associative algebras of dimension 3 over an algebraically closed field of characteristic not equal to 2. Comparing with the discussion in [Ivanova N.M. and…
The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the…
In this paper we give classification of two-dimensional real evolution algebras. For several chains of evolution algebras we study their classification dynamics.
A classification of (countable) direct limits of finite dimensional involution simple associative algebras over an algebraically closed field of arbitrary characteristic is obtained. This also classifies the corresponding dimension groups.…
The affine group scheme of automorphisms of an evolution algebra that is equal to its square, is shown to lie in an exact sequence, such that the other terms depend solely on the directed graph associated to the algebra. As a consequence,…
The paper is devoted to the investigation of finite dimensional commutative nilpotent (associative) algebras N over an arbitrary base field of characteristic zero. Due to the lack of a general structure theory for algebras of this type (as…
We give a classification of $5$- and $6$-dimensional complex one-generated nilpotent bicommutative algebras.
We consider evolution algebras and their related substructures: evolution ideals and evolution subalgebras. After exposing some of the concepts related to them in the literature, we explore the order structures that arise in the sets of…
We classify the $4$-dimensional nilpotent bicommutative algebras over $\mathbb C$ from both algebraic and geometric approaches.
We prove that the natural permutation representation of highly transitive finite groups cannot be realized as the full automorphism group of an idempotent, finite-dimensional evolution algebra acting on the set of lines spanned by its…
Associative algebras with involution over a field of zero characteristic are considered. It is proved that in this case for any finitely generated associative algebra with involution there exists a finite dimensional algebra with involution…
An algebraic definition of Gardner's deformations for completely integrable bi-Hamiltonian evolutionary systems is formulated. The proposed approach extends the class of deformable equations and yields new integrable evolutionary and…
I show that simple finite vertex algebras are commutative, and that the Lie conformal algebra structure underlying a reduced (i.e., without nilpotent elements) finite vertex algebra is nilpotent.
The problem of whether a metabolic idempotent of a central simple algebra with involution is contained in an invariant quaternion subalgebra is investigated. As an application, the similar problem is studied for skew-symmetric elements…
We consider the intersection $\mathfrak{M}(A)$ of all maximal ideals of an evolution algebra $A$ and study the structure of the quotient $A/\M(A)$. In a previous work, maximal ideals have been related to hereditary subsets of a graph…
We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie-Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain…
The leitmotiv of this paper is linking algebraic properties of an evolution algebra with combinatorial properties of the (possibly several) graphs that one can associate to the algebra. We link nondegeneracy, zero annihilator, absorption…
We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or…