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We show that for algebraic groups over local fields of characteristic zero, the following are equivalent: Every homomorphism has a closed image, every unitary representation decomposes into a direct sum of finite-dimensional and mixing…

We investigate homological properties of perfect algebras of prime characteristic. The principle is as follows: perfect algebras resolve the singularities. For example, we show any module over the ring of absolute integral closure has…

Let $A$ be a finite-dimensional algebra with two simple modules. It is shown that if the derived category of $A$ admits a stratification with simple factors being the base field $k$, then $A$ is derived equivalent to a quasi-hereditary…

Any nonempty, compact, semi-algebraic set in [0, 1] n is the projection of the set of mixed equilibria of a finite game with 2 actions per player on its first n coordinates. A similar result follows for sets of equilibrium payoffs. The…

Let $B\rightarrow A$ be a homomorphism of Hopf algebras and let $C$ be an algebra. We consider the induction from $B$ to $A$ of $C$ in two cases: when $C$ is a $B$-interior algebra and when $C$ is a $B$-module algebra. Our main results…

We give a Clifford correspondence for an algebra A over an algebraically closed field, that is an algorithm for constructing some finite-dimensional simple A-modules from simple modules for a subalgebra and endomorphism algebras. This…

The isomorphism problem means to decide if two given finite-dimensional simple algebras over the same centre are isomorphic and, if so, to construct an isomorphism between them. A solution to this problem has applications in computational…

Let $A$ be an Artin algebra, $M$ be a Gorenstein projective $A$-module and $B =$ End$_A M$, then $M$ is a $A$-$B$-bimodule. We use the restricted flat dimension of $M_B$ to give a characterization of the homological dimensions of $A$ and…

Let $p$ be a prime number. We define the notion of $F$-finiteness of homomorphisms of $\mathbb F_p$-algebras, and discuss some basic properties. In particular, we prove a sort of descent theorem on $F$-finiteness of homomorphisms of…

We consider pairs of operators $A,B\in B(H)$, where $H$ is a Hilbert space, such that there exist a linear isometry $f$ from the span of $\{A,B\}$ into $\mathbb{C}^2$ mapping $A,B$ into orthonormal vectors. We prove some necessary…

We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some…

A semi-algebraic set is a subset of $\mathbb{R}^n$ defined by a finite collection of polynomial equations and inequalities. In this paper, we investigate the problem of determining whether two points in such a set belong to the same…

Semi-algebraic set is a subset of the real space defined by polynomial equations and inequalities. In this paper, we consider the problem of deciding whether two given points in a semi-algebraic set are connected. We restrict to the case…

In this paper, we describe an elementary method for counting the number of non-isomorphic algebras of a fixed dimension over a given finite field. We show how this method works for the explicit example of $2$-dimensional algebras over the…

Let $\ell$ be a commutative ring with unit. To every pair of $\ell$-algebras $A$ and $B$ one can associate a simplicial set $\hom(A,B^\Delta)$ so that $\pi_0\hom(A,B^\Delta)$ equals the set of polynomial homotopy classes of morphisms from…

Let $\mathcal A$ and $\mathcal B$ be two (complex) algebras. A linear map $\phi:{\mathcal A}\to{\mathcal B}$ is called $n$-homomorphism if $\phi(a_{1}... a_{n})=\phi(a_{1})...\phi(a_{n})$ for each $a_{1},...,a_{n}\in{\mathcal A}.$ In this…

Let $R$ be a finite-dimensional algebra over an algebraically closed field $F$ graded by an arbitrary group $G$. We prove that $R$ is a graded division algebra if and only if it is isomorphic to a twisted group algebra of some finite…

We provide a clarification of the classification of two-dimensional algebras over an arbitrary base field. Using this clarification, we determine the number of non-isomorphic two-dimensional algebras over a finite field.

Let $\F$ be an algebraically closed field. Let $\V$ be a vector space equipped with a non-degenerate symmetric or symplectic bilinear form $B$ over $\F$. Suppose the characteristic of $\F$ is \emph{large}, i.e. either zero or greater than…

Let $\mathrm{R}$ be a real closed field and $\mathrm{C}$ the algebraic closure of $\mathrm{R}$. We give an algorithm for computing a semi-algebraic basis for the first homology group, $\mathrm{H}_1(S,\mathbb{F})$, with coefficients in a…