Related papers: On small matrix subalgebras with a trivial central…

Let K be a field of characteristic different from 2 and let V be a vector space of dimension n over K. Let M be a non-zero subspace of symmetric bilinear forms defined on V x V and let r=rank(M) denote the set of different positive integers…

The maximal dimension of a commutative subalgebra of the Grassmann algebra is determined. It is shown that for any commutative subalgebra there exists a commutative subalgebra which is spanned by monomials and has the same dimension. It…

For monomial special multiserial algebras, which in general are of wild representation type, we construct radical embeddings into algebras of finite representation type. As a consequence, we show that the representation dimension of…

We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and…

Let $k$ be an algebraically closed field. The generalized or $n$-Kronecker quiver $K(n)$ is the quiver with two vertices, called a source and a sink, and $n$ arrows from source to sink. Given a finite-dimensional module $M$ of the path…

A subfactor is an inclusion $N \subset M$ of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action $M^G \subset M$, and subfactors can be thought of as fixed points of more general…

A classification of semisimple algebras of vector fields on C^N that have a Cartan subalgebra of dimension N is given. The proof uses basic representation theory and the local canonical form of semisimple Lie algebras of vector fields.

Let $1\leq m \leq n$ be integers with $m|n$ and $\cat{Alg}_{n,m}$ the class of central simple algebras of degree $n$ and exponent dividing $m$. In this paper, we find new, improved upper bounds for the essential dimension and 2-dimension of…

We describe all central extensions of all $3$-dimensional non-zero complex Zinbiel algebras. As a corollary, we have a full classification of $4$-dimensional non-trivial complex Zinbiel algebras and a full classification of $5$-dimensional…

We consider simple superalgebras which are a supersymmetric extension of $\fspin(s,t)$ in the cases where the number of odd generators does not exceed 64. All of them contain a super Poincar\'e algebra as a contraction and another as a…

We complete the solution of the problem of finding the universal central extension of the matrix superalgebras $\mathfrak{sl}(m, n, A)$ where $A$ is an associative superalgebra and computing $H_2\big(\mathfrak{sl}(m, n, A)\big)$. The…

Let $K$ be a field of characteristic zero and let $\mathfrak{M}_5$ be the variety of associative algebras over $K$, defined by the identity $[x_1,x_2][x_3,x_4,x_5]$. It is well-known that such variety is a minimal variety and that is…

A question of interest in Linear Algebra is whether all n x n complex matrices can be unitarily tridiagonalised. The answer for all n not equal to 4 (affirmative or negative) has been known for a while, whereas the case n=4 seems to have…

Supersymmetry is deeply related to division algebras. Nonabelian Yang-Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green-Schwarz…

We compare, for L a Lie ring over the integers, its lower central series (\gamma_n(L))_{n>0} and its dimension series defined by \delta_n(L):=L\cap \varpi^n(L) in the universal enveloping algebra of L. We show that \gamma_n(L)=\delta_n(L)…

All matrices we consider have entries in a fixed algebraically closed field $K$. A minor of a square matrix is principal means it is defined by the same row and column indices. We study the ideal generated by size $t$ principal minors of a…

Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that a left Leibniz algebra, all of whose maximal subalgebras are right ideals, is nilpotent. A…

The centralizer of an endomorphism of a finite dimensional vector space is known when the endomorphism is nonderogatory or when its minimal polynomial splits over the field. It is also known for the real Jordan canonical form. In this paper…

For a given pair of numbers $(d,k)$, we establish the minimal number of vertices in pure $d$-dimensional simplicial complexes with non-trivial homology in dimension $k$. Furthermore, we solve the problem under the additional constraint of…

We complete the problem of finding the universal central extension in the category of Leibniz superalgebras of $\mathfrak{sl}(m, n, D)$ when $m+n \geq 3$ and $D$ is a superdialgebra, solving in particular the problem when $D$ is an…