Narrow Results

This paper is devoted to the study of local and 2-local derivations of null-filiform, filiform and naturally graded quasi-filiform associative algebras. We prove that these algebras as a rule admit local derivations which are not derivations. We show that filiform and naturally graded quasi-filiform associative algebras admit 2-local derivations which are not derivations and any 2-local derivation of null-filiform associative algebras is a derivation.

In this paper we shall find a matrix realization of the quantum group 𝔤_p,q introduced in [B. L. Feigin et al., Nucl. Phys. B757 (2006) 303–343]. For this purpose, we construct all primitive idempotents and a basis of 𝔤_p,q. We determine the action of elements of the basis on indecomposable projective modules, that gives rise to a matrix realization of 𝔤_p,q. By using this result, we obtain a basis of the space of symmetric linear functions on 𝔤_p1,p2 and express the symmetric linear functions obtained by the left integral, the balancing element, and the center of 𝔤_p,q in terms of this basis.

For any integer d≥2 and over any field , we construct a residually finite nonnilpotent nil algebra over generated by d elements such that any d-1 elements generate a nilpotent subalgebra. This leads to analogous results for nonassociative algebras and groups. In our proof we do not use the well-known Golod-Šafareviš lemma.

A graded associative algebra generated by m elements of degree one is called Golod–Shafarevich (GS) if it is presented with less than m²/4 relators of degree at least two. We explore conditions under which subalgebras of graded GS algebras are themselves GS. We prove that infinitely many Veronese powers of an algebra presented by m generators and r relators are GS if . For quadratic algebras, the bound is improved to . We also show that if A is a generic quadratic algebra presented by m generators and r relators, then all Veronese powers of A are GS if , and all but finitely many Veronese powers of A are not GS if .

We describe an effective algorithm to determine the maximal class-c quotient (or the maximal commutative class-c quotient) of a finitely presented associative algebra over an arbitrary field. As application, we investigate the relatively free d-generator algebras satisfying the identity xⁿ = 0. In particular, we consider the identity x³ = 0 and the cases (d, n) = (2, 4) and (d, n) = (2, 5).

We study the nilpotency degree of a relatively free finitely generated associative algebra with the identity xⁿ = 0 over a finite field 𝔽 with q elements. In the case of q ≥ n the nilpotency degree is proven to be the same as in the case of an infinite field of the same characteristic. In the case of q = n - 1 it is shown that the nilpotency degree differs from the nilpotency degree for an infinite field of the same characteristic by at most one. The nilpotency degree is explicitly computed for n = 3.

We consider the quantum analog of the Lie commutator ${[x,y]}_q=xy-qyx$ for an invertible element $q$ of the ground field and prove lower and upper bounds for the nilpotence degree of an associative algebra satisfying an identity of the form $[\dots [{[x_1,x_2]}_q,x_3{]}_q,\dots ,x_{s+1}{]}_q=0$.

In this paper, we study the moduli space of 1|1-dimensional complex associative algebras. We give a complete calculation of the cohomology of every element in the moduli space, as well as compute their versal deformations.

In this paper, we translate the problem of extending an associative algebra by another associative algebra into the language of codifferentials. The authors have been constructing moduli spaces of algebras and studying their structure by constructing their versal deformations. The codifferential language is very useful for this purpose. The goal of this paper is to express the classical results about extensions in a form which leads to a simpler construction of moduli spaces of low-dimensional algebras.

We introduce an associative algebra $A^{\infty }(V)$ using infinite matrices with entries in a grading-restricted vertex algebra $V$ such that the associated graded space $\mathrm{\text{Gr}}(W)={\coprod }_{n\in \mathbb{N}}{\mathrm{\text{Gr}}}_n(W)$ of a filtration of a lower-bounded generalized $V$-module $W$ is an $A^{\infty }(V)$-module satisfying additional properties (called a nondegenerate graded $A^{\infty }(V)$-module). We prove that a lower-bounded generalized $V$-module $W$ is irreducible or completely reducible if and only if the nondegenerate graded $A^{\infty }(V)$-module $\mathrm{\text{Gr}}(W)$ is irreducible or completely reducible, respectively. We also prove that the set of equivalence classes of the lower-bounded generalized $V$-modules is in bijection with the set of the equivalence classes of nondegenerate graded $A^{\infty }(V)$-modules. For $N\in \mathbb{N}$, there is a subalgebra $A^N(V)$ of $A^{\infty }(V)$ such that the subspace ${\mathrm{\text{Gr}}}^N(W)={\coprod }_{n=0}^N{\mathrm{\text{Gr}}}_n(W)$ of $\mathrm{\text{Gr}}(W)$ is an $A^N(V)$-module satisfying additional properties (called a nondegenerate graded $A^N(V)$-module). We prove that $A^N(V)$ are finite-dimensional when $V$ is of positive energy (CFT type) and $C_2$-cofinite. We prove that the set of the equivalence classes of lower-bounded generalized $V$-modules is in bijection with the set of the equivalence classes of nondegenerate graded $A^N(V)$-modules. In the case that $V$ is a Möbius vertex algebra and the differences between the real parts of the lowest weights of the irreducible lower-bounded generalized $V$-modules are less than or equal to $N\in \mathbb{N}$, we prove that a lower-bounded generalized $V$-module $W$ of finite length is irreducible or completely reducible if and only if the nondegenerate graded $A^N(V)$-module ${\mathrm{\text{Gr}}}^N(W)$ is irreducible or completely reducible, respectively.

We shall define the associative algebra structure closest to a given algebra structure. When we face a new multiplication, being caused by noise and so on, it must be useful to compute with the closest associative multiplication to such a perturbed one. In this paper we shall give a procedure to find the closest associative structure and demonstrate our strategy for the 2-dimensional algebras over the field of real numbers. Then we trace the points of the sets of structure constants of the deformed algebras.

Let $A$ be a finitely generated associative algebra over a field $F$ of characteristic $\ne 2$ and let $A^{(-)}=(A,[a,b]=ab-ba)$ be its associated Lie algebra. In this paper, we investigate relations between the growth functions of $A$ and the Lie algebra $[A,A]$. We prove that if A is generated by a finite collection of nilpotent elements, then the growth functions are asymptotically equivalent.

The analysis of an open quantum system can be by far difficult if the dimension of the system Hilbert space is large or infinite. However, in some cases the dynamics on a finite-dimensional Hilbert space can be decomposed into a block-diagonal form, which simplifies the system structure. In this presentation, we will study several criteria for the complete reducibility and, in addition, a computational method for a basis of each simplified component to apply for the analysis of open quantum systems. An important point of these tools is that they are “effective” methods (one can complete the task in a finite number of steps).

We provide a complete classification of three-dimensional associative algebras over the real and complex number fields based on a completely elementary proof. We list up all the multiplication tables of the algebras up to isomorphism. We compare our results with those given by mathematicians in the 19th century and this century.

In this paper, we study the geography of the algebraic set . We have three subjects. The first is the point on closest to a given point. This point represents the closest associative algebra structure to any given algebra structure. The second is a path on from a point to another point, which represents an algebraic deformation of an associative algebra. The third is the orbits. We discuss how the 7 orbits of pair-wise non-isomorphic associative algebras over ℝ of 2-dimension cover our algebraic set . First two of them are based on our papers [5] and [6], and the last is our further work.