Related papers: Counting finite-dimensional algebras over finite f…

Over fields of characteristic zero, we determine all absolutely irreducible Yetter-Drinfeld modules over groups that have prime dimension and yield a finite-dimensional Nichols algebra. To achieve our goal, we introduce orders of braided…

Let F_q be the finite field of q elements. Let H be a multiplicative subgroup of F_q^*. For a positive integer k and element b\in F_q, we give a sharp estimate for the number of k-element subsets of H which sum to b.

A very first step to develop non-commutative algebraic geometry is the arithmetic of polynomials in non-commuting variables over a commutative field, that is, the study of elements in free associative algebras. This investigation is…

We classify finite-dimensional Hopf algebras whose coradical is isomorphic to the algebra of functions on S_3. We describe a new infinite family of Hopf algebras of dimension 72.

In this paper we study the number of finite topologies on an $n$-element set subject to various restrictions.

We show that if a field k contains sufficiently many elements(for instance, if k is infinite), and K is an algebraically closed field containing k, then every linear algebraic k-group over K is k-isomorphic to Aut(A\otimes_kK), where A is a…

Let $\Lambda$ be an $n$-Auslander algebra with global dimension $n+1$. In this paper, we prove that $\Lambda$ is representation-finite if and only if the number of non-isomorphic indecomposable $\Lambda$-modules with projective dimension…

In this paper we describe all, up to isomorphism, left unital, right unital and unital algebra structures on two-dimensional vector space over any algebraically closed field and $\mathbb{R}$. We tabulate the algebras with the units.

To a regular projection of a knot we associate a finite dimensional non-commutative associative algebra which is self-injective and special biserial.

Let $k$ be a field of characteristic not two or three. We classify up to isomorphism all finite-dimensional Lie superalgebras $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ over $k$, where $\mathfrak{g}_0$ is a three-dimensional simple…

We implement methods from the geometry of numbers to give explicit estimates for the number of integral ideals in a number field. We pay particular attention to minimising the effect of the degree $n$ of the number field on the error term…

In this paper we study the isomorphism problem for the universal enveloping algebras of nilpotent Lie algebras. We prove that if the characteristic of the underlying field is not~2 or~3, then the isomorphism type of a nilpotent Lie algebra…

This paper introduces two new algorithms for Lie algebras over finite fields and applies them to the investigate the known simple Lie algebras of dimension at most $20$ over the field $\mathbb{F}_2$ with two elements. The first algorithm is…

The irreducible alternative superbimodules are studied. The complete classification is obtained for even bimodules of arbitrary dimension and for finite-dimensional irreducible superbimodules over an algebraically closed field.

This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups…

We describe the structure of finite dimensional selfinjective algebras over an arbitrary field without short cycles of indecomposable modules.

Nilpotent Leibniz algebras with isomorphic maximal subalgebras are considered. The algebras are classified for coclass zero, one, and two. The results are field dependent.

We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry…

In the paper I considered methods for solving equations of the form axb+cxd=e in the algebra which is finite dimensional over the field.

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…