Ilya Zhdanovskiy
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.
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A homotope, or a mutation, of a $k$-algebra is a new algebra with the same underlying space, but with the multiplication law dependent on the multiplication law of the original algebra. In this paper, we show that a generic finite-dimensional algebra of dimension greater than 3 has infinitely many non-isotopic homotopes, and that, more generally, a similar result is true for generic trilinear forms. We also study a particular class of homotopes called $Δ$-homotopes, where $Δ$ is an element of the algebra, and show that there are algebras with infinitely many non-isomorphic homotopes even under…
Finiteness conditions on the Yoneda algebra of a monomial algebra
设A是连通分次非交换单项代数。我们把一个有限图Γ（A）与A联系起来，称之为A的CPS图.利用Γ（A）刻画了Yoneda代数Ext_A（k，k）的Noether性、有限GK维数和有限生成等性质.我们表明，这些属性，特别是有限代，可以检查通过终止算法。
We observe that a finitely generated algebraic algebra R (over a field) is finite dimensional if and only if the associated graded ring grR is right noetherian, if and only if grR has right Krull dimension, if and only if grR satisfies a polynomial identity.
Finitely generated invariants of Hopf algebras on free associative algebras
证明了有限秩自由结合代数在由群元和斜本原元生成的有限维霍普夫代数的线性作用下的不变量，当作用是纯量时，恰好形成一个有限维Hopf代数的线性生成代数.这推广了一个类似的结果群作用的自同构得到的迪克斯和Formanek，和Kharchenko。
Finitely presented quadratic algebras of intermediate growth
本文给出了两个中间增长的二次代数（由二次关系表示的代数）的例子。
The identities and central polynomials of the finite dimensional unitary Grassmann algebras over a finite field
本文描述了有限域上酉有限维Grassmann代数的恒等式的T-理想和中心多项式的T-空间。
On algebras that almost have finite dimensional representations
引入了代数的几乎有限维的概念，并研究了它与经典有限性条件的关系。
Given a finite connected bipartite graph, finite-dimensional indecomposable semisimple Leibniz algebras are constructed. Furthermore, any finite-dimensional indecomposable semisimple Leibniz algebra admits a similar construction.
Invariants of finite Hopf algebras
本文将有限群不变理论和有限群概型中的经典结果推广到有限霍普夫代数在交换环上的作用.
For an effect algebra $A$, we examine the category of all morphisms from finite Boolean algebras into $A$. This category can be described as a category of elements of a presheaf $R(A)$ on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra $A$ can be characterized by properties of the category of elements of the presheaf $R(A)$. We prove that the tensor product of of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inc…
We study a Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the group ring of finitely generated abelian groups. The Batalin-Vilkovisky algebra structure for finite abelian groups comes from the fact that the group ring of finite groups is a symmetric algebra, and the Batalin-Vilkovisky algebra structure for free abelian groups of finite rank comes from the fact that its group ring is a Calabi-Yau algebra.
Finitely generated algebras with involution and their identities
考虑零特征域上具有对合的结合代数。证明了在这种情况下，对任意一个带对合的有限维生成结合代数，存在一个带对合的有限维代数，它满足与带对合的恒等式完全相同的恒等式.
Group gradings on finite-dimensional incidence algebras. II
本文完成了有限维关联代数上群分次的描述，并对能实现为带有群分次的关联代数的有限维分次代数进行了分类。
The Fibonacci Lie algebra is not finitely presented
本文证明了斐波那契李代数及相关的恰无限自相似李代数均不具有有限表示性质。
Finite-dimensional subalgebras of the Virasoro algebra
确定了无中心Virasoro代数的所有二维李子代数，完成了复Virasoro代数的所有有限维李子代数的刻画。
In this paper, the complete algebraic structure of finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group algebras of various normally monomial groups. The automorphism groups of these group algebras are also determined.
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.
Finite dimensional graded simple algebras
设$R$是代数闭域$F$上的有限维代数，该域被任意群$G$分次.证明了$R$是分次除代数当且仅当它同构于$G$的某个有限子群的扭群代数.若$F$的特征为零或${\rm char} F$不整除$G$的任何有限子群的阶，则证明了$R$是分次单的当且仅当它是有限维分次除代数上的矩阵代数.
A group graded $K$-algebra $A=\bigoplus\limits_{g\in G} A_g$ is called "locally finite" if $\dim_K A_g < \infty$ for every $g\in G$. We characterise the weighted graphs $(E,w)$ for which the weighted Leavitt path algebra $L_K(E,w)$ is locally finite with respect to its standard grading. We also prove that the locally finite weighted Leavitt path algebras are precisely the Noetherian ones and that $L_K(E,w)$ is locally finite iff $(E,w)$ is finite and the Gelfand-Kirillov dimension of $L_K(E,w)$ equals $0$ or $1$. Further it is shown that a locally finite weighted Leavitt path algebra is isomor…
Let $A$ be an associative algebra over a field of characteristic $\neq 2$ that is generated by a finite collection of nilpotent elements. We prove that all Lie derived powers of $A$ are finitely generated Lie algebras.
Structure Theory for One Class of Locally Finite Lie Algebras
本文考虑特征为零的局部有限李代数，它满足这样一个条件：对于这样的代数$L$的每个有限个数的元素$x_{1}, x_{2},..., x_{k}$，存在包含这些元素的有限维子代数$A$，对于某个整数$n$，存在包含这些元素的有限维子代数$L(adA)^n\subset A$.对于这样的代数，我证明了几个结构定理，可以被视为有限维李代数理论的经典结构定理的推广。
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…
Local derivations on finite-dimensional Lie algebras
证明了特征为零的代数闭域上的有限维半单李代数上的每一个局部导子都是导子。我们还给出了有限维幂零李代数$\mathcal{L}$与$\dim\mathcal{L}\geq 3$的例子，其中允许局部导子不是导子。
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.
On the structure of finite Novikov conformal algebras
本文分类了特征零代数闭域上的有限单Novikov共形代数，包括基域上的当前共形代数和一类单参数Virasoro型共形代数，并证明了半单有限Novikov共形代数是单代数的直和，同时描述了其形变与中心扩张。
Ext-finite modules for weakly symmetric algebras with radical cube zero
假设$A$是弱对称的，不可分解的，根式立方为零，根式平方不为零。我们证明了这样的野生表示型代数不存在外代数为有限维的非投射模$M$。给出了具有非投射模且其外代数为有限维的弱对称不可分解代数的一个完全分类。
Finite dimensional factor algebras of F_2[X_1,...,X_n] and their fixed point subalgebras
给出了有限域$\mathbb F_2$上n元多项式代数的有限维因子代数的不动点子代数（关于所有$\mathbb F_2$ -代数自同构）.
We determine the Gelfand-Kirillov dimension of a weighted Leavitt path algebra $L_K(E,w)$ where $K$ is a field and $(E,w)$ a finite weighted graph. Further we show that a finite-dimensional weighted Leavitt path algebra over a field $K$ is isomorphic to a finite product of matrix rings over $K$.
An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension
构造了有限非零Gelfand-Kirillov维数的可数域上的诣零代数。
The paper deals with a construction of a separating system of rational invariants for finite dimensional generic algebras. In the process of dealing an approach to a rough classification of finite dimensional algebras is offered by attaching them some quadratic forms.
In this paper, we classify all finite irreducible conformal modules over a class of Lie conformal algebras $\mathcal{W}(b)$ with $b\in\mathbb{C}$ related to the Virasoro conformal algebra. Explicitly, any finite irreducible conformal module over $\mathcal{W}(b)$ is proved to be isomorphic to $M_{Δ,α,β}$ with $Δ\neq 0$ or $β\neq 0$ if $b=0$, or $M_{Δ,α}$ with $Δ\neq 0$ if $b\neq0$. As a byproduct, all finite irreducible conformal modules over the Heisenberg-Virasoro conformal algebra and the Lie conformal algebra of $\mathcal{W}(2,2)$-type are classified. Finally, the same thing is done for the…
On finite-dimensional homogeneous Lie algebras of derivations of polynomial rings
该研究针对非局部幂零的有限个齐次导子生成的李代数，推导得出其有限维判定准则，补充了以往仅针对局部幂零齐次导子相关结论的研究空白。
Distinguishing division algebras by finite splitting fields
本文研究了有限分裂域集合上可除代数个数的确定问题。作为一个推论，我们能够确定当两个中心司代数可以区分他们的有限分裂领域在某些领域。
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.
Quasi-finite algebras graded by Hamiltonian and vertex operator algebras
引入了拟有限代数的一般概念，拟有限代数是由满足某些性质的齐次子空间上的所有具有拓扑的整数的集合所分次的代数。引入了正则双模的一个类似物，并描述了拟有限代数上的各种模范畴。当应用于满足朱的$C_2$ -有限性条件的顶点算子代数的当前代数（泛包络代数）时，我们的一般考虑在这样的顶点算子代数的表示论上得到了重要的结果.特别地，证明了这种顶点算子代数上的模范畴等价于有限维结合代数上的模范畴.
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 $\mathbb{F}_{2}$.
Classification of the d-representation-finite symmetric k-algebras of finite representation type
本文基于d-表示有限对称Nakayama代数和路径代数的平凡延伸，对有限表示类型的d-表示有限对称代数进行分类。
Construction of Koszul algebras by finite Galois covering
证明了代数和模的拟Koszu算子是Morita不变性的。有限维$K$ -代数$A$是拟Koszul当且仅当斜群代数$A \ast G$是拟Koszul，其中$G$是满足$\char K \nmid |G|$的有限群。有限维$G$ -分次$K$ -代数$A$是拟Koszul当且仅当Smash积$A # G^*$是拟Koszul，其中$G$是满足$\char K \nmid |G|$的有限群.应用这些结果证明了：如果有限维连通的Koszul代数是Koszul代数，那么它的有限伽罗瓦群$G$满足$\char K \nmid |G|$的Galois覆盖代数也是Koszul代数.因此，可以利用有限Galois覆盖构造Koszul代数.利用有限循环伽罗瓦群的伽罗瓦覆盖构造了Koszul代数.作为例子，许多Koszul代数是由外代数和Koszul预投射代数通过有限的循环或非循环伽罗瓦群覆盖而构造的。
Representation theory of rectangular finite $W$-algebras
我们对矩形有限$W$ -代数的有限维不可约表示进行了分类，即，有限$W$ -代数$U(\mathfrak{g}, e)$，其中$\mathfrak{g}$是辛或正交李代数，$e \in \mathfrak{g}$是具有相同大小的Jordan块的幂零元幂零元。
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.
On $τ$-tilting finiteness of symmetric algebras of polynomial growth
本文研究代数闭域上多项式增长对称代数等有限维代数的$τ$-倾斜有限性，发现导出等价保持该性质，并证明了相关代数中表示有限性与$τ$-倾斜有限性的一致性。
(Finite) presentations of Bi-Zassenhaus loop algebras
证明了Bi-Zassenhaus loop代数是可表示到中心元和第二中心元的.特别是，我们展示了一个明确的有限介绍李代数的商在其第二个中心是同构的双Zassenhaus圈代数。
All finite-dimensional Leibniz algebra bimodules of a Lie algebra $\mathfrak{sl}_2$ over a field of characteristic zero are described.
Derived dimensions of representation-finite algebras
证明了任意表示有限Artin代数的导维数至多为1。
Injectives in the variety generated by a finite subdirectly irreducible Heyting algebra with involution
证明了任何具有对合的有限次直不可约Heyting代数都是拟素的，并且有限次直不可约Heyting代数生成的簇中的内射代数恰是该代数的某个直幂的对角子代数，它们是完备的格.
Cosemisimple Hopf algebras with antipode of arbitrary finite order
证明了存在任意有限偶数阶的余半单霍普夫代数.我们还讨论了这类霍普夫代数的Schur指标。
On finite representations of conformal algebras
证明了有限无挠结合共形代数具有有限忠实共形表示。作为一个推论，它表明，一个可以加入共形单位这样的代数。举例说明了一个共形单位不能与任何无挠结合共形代数相连接。特别是，存在线性增长的结合共形代数，甚至局部有限的，没有有限的忠实表示。我们还考虑了无挠有限李共形代数的有限忠实表示的存在性问题（类似于Ado定理）。证明了Poincare-Birkhoff-Witt定理的共形类似物将蕴涵有限共形李代数的Ado定理。证明了任意挠自由有限可解李共形代数都有有限忠实表示。
In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg-Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.
Associative nil-algebras over finite fields
研究了有限域上具有单位元$x^n=0$的相对自由的双生成结合代数的幂零度.
In this paper we describe finite homogeneous effect algebras whose sharp elements are only $0$ and $1$.
On the Ado Theorem for finite Lie conformal algebras with Levi decomposition
证明了具有分裂可解根的有限无挠共形李代数具有有限忠实共形表示。
Hall polynomials for representation-finite cluster-tilted algebras
我们证明了表示有限簇倾斜代数的Hall多项式的存在性。
Keller's conjecture for split finite-dimensional algebras
本文证明了任意域上所有分裂有限维代数均满足Keller猜想，推广了该猜想的适用范围。
We show how to study a certain associative algebra recently discovered by Iyudu and Shkarin using the Anick resolution. This algebra is a counterexample to the conjecture of Positselski on Koszul algebras of finite global dimension.
A double algebra is a linear space $V$ equipped with linear map $V\otimes V\to V\otimes V$. Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double algebras do not exist over an arbitrary field, and all simple finite-dimensional associative double algebras over an algebraically closed field are trivial. Over an arbitrary field, every simple finite-dimensional associative double algebra is commutative. A double algebra structure on a finite-dimensional space $V$ is naturally described by a linear operator $…
Finite order automorphisms and real forms of affine Kac-Moody algebras in the smooth and algebraic category
研究了“光滑”仿射Kac-Moody代数的有穷级和实形式的自同构，即光滑环代数在简单李代数中的二维扩张.它表明，它们可以参数化的某些不变量，特别是对合的分类基本上遵循从嘉当的分类在有限维。我们还证明了我们的方法同样适用于通常的代数设置，并导致相同的结果。
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.
The Cartan determinant conjecture for representation-finite algebras
本文针对代数闭域上的有限表示型代数，证明了满足特定条件的自同态代数的嘉当行列式为1，从而验证了该类代数的嘉当行列式猜想成立。
The finite dual $H^{\circ}$ of an affine commutative-by-finite Hopf algebra $H$ is studied. Such a Hopf algebra $H$ is an extension of an affine commutative Hopf algebra $A$ by a finite dimensional Hopf algebra $F$. The main theorem gives natural conditions under which $H^{\circ}$ decomposes as a crossed or smash product of $F^{\ast}$ by the finite dual of $A$. This decomposition is then further analysed using the Cartier- Gabriel-Kostant theorem to obtain component Hopf subalgebras of $H^{\circ}$ mapping onto the classical components of $A^{\circ}$. The detailed consequences for a number of f…
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