Let $ F $ be a finite field and consider $ UT_n $ the algebra of $ n\times n $ upper triangular matrices over $ F $. In [1], it was proved that every $ G $-grading is elementary. In [2], the authors classified all nonisomorphic elementary $ G $-gradings. They also described the set of all $G $-graded polynomial identities for $ UTn $ when $ F $ is an infinite field. In [3], was described the all $ G $-graded polynomial identities for $ UT_n $ when $ F $ is a finite field. In this work, we will discuss the case when $G = \mathbb{Z}_2$, $ n=2, 3 $ and $ F $ is a finite field.
In this article we study homotopes of finite-dimensional algebras (not necessarily, associative). In the case of associative algebras we study homotopes by methods of Category theory and give description of so-called well-tempered elements of finite-dimensional associative algebra in algebraic terms.
In this paper, we introduce the class of finitely semi-graded algebras which extends the connected graded algebras finitely generated in degree one. The Koszul behavior of finitely semi-graded algebras is investigated by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series and the Yoneda algebra are defined for this class of algebras. Finitely semi-graded algebras include many important examples of non $\mathbb{N}$-graded algebras finitely generated in degree one coming from mathematical physics, and for these concrete examples the Koszulity will be …
In 1999, Y. Kashina introduced the exponent of a Hopf algebra. In this paper, we prove that the exponent of a finite dimensional non-cosemisimple Hopf algebra with Chevalley property in characteristic 0 is infinite, and the exponent of a finite dimensional non-cosemisimple pointed Hopf algebra in positive characteristic is finite.
In this paper we show that evolution algebras over any given field $\Bbbk$ are universally finite. In other words, given any finite group $G$, there exist infinitely many regular evolution algebras $X$ such that $Aut(X)\cong G$. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme $\mathbf{G}$ over $\Bbbk$ is isomorphic to the algebraic affine group scheme of automorphisms of a re…
We investigate criteria for von-Neumann finiteness and reversibility in some classes of non-associative algebras. We show that all finite-dimensional alternative algebras, as well as all algebras obtained from the real numbers via the standard Cayley-Dickson doubling process, are von-Neumann finite. Precise criteria for von-Neumann finiteness and reversibility of involutive algebras are given, in terms of isomorphism types of their 3-dimensional subalgebras.
Classification of finite irreducible conformal modules over some Lie conformal algebras related to the Virasoro conformal algebra
本文对一类李共形代数$\mathcal{W}(b)$上的有限不可约共形模进行了分类，其中$b\in\mathbb{C}$与Virasoro共形代数有关.证明了$\mathcal{W}(b)$上的有限不可约共形模同构于$M_{Δ,α,β}$ with $Δ\neq 0$或$β\neq 0$ if $b=0$，或$M_{Δ,α}$ with $Δ\neq 0$ if $b\neq0$ .作为副产品，我们对Heisenberg-Virasoro共形代数和$\mathcal{W}(2,2)$型李共形代数上的有限不可约共形模进行了分类.最后，对薛定谔-维拉索罗共形代数也做了同样的事情。
It is well known that central simple algebras are split by suitable finite Galois extensions of their centers. A counterpart of this result was studied by Juan and Magid in the set up of differential matrix algebras, wherein Picard-Vessiot extensions that split matrix differential algebras were constructed. In this article, we exhibit instances of differential matrix algebras which are split by finite extensions. In some cases, we relate the existence of finite splitting extensions of a differential matrix algebra to the triviality of its tensor powers, and show in these cases, that orders of …
Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple A-modules have dimension at least 6. This is a revised version.
We discuss finiteness/infiniteness of $τ$-tilting modules over tensor products of two symmetric algebras. As an application, we discuss that over block algebras of direct products of finite groups.
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 field $\mathbb{F}_{2}$.
First, we give a new example of silting-discrete algebras. Second, one explores when the algebra of triangular matrices over a finite dimensional algebra is $τ$-tilting finite. In particular, we classify algebras over which triangular matrix algebras are $τ$-tilting finite. Finally, we investigate when a triangular matrix algebra is silting-discrete.
The objects of study in this paper are Hopf algebras $H$ which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra by a finite dimensional Hopf algebra. Basic structural and homological properties are recalled and classes of examples are listed. Bounds are obtained on the dimensions of simple $H$-modules, and the structure of $H$ is shown to be severely constrained when the finite dimensional extension is semisimple and cosemisimple.
