On associative conformal algebras of linear growth II

Hochshild cohomology of the associative conformal algebra Cend_{1,x}

It is established in this work that second Hochshild cohomology group of the associative conformal algebra Cend_{1,x} is zero. As a corollary, this algebra split off in each extension with a nilpotent kernel. Key words: associative conformal algebra, splitting off radical, Hochshild cohomology.

Composition-Diamond lemma for associative $n$-conformal algebras

In this paper, we study the concept of associative $n$-conformal algebra over a field of characteristic 0 and establish Composition-Diamond lemma for a free associative $n$-conformal algebra. As an application, we construct Gröbner-Shirshov bases for Lie $n$-conformal algebras presented by generators and defining relations.

Simple associative conformal algebras of linear growth

We describe simple finitely generated associative conformal algebras of Gel'fand--Kirillov dimension one.

Standard bases for the universal associative conformal envelopes of Kac--Moody conformal algebras

We study the universal enveloping associative conformal algebra for the central extension of a current Lie conformal algebra at the locality level $N=3$. A standard basis of defining relations for this algebra is explicitly calculated. As a corollary, we find a linear basis of the free commutative conformal algebra relative to the locality $N=3$ on the generators.

Rota-Baxter family $Ω$-associative conformal algebras and their cohomology theory

In this paper, we first propose the concept of Rota-Baxter family $Ω$-associative conformal algebras, then we study the cohomology theory of Rota-Baxter family $Ω$-associative conformal algebras of any weight and justify it by interpreting the lower degree cohomology groups as formal deformations.

On finite representations of conformal algebras

We prove a finite torsion-free associative conformal algebra to have a finite faithful conformal representation. As a corollary, it is shown that one may join a conformal unit to such an algebra. Some examples are stated to demonstrate that a conformal unit can not be joined to any torsion-free associative conformal algebra. In particular, there exist associative conformal algebras of linear growth and even locally finite ones that have no finite faithful representation. We also consider the problem of existence of a finite faithful representation for a torsion-free finite Lie conformal algebr…

On antisymmetric infinitesimal conformal bialgebras

In this paper, we construct a bialgebra theory for associative conformal algebras, namely antisymmetric infinitesimal conformal bialgebras. On the one hand, it is an attempt to give conformal structures for antisymmetric infinitesimal bialgebras. On the other hand, under certain conditions, such structures are equivalent to double constructions of Frobenius conformal algebras, which are associative conformal algebras that are decomposed into the direct sum of another associative conformal algebra and its conformal dual as $\mathbb{C}[\partial]$-modules such that both of them are subalgebras an…

Extending structures for associative conformal algebras

In this paper, we give a study of the $\mathbb{C}[\partial]$-split extending structures problem for associative conformal algebras. Using the unified product as a tool, which includes interesting products such as bicrossed product, cocycle semi-direct product and so on, a cohomological type object is constructed to characterize the $\mathbb{C}[\partial]$-split extending structures for associative conformal algebras. Moreover, using this theory, the extending structures of an associative conformal algebra $A$ which is free as a $\mathbb{C}[\partial]$-module by the $\mathbb{C}[\partial]$-module …

系数取所有有限模的Virasoro李共形代数的泛结合共形包络的Hochschild上同调

Hochschild cohomology of the universal associative conformal envelope of the Virasoro Lie conformal algebra with coefficients in all finite modules

本研究通过代数离散Morse理论与Gröbner–Shirshov基构造Anick分解，求出了生成元上结合局部性N=3的Virasoro李共形代数的泛结合共形包络$U(3)$在所有有限模系数下的Hochschild上同调群。

结合共形代数上扭曲 Rota-Baxter 算子的上同调

Cohomology of Twisted Rota-Baxter operators on Associative~Conformal Algebra

本文构造由 H-TRB 算子的 Maurer-Cartan 元给出的 L_∞-代数，将共形 H-TRB 算子上同调刻画为相应 Hochschild 上同调，并通过线性与形式形变展示其应用。

A new Composition-Diamond lemma for associative conformal algebras

Let $C(B,N)$ be the free associative conformal algebra generated by a set $B$ with a bounded locality $N$. Let $S$ be a subset of $C(B,N)$. A Composition-Diamond lemma for associative conformal algebras is firstly established by Bokut, Fong, and Ke in 2004 \cite{BFK04} which claims that if (i) $S$ is a Gröbner-Shirshov basis in $C(B,N)$, then (ii) the set of $S$-irreducible words is a linear basis of the quotient conformal algebra $C(B,N|S)$, but not conversely. In this paper, by introducing some new definitions of normal $S$-words, compositions and compositions to be trivial, we give a new Co…

Gerstenhaber algebra of an associative conformal algebra

We define a cup product on the Hochschild cohomology of an associative conformal algebra $A$, and show the cup product is graded commutative. We define a graded Lie bracket with the degree $-1$ on the Hochschild cohomology $\HH^{\ast}(A)$ of an associative conformal algebra $A$, and show that the Lie bracket together with the cup product is a Gerstenhaber algebra on the Hochschild cohomology of an associative conformal algebra. Moreover, we consider the Hochschild cohomology of split extension conformal algebra $A\hat{\oplus}M$ of $A$ with a conformal bimodule $M$, and show that there exist an…

$\mathcal{O}$-operators and Nijenhius operators of associative conformal algebras

We study $\mathcal{O}$-operators of associative conformal algebras with respect to conformal bimodules. As natural generalizations of $\mathcal{O}$-operators and dendriform conformal algebras, we introduce the notions of twisted Rota-Baxter operators and conformal NS-algebras. We show that twisted Rota-Baxter operators give rise to conformal NS-algebras, the same as $\mathcal{O}$-operators induce dendriform conformal algebras. And we introduce a conformal analog of associative Nijenhius operators and enumerate main properties. By using derived bracket construction of Kosmann-Schwarzbach and a …

Rota-Baxter算子与BiHom-结合共形代数上的Loday型代数

Rota-Baxter operators and Loday-type algebras on the BiHom-associative conformal algebras

本文在BiHom-结合共形代数框架下系统研究了(三)树状代数、NS代数与(扭曲)Rota-Baxter算子，揭示了它们之间的内在联系，并建立了BiHom-四元共形代数与Rota-Baxter系统的关联。

同伦共形代数

Homotopy conformal algebras

本文引入 $A_\infty$-共形代数与结合共形 $2$-代数，证明同伦迁移定理及二者范畴等价，并探讨其与 $L_\infty$-共形代数的关系。

Weyl共形代数在有限模中系数的Hochschild上同调

Hochschild cohomology of the Weyl conformal algebra with coefficients in finite modules

本文通过将代数离散Morse理论推广到微分代数，完整计算了Weyl结合共形代数在所有有限模中系数的Hochschild上同调群。

具有二次余维数增长的基础代数与簇

Varieties of associative algebras with quadratic codimension growth

该研究对特征零域上余维序列二次增长的结合代数按PI等价分类，通过构造检测器代数证明相关等价关系，得出此类代数的结构结论。

Universal enveloping Poisson conformal algebras

Lie conformal algebras are useful tools for studying vertex operator algebras and their representations. In this paper, we establish close relations between Poisson conformal algebras and representations of Lie conformal algebras. We also calculate explicitly Poisson conformal brackets on the associated graded conformal algebras of universal associative conformal envelopes of Virasoro conformal algebra and Neveu--Schwartz conformal superalgebra.

Defining relations and Gröbner--Shirshov bases of Poisson algebras as of conformal modules

We study the relation between Poisson algebras and representations of Lie conformal algebras. We establish a setting for the calculation of a Gröbner--Shirshov basis in a module over an associative conformal algebra and apply this technique to Poisson algebras considered as conformal modules over appropriate associative envelopes of current Lie conformal algebras. As a result, we obtain a setting for the calculation of a Gröbner--Shirshov basis in a Poisson algebra.

On universal conformal envelopes for quadratic conformal algebras

We prove that every quadratic Lie conformal algebra constructed on a special Gelfand-Dorfman algebra embeds into the universal enveloping associative conformal algebra with a locality function bound N = 3.

On the Wedderburn principal theorem in conformal algebras

We investigate an analogue of the Wedderburn principal theorem for associative conformal algebras with finite faithful representations. It is shown that the radical splitting property for an algebra of this kind holds if the maximal semisimple factor of this algebra is unital, but does not hold in general.

Unital associative pseudoalgebras and their representations

Pseudoalgebras, introduced in [BDK], are multi-dimensional analogues of conformal algebras, which provide an axiomatic description of the singular part of the operator product expansion. Our main interest in this paper is the pseudoalgebra Cend_n, which is the analogue of an algebra of endomorphisms of a finite module. We study its algebraic properties. In particular, we introduce the class of unital pseudoalgebras and describe their structure and representations. Also, we classify pseudoalgebras algebraically similar to Cend_n.

由相关代数结构导出的BiHom-Poisson共形超代数

BiHom-Poisson conformal superalgebras arising from related algebraic structures

本文针对BiHom-Poisson共形超代数，给出其构造定理，证明其在扭转变换下封闭，还推导了其与正则BiHom-结合共形代数、Rota-Baxter算子、BiHom-结合超代数的关联及张量积性质，补充了从Hom-post-Poisson共形超代数的构造方法。

Universally defined representations of Lie conformal superalgebras

We distinguish a class of irreducible finite representations of conformal Lie (super)algebras. These representations (called universally defined) are the simplest ones from the computational point of view: a universally defined representation of a conformal Lie (super)algebra $L$ is completely determined by commutation relations of $L$ and by the requirement of associative locality of generators. We describe such representations for conformal superalgebras $W_n$, $n\ge 0$, with respect to a natural set of generators. We also consider the problem for superalgebras $K_n$. In particular, we find …

Codimensions growth estimate of varieties of dialgebras

In the paper, we obtain the estimates connecting codimensions of varieties of non-associative algebras and corresponding varieties of dialgebras.

Context-free languages and associative algebras with algebraic Hilbert series

In this paper, homological methods together with the theory of formal languages of theoretical computer science are proved to be effective tools to determine the growth and the Hilbert series of an associative algebra. Namely, we construct a class of finitely presented associative algebras related to a family of context-free languages. This allows us to connect the Hilbert series of these algebras with the generating functions of such languages. In particular, we obtain a class of finitely presented graded algebras with non-rational algebraic Hilbert series.

Derivations, gradings, actions of algebraic groups, and codimension growth of polynomial identities

Suppose a finite dimensional semisimple Lie algebra $\mathfrak g$ acts by derivations on a finite dimensional associative or Lie algebra $A$ over a field of characteristic $0$. We prove the $\mathfrak g$-invariant analogs of Wedderburn - Mal'cev and Levi theorems, and the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial identities with derivations of $A$. It turns out that for associative algebras the differential PI-exponent coincides with the ordinary one. Also we prove the analog of Amitsur's conjecture for finite dimensional associative algebras with an …

Classifying complements for conformal algebras

Let $R\subseteq E$ be two Lie conformal algebras and $Q$ be a given complement of $R$ in $E$. Classifying complements problem asks for describing and classifying all complements of $R$ in $E$ up to an isomorphism. It is known that $E$ is isomorphic to a bicrossed product of $R$ and $Q$. We show that any complement of $R$ in $E$ is isomorphic to a deformation of $Q$ associated to the bicrossed product. A classifying object is constructed to parameterize all $R$-complements of $E$. Several explicit examples are provided. Similarly, we also develop a classifying complements theory of associative …

Codimension Growth of Lie algebras with a generalized action

Let $L$ be a finite dimensional Lie $F$-algebra endowed with a generalized action by an associative algebra $H$. We investigate the exponential growth rate of the sequence of $H$-graded codimensions $c_n^H(L)$ of $L$ which is a measure for the number of non-polynomial $H$-identities of $L$. More precisely, we construct the first example of an $S$-graded Lie algebra having a non-integer, even irrational, exponential growth rate $\lim_{n\rightarrow \infty} \sqrt[n]{c_n^{S}(L)}$. Hereby $S$ is a semigroup and an exact value is given. On the other hand, returning to general $H$, if $L$ is semisimp…

Algebras and semigroups of locally subexponential growth

We prove that a countable dimensional associative algebra (resp. a countable semigroup) of locally subexponential growth is $M_\infty$-embeddable as a left ideal in a finitely generated algebra (resp. semigroup) of subexponential growth. Moreover, we provide bounds for the growth of the finitely generated algebra (resp. semigroup). The proof is based on a new construction of matrix wreath product of algebras.

On the codimension growth of G-graded algebras

Let W be an associative PI-affine algebra over a field F of characteristic zero. Suppose W is G-graded where G is a finite group. Let exp(W) and exp(W_e) denote the codimension growth of W and of the identity component W_e, respectively. We prove: exp(W) \leq |G|^2 exp(W_e). This inequality had been conjectured by Bahturin and Zaicev.

具有双曲正弦生成元的Weyl型代数的结构与不变量

Structure and Invariants of Weyl-Type Algebras with Hyperbolic Sine Generators

本文研究了具有双曲正弦生成元的Weyl型代数的结构与不变量，证明其为Azumaya代数并探讨了其Brauer群中的表示，同时计算了Gelfand-Kirillov维度并提出了两个开放问题。

关于李代数与若尔当代数的增长

On the growth of Lie algebras and Jordan algebras

本文证明多项式有界函数可实现为李代数增长函数，并证明若尔当代数与结合代数增长函数类一致，肯定回答Martínez和Zelmanov的问题。

An uncountable family of almost nilpotent varieties of polynomial growth

A non-nilpotent variety of algebras is almost nilpotent if any proper subvariety is nilpotent. Let the base field be of characteristic zero. It has been shown that for associative or Lie algebras only one such variety exists. Here we present infinite families of such varieties. More precisely we shall prove the existence of 1) a countable family of almost nilpotent varieties of at most linear growth and 2) an uncountable family of almost nilpotent varieties of at most quadratic growth.

Actions of Ore extensions and growth of polynomial $H$-identities

We show that if $A$ is a finite dimensional associative $H$-module algebra for an arbitrary Hopf algebra $H$, then the proof of the analog of Amitsur's conjecture for $H$-codimensions of $A$ can be reduced to the case when $A$ is $H$-simple. (Here we do not require that the Jacobson radical of $A$ is an $H$-submodule.) As an application, we prove that if $A$ is a finite dimensional associative $H$-module algebra where $H$ is a Hopf algebra $H$ over a field of characteristic $0$ such that $H$ is constructed by an iterated Ore extension of a finite dimensional semisimple Hopf algebra by skew-pri…

Nil restricted Lie algebras of oscillating intermediate growth

The Grigorchuk and Gupta-Sidki groups are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2, Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic. It is known that the famous construction of Golod yields finitely generated associative nil-algebras of exponential growth. Recent extensions of that approach allowed to construct finitely generated associative nil-algebras of polynomial and intermediate growth. Another motivation of the paper is a cons…

No dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2

The Gelfand-Kirillov dimension measures the asymptotic growth rate of algebras. For every associative dialgebra $\mathcal{D}$, the quotient $\mathcal{A}_\mathcal{D}:=\mathcal{D}/\mathsf{Id}(S)$, where $\mathsf{Id}(S)$ is the ideal of $\mathcal{D}$ generated by the set $S:=\{x \vdash y-x\dashv y \mid x,y\in \mathcal{D}\}$, is called the associative algebra associated to $\mathcal{D}$. Here we show that the Gelfand--Kirillov dimension of $\mathcal{D}$ is bounded above by twice the Gelfand--Kirillov dimension of $\mathcal{A}_\mathcal{D}$. Moreover, we prove that no associative dialgebra has Gelfa…

Scale for codimension growth of Poisson PI-algebras

A.Regev proved that the codimension growth of an associative PI-algebra is at most exponential. The author established a scale for the codimension growth of Lie PI-algebras, which includes a series of functions between exponential and factorial functions. We prove that the same scale stratifies the ordinary codimension growth of Poisson PI-algebras satisfying Lie identical relations. As a byproduct, we obtain a new bound on the codimension growth of Lie PI-algebras in terms of complexity functions. We also study very fast codimension growth of some Poisson algebras without Lie identities. We e…

Group graded PI-algebras and their codimension growth

Let W be an associative PI-algebra over a field F of characteristic zero. Suppose W is G-graded where G is a finite group. Let exp(W) and exp(W_e) denote the codimension growth of W and of the identity component W_e, respectively. The following inequality had been conjectured by Bahturin and Zaicev: exp(W)\leq |G|^2 exp(W_e). The inequality is known in case the algebra W is affine (i.e. finitely generated). Here we prove the conjecture in general.

关于$H$-简单的非必然结合代数

On $H$-simple not necessarily associative algebras

本文研究了具有广义H-作用的非必然结合代数，证明了特征0域上有限维H-简单代数的多项式H-恒等式余维数增长指数的存在性，并探讨了对应的自由-遗忘伴随。

An upper bound for the lower central series quotients of a free associative algebra

Feigin and Shoikhet conjectured in math/0610410 that successive quotients $B_m(A_n)$ of the lower central series filtration of a free associative algebra $A_n$ have polynomial growth. In this paper we give a proof of this conjecture, using the structure of $W_n$-representation on $B_m(A_n)$ which was defined in math/0610410 . We also prove that the number of squares in a Young diagram $D$ corresponding to an irreducible $W_n$-module in the Jordan-Holder series of $B_m(A_n)$ is bounded above by the integer $(m-1)^2+2[(n-2)/2](m-1)$. This bound combined with MAGMA computations by Rains in math/0…

结合代数的随机生成

Random generation of associative algebras

本文研究了有限和profinite结合代数的随机生成问题，将单代数的概率生成结果推广到一大类有限代数，并证明了profinite代数正有限生成的充要条件为具有多项式极大子代数增长。

Actions of Maximal Growth

We study acts and modules of maximal growth over finitely generated free monoids and free associative algebras as well as free groups and free group algebras. The maximality of the growth implies some other specific properties of these acts and modules that makes them close to the free ones; at the same time, we show that being a strong "infiniteness" condition, the maximality of the growth can still be combined with various finiteness conditions, which would normally make finitely generated acts finite and finitely generated modules finite-dimensional.

群格氏代数与二次码维增长的变种

Group graded algebras and varieties with quadratic codimension growth

本文研究了群格氏代数的二次码维增长变种，通过分类和描述方法，揭示了单位格氏变种的结构特性。

真中心指数等于二的代数簇的刻画

A characterization of varieties of algebras of proper central exponent equal to two

本文旨在刻画 $c_n^\delta(\mathcal V)$ 的指数增长大于 2 的结合代数簇，并作为推论得到了指数增长等于 2 的代数簇的刻画。

On Jordan doubles of slow growth of Lie superalgebras

To an arbitrary Lie superalgebra $L$ we associate its Jordan double ${\mathcal Jor}(L)$, which is a Jordan superalgebra. This notion was introduced by the second author before. Now we study further applications of this construction. First, we show that the Gelfand-Kirillov dimension of a Jordan superalgebra can be an arbitrary number $\{0\}\cup [1,+\infty]$. Thus, unlike associative and Jordan algebras, one hasn't an analogue of Bergman's gap $(1,2)$ for the Gelfand-Kirillov dimension of Jordan superalgebras. Second, using the Lie superalgebra $\mathbf R$ constructed before, we construct a Jor…

多项式方程的秩稳定性

Rank-stability of polynomial equations

本文扩展人群稳定性理论至结合代数与李代数，研究多项式方程近似解的秩逼近障碍，发展秩稳定性测试，证明有限维结合代数秩稳定而多数有限维李代数不稳定。

The Polynomial Part of the Codimension Growth of Affine PI Algebras

Let $F$ be a field of characteristic zero and $W$ be an associative affine $F$-algebra satisfying a polynomial identity (PI). The codimension sequence associated to $W$, $c_n(W)$, is known to be of the form $Θ(c n^t d^n)$, where $d$ is the well known (PI) exponent of $W$. In this paper we establish an algebraic interpretation of the polynomial part (the constant $t$) by means of Kemer's theory. In particular, we show that in case $W$ is a basic algebra, then $t = \frac{d-q}{2} + s$, where $q$ is the number of simple component in $W/J(W)$ and $s+1$ is the nilpotency degree of $J(W)$. Thus provi…

具有分级反演的代数结构分类及二次余维增长

A structural classification of algebras with graded involution and quadratic codimension growth

本文研究了具有分级反演的代数结构，通过分类具有二次余维增长的代数，揭示了变种生成与最小变种分解之间的对应关系。

Growth and Relations in Graded Rings

Suppose $A$ is a graded associative algebra over a field, $I$ is its ideal generated by a set $α$ of homogeneous elements, and B = A/I. In this note, some inequalities between Hilbert series of algebras $A,B$ and the number of elements of the set $α$ are announced. As in the Golod--Shafarevich inequality as in our case the equality in every estimate is exact iff the set $α$ is strongly free: so we obtain some new characterizations of such sets. As a consequence it is proved that over a field of zero characteristic for the class of finitely defined graded algebras there is no algorithm to answe…

Growth of differential identities

In this paper we study the growth of the differential identities of some algebras with derivations, i.e., associative algebras where a Lie algebra $L$ (and its universal enveloping algebra $U(L)$) acts on them by derivations. In particular, we study in detail the differential identities and the cocharacter sequences of some algebras whose sequence of differential codimensions has polynomial growth. Moreover, we shall give a complete description of the differential identities of the algebra $UT_2$ of $2\times 2$ upper triangular matrices endowed with all possible action of a Lie algebra by deri…

二次余维增长与带有超逆运算的最小变分

Quadratic codimension growth and minimal varieties of unitary algebras with superinvolution

本文研究了带有超逆运算的单位代数的最小变种，通过二次余维增长的结构特征，揭示了其生成机制。

三元幂等Γ半环、不可约性与高阶路径代数

Ternary Idempotent $Γ$-Semirings, Non-Reducibility, and Higher-Order Path Algebras

本文提出三元幂等Γ半环，证明其严格扩展经典半环路径代数，并展示高阶路径问题的不可约性与复杂性类差异。

关联单项式代数的增长及其在Manturov群中的应用

Growth of associated monomial algebras with application to Manturov groups

本文研究了关联单项式代数与原代数的GK维度关系，并应用该结果分析了Manturov (k,n)-群的增长性，发现其在特定条件下具有指数增长。