The Anick automorphism of free associative algebras

On the automorphisms of a free Lie algebra of rank 3 over an integral domain

We prove that the group of tame automorphisms of a free Lie algebra (as well as of a free anticommutative algebra) rank 3 over an arbitrary integral domain has the structure of an amalgamated free product. We construct an example of a wild automorphism of a free Lie algebra (as well as of a free anticommutative algebra) of rank 3 over an arbitrary Euclidean ring analogous to the Anick automorphism for free associative algebras.

Automorphisms of Weyl Algebra and a Conjecture of Kontsevich

We outline the proof of a conjecture of Kontsevich on the isomorphism between the group of polynomial symplectomorphisms in $2n$ variables and the group of automorphisms of the $n$-th Weyl algebra over complex numbers. Our proof uses lifting of polynomial symplectomorphisms to Weyl algebra automorphisms by means of approximation by tame symplectomorphisms and gauging of the lifted morphism. Approximation by tame symplectomorphisms is the symplectic version of the well-known theorem of D. Anick and is a result of our prior work.

Automorphisms fixing a variable of K<x,y,z>

We study automorphisms of the free associative algebra K<x,y,z> over a field K which fix z and such that the images of x, y are linear with respect to x, y. We prove that some of these automorphisms are wild in the class of all automorphisms fixing z, including the well known automorphism discovered by Anick, and show how to recognize the wild ones. This class of automorphisms induces tame automorphisms of the polynomial algebra K[x,y,z]. For n>2 the automorphisms of K<x_1,...,x_n,z> which fix z and are linear in the x's are tame.

The group of automorphisms of the category of free associative algebras

In this paper, the problem formulated in [8] is solved. We prove, that the group of automorphisms of the category of free associative algebras is generated by semi-inner and mirror automorphisms

Tame Automorphisms Fixing a Variable of Free Associative Algebras of Rank Three

We study automorphisms of the free associative algebra K<x,y,z> over a field K which fix the variable z. We describe the structure of the group of z-tame automorphisms and derive algorithms which recognize z-tame automorphisms and z-tame coordinates.

有限维幂零结合代数的局部与2-局部自同构

Local and 2-local automorphisms of finite-dimensional nilpotent associative algebras

本文研究了$n$维null-filiform和filiform结合代数的自同构、局部与2-局部自同构，明确了其矩阵一般形式，发现局部自同构不全为自同构，并证明了所有2-局部自同构均为自同构。

结合代数的导子与自同构群的计算方法

Computational Approaches to Derivations and Automorphism Groups of Associative Algebras

本文利用低维结合代数的分类结果和符号计算工具，系统描述了二维至四维复数结合代数的导子与自同构群，揭示了其结构特征与对称性。

Automorphisms of free braided associative algebras in two variables

We describe the groups of automorphisms of two generated free braided associative algebras with involutive diagonal braidings over a field of characteristic $\neq 2$. Depending on the form of the diagonal involutive braiding, five different automorphism groups arise as automorphism groups of two generated free braided associative algebras. This list covers all three automorphism groups that arise in the case of quantum planes \cite{AC}.

Coordinates and Automorphisms of Polynomial and Free Associative Algebras of Rank Three

We study z-automorphisms of the polynomial algebra K[x,y,z] and the free associative algebra K<x,y,z> over a field K, i.e., automorphisms which fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K<x,y,z> we include also results about the structure of the z-tame automorphisms and algorithms which recognize z-tame automorphisms and z-tame coordinates.

结合代数上的精细分次与自同构群

Fine gradings and automorphism groups on associative algebras

本文研究结合代数的精细分次与自同构群，给出了分次精细的充要条件，并计算了相关自同构群，特别在实数域上有限维单代数情形下给出具体结果。

Groups of triangular automorphisms of a free associative algebra and a polynomial algebra

We study a structure of the group of unitriangular automorphisms of a free associative algebra and a polynomial algebra and prove that this group is a semi direct product of abelian groups. Using this decomposition we describe a structure of the lower central series and the series of derived subgroups for the group of unitriangular automorphisms and prove that every element from the derived subgroup is a commutator. In addition we prove that the group of unitriangular automorphisms of a free associative algebra of rang more than 2 is not linear and describe some two-generated subgroups from th…

结合中心单可构造代数的 Kantor 对的自同构群概型

Automorphism group schemes of Kantor pairs of associative central simple structurable algebras

研究结合中心单可构造代数的 Kantor 对及相关三元系在特定条件下的自同构群概型，针对三种情形中的两种，将自同构群概型的中心积分解与 Kantor 对的张量积分解相联系。

局部导出与某些结合代数局部自同构的关系

The relationship between local derivations and local automorphisms of some associative algebras

本文研究了五维自然分级 nilpotent 结合代数的局部导出与局部自同构，揭示了 π₂ 和 π₃ 代数的矩阵一般形式，并证明这些代数的局部自同构（导出）不等于自同构（导出），同时建立了局部自同构与局部导出之间的指数关系，展示了局部导出集构成李代数。

Exact sequences of inner automorphisms of tensors

We produce a long exact sequence whose terms are unit groups of associative algebras that behave as inner automorphisms of a given tensor. Our sequence generalizes known sequences for associative and non-associative algebras. In a manner similar to those, our sequence facilitates inductive reasoning about, and calculation of the groups of symmetries of a tensor. The new insights these methods afford can be applied to problems ranging from understanding algebraic structures to distinguishing entangled states in particle physics.

Automorphisms of the semigroup of endomorphisms of free associative algebras

Let $A=A(x_{1},...,x_{n})$ be a free associative algebra in $\mathcal{A}$ freely generated over $K$ by a set $X=\{x_{1},...,x_{n}\}$, $End A$ be the semigroup of endomorphisms of $A$, and $Aut End A$ be the group of automorphisms of the semigroup $End A$. We investigate the structure of the groups $Aut End A$ and $Aut \mathcal{A}^{\circ}$, where $\mathcal{A}^{\circ}$ is the category of finitely generated free algebras from $\mathcal{A}$. We prove that the group $Aut End A$ is generated by semi-inner and mirror automorphisms of $End F$ and the group $Aut \mathcal{A}^{\circ}$ is generated by sem…

自由代数自同构群的切李代数

Tangent Lie Algebras of Automorphism Groups of Free Algebras

本文为自由代数自同构群引入切李代数，证明其包含于常散度导子李代数，并利用该工具证明几乎所有已知野生自同构（除Nagata和Anick外）均为绝对野生，包括Bergman自同构及多数多项式幂零李代数簇中的自由代数。

The automorphisms group and the classification of gradings of finite dimensional associative algebras

Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all invertible group-like elements of the finite dual $a (A)^{\rm o}$. For a group $G$, all $G$-gradings on $A$ are explicitly described and classified: the set of isomorphisms classes of all $G$-gradings on $A$ is in bijection with the quotient set $ {\rm Hom}_{\rm BiAlg} \, \bigl( a (A) , \, k[G] \bigl)/\approx$ of all bialgebra maps $a (A) \, \to k[G]$, via th…

第二类结合三元系的分次

Gradings on associative triple systems of the second kind

本文研究第二类结合三元系，通过建立单三元系与 Loos 带对合 $3$-分次结合代数的自同构群概形同构，实现可结构化代数分次的完全同构分类。

十六元数的自同构

Automorphisms of Sedenions

研究十六元数自同构问题，借助克利福德代数，通过扩展7顶点法诺平面为15顶点法诺体，结合新校准与现有结合校准，明确推导其自同构，扩展前人结果并解决文献差异，还推测法诺超体可揭示凯莱 - 迪克森代数结构。

特征 $p$ 的微分扩张的自同构

The automorphisms of differential extensions of characteristic $p$

本文研究特征 $p$ 情形下非结合微分扩张的自同构，通过限制构造代数所用微分多项式环的自同构，显式刻画其结构，并推出 $D$ 与 $K$ 的结合微分扩张的内自同构描述。

Automorphisms and derivations of algebras of infinite matrices

We describe automorphisms and derivations of several important associative and Lie algebras of infinite matrices over a field.

Automorphisms and derivations of affine commutative and PI-algebras

We prove analogs of A.~Selberg's result for finitely generated subgroups of $\text{Aut}(A)$ and of Engel's theorem for subalgebras of $\text{Der}(A)$ for a finitely generated associative commutative algebra $A$ over an associative commutative ring. We prove also an analog of the theorem of W.~Burnside and I.~Schur about locally finiteness of torsion subgroups of $\text{Aut}(A)$.

Stable tameness of automorphisms of F<x,y,z> fixing z

It is proved that every z-automorphism (z-coordinate, respectively) of the free associative algebra F<x,y,z> over an arbitrary field F is stably tame.

Lie properties in associative algebras

Let K be a field, then we exhibit two matrices in the full nxn matrix algebra M_{n}(K) which generate M_{n}(K) as a Lie K-algebra with the commutator Lie product. We also study Lie centralizers of a not necessarily commutative unitary algebra and obtain results which we hope will eventually be a step in the direction of, firstly, proving that a Lie-nilpotent K-subspace (or a sub Lie K-algebra) of a finite-dimensional associative algebra over K of index k (say) generates a Lie-nilpotent associative subalgebra of much higher nilpotency index, and secondly, in the light of the sharp upper bound f…

Squares and associative representations of two dimensional evolution algebras

We associate an square to any two dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behaviour of the algebra. We determine the identities of degrees at most four, as well as derivations and automorphisms. We look at the group of automorphisms as an algebraic group, getting in this form a new algebraic invariant. The study of associative representations of evolution algebras is also started and we get faithful representations for most two-dimensional evolution algebras. In some cases we prove that faithf…

On the graded automorphisms of upper triangular matrix algebras

We compute the graded automorphisms of the upper triangular matrices, viewed as associative, Lie and Jordan algebras. We compute also the so called self-equivalences and Weyl and diagonal groups for every grading.

与交换结合代数自同构相关的内部同痕

Inner isotopes associated with automorphisms of commutative associative algebras

本文利用由代数自同态定义的内部同痕，从交换结合多项式代数构造新的交换非结合代数，建立其同构类与维数整数分拆的双射，并完整刻画幂等元及其谱。

A short proof that the free associative algebra is Hopfian

A short proof is given of the fact that various classes of algebras including the free associative algebra are Hopfian, i.e., every epimorphism is an automorphism. This further simplifies the Dicks-Lewin solution of the Jacobian conjecture for the free associative algebra.

On automorphisms of categories of free algebras of some varieties

Given a variety of universal algebras. A method is suggested for describing automorphisms of a category of free algebras of this variety. Applying this general method all automorphisms of such categories are found in two cases: 1) for the variety of all free associative K-algebras over an infinite field K and 2) for the variety of all representations of groups in unital R-modules over a commutative associative ring R with unit. It turns out that they are almost inner in a sense.

Automorphic equivalence problem for free associative algebras of rank two

Let K<x,y> be the free associative algebra of rank 2 over an algebraically closed constructive field of any characteristic. We present an algorithm which decides whether or not two elements in K<x,y> are equivalent under an automorphism of K<x,y>. A modification of our algorithm solves the problem whether or not an element in K<x,y> is a semiinvariant of a nontrivial automorphism. In particular, it determines whether or not the element has a nontrivial stabilizer in Aut K<x,y>. An algorithm for equivalence of polynomials under automorphisms of C[x,y] was presented by Wightwick. Another, much s…

Finitely generated invariants of Hopf algebras on free associative algebras

We show that the invariants of a free associative algebra of finite rank under a linear action of a finite-dimensional Hopf algebra generated by group-like and skew-primitive elements form a finitely generated algebra exactly when the action is scalar. This generalizes an analogous result for group actions by automorphisms obtained by Dicks and Formanek, and Kharchenko.

有限维代数的典范滤过

Canonical Filtrations of Finite-Dimensional Algebras

本文利用 GIT 最优失稳单参数子群定义有限维结合代数与李代数的典范滤过，刻画半单性并给出计算方法及相伴分次代数的结构结果。

On the lifting of the Nagata automorphism

It is proved that the Nagata automorphism (Nagata coordinates, respectively) of the polynomial algebra $F[x,y,z]$ over a field $F$ cannot be lifted to a $z$-automorphism ($z$-coordinate, respectively) of the free associative algebra $K<x,y,z>$. The proof is based on the following two new results which have their own interests: degree estimate of ${Q*_FF<x_1,...,x_n>}$ and tameness of the automorphism group ${\text{Aut}_Q(Q*_FF<x,y>)}$.

Asymptotics of H-identities for associative algebras with an H-invariant radical

We prove the existence of the Hopf PI-exponent for finite dimensional associative algebras $A$ with a generalized Hopf action of an associative algebra $H$ with $1$ over an algebraically closed field of characteristic $0$ assuming only the invariance of the Jacobson radical $J(A)$ under the $H$-action and the existence of the decomposition of $A/J(A)$ into the sum of $H$-simple algebras. As a consequence, we show that the analog of Amitsur's conjecture holds for $G$-codimensions of finite dimensional associative algebras over a field of characteristic $0$ with an action of an arbitrary group $…

Automorphisms of the semigroup of invertible matrices with nonnegative elements

In this paper we prove that every automorphism of the semigroup of invertible matrices with nonnegative elements over a linearly oredered associative ring on some specially defined subgroup concides with the composition of an inner automorphism of the semigroup, an automorphism of the ring preserving the order, and a central homothety.

Planar Prop of Differential Operators

We propose a definition of differential operators of an associative algebra $A$ in the spirit of Hochschild cohomology. Specifically we define $D(A)$ as the zero cohomology of a certain bicomplex formed by Hom-spaces $\mathrm{Hom}(A^{\otimes q}, A^{\otimes p})$. We show that it has a structure of a planar prop, i.e. each differential operator has multiple inputs and outputs and they can be composed along planar graphs. Furthermore, for a formally smooth algebra we have the surjective symbol map from $D(A)$ to the space of poly-derivations. We also consider another planar prop $E(A)$ generated …

On the automorphism groups of some Leibniz algebras

We study the automorphism groups of finite-dimensional cyclic Leibniz algebras. In this connection, we consider the relationships between groups, modules over associative rings and Leibniz algebras.

低维结合代数的相容对及其不变量

Compatible Pairs of Low-Dimensional Associative Algebras and Their Invariants

本文分类了复维数小于四的结合代数的相容对，并系统计算了其导子、中心化子、自同构群及多种算子等代数不变量。

代数的代数导子与自同构群的根

Algebraic derivations and radicals of automorphism groups of algebras

本文研究半素结合代数的代数导子与相关自同构群的根性质，证明其素根与弱可解根重合，并联系Martindale商环内导子描述。

八元数代数的子代数

Subalgebras of Octonion Algebras

本文确定了任意八元数代数的所有子代数，揭示了其结合性与维数的关系，并证明了同构子代数在自同构下的轨道性质。

Almost nilpotency of an associative algebra with an almost nilpotent fixed-point subalgebra

Let A be an associative algebra of arbitrary dimension over a field F and G a finite soluble group of automorphisms of A oforder n, prime to the characteristic of F. We prove that if the fixed-point subalgebra of A under the action of G contains a two-sided nilpotent ideal of nilpotency index d and of finite codimension m in I, then A contains a nilpotent two-sided ideal of nilpotency index bounded by a function of n and d and of finite codimension bounded by a function of m, n and d.

Petit Algebras and their Automorphisms

In this thesis, we study the properties of a nonassociative algebra construction from skew polynomial rings. This construction was introduced by Petit in the 1960s but largely ignored until recently. In particular, the automorphism groups of these algebras are studied, paying particular attention to the case when the construction yields a finite semifield. The thesis is concluded by revisiting a result on associative solvable crossed product algebras by both Petit and Albert. We show a crossed product algebra is solvable if and only if it can be written as a chain of Petit algebras satisfying …

Automorphisms of the semigroup of invertible matrices with nonnegative elements over commutative partially ordered rings

Suppose that R is an ordered ring, G_n(R) is a subsemigroup of $GL_n(R)$, consisting of all matrices with nonnegative elements. A.V. Mikhalev and M.A. Shatalova described all automorphisms of G_n(R), where R is a linearly ordered skewfield and n>1. E.I. Bunina and A.V. Mikhalev found all automorphisms of G_n(R), if R is an arbitrary linearly ordered associative ring with 1/2, n>2. In this paper we describe automorphisms of G_n(R), if R is a commutative partially ordered ring, containing Q, n>2.

Algebres non associatives normees de division. Classification des algebres reelles de Jordan non commutatives de division lineaire de dimension 8

In this work we are interested in the general problem of the determination of the normed division algebras. Our fundamental results are obtained in the particular subclass of those 8-dimensional quadratic flexible real division algebras. We give a new process which generalizes that of Cayley-Dickson and which allows the obtaining of a new family of eight-dimensional quadratic flexible real division algebras. We give examples of 8-dimensional quadratic flexible real division algebras which cannot be obtained by this first process of duplication and by means of a second process, which consists i…

Non-associative Frobenius algebras for simply laced Chevalley groups

We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.

关联Rota-Baxter算子在Sweedler代数$H_4$上的研究

Associative Rota--Baxter operators on the Sweedler algebra $H_4$

本文研究了Sweedler代数$H_4$上的Rota-Baxter算子，通过分类和核的分析，给出了所有算子的完整描述。

An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange

The inner automorphisms of a group G can be characterized within the category of groups without reference to group elements: they are precisely those automorphisms of G that can be extended, in a functorial manner, to all groups H given with homomorphisms G --> H. Unlike the group of inner automorphisms of G itself, the group of such extended systems of automorphisms is always isomorphic to G. A similar characterization holds for inner automorphisms of an associative algebra R over a field K; here the group of functorial systems of automorphisms is isomorphic to the group of units of R modulo …

相容结合代数与若干不变量

Compatible Associative Algebras and Some Invariants

本文研究相容结合代数，给出维数小于四的该类代数及其对应导子、中心等结构的分类，并针对具体例子刻画罗塔-巴克斯特算子、第二上同等不变量，推进了低维相容结合代数的结构分析。

幂零李代数的局部导子与自同构

Local derivations and automorphisms of nilpotent Lie algebra

本文研究幂零李代数的局部导子与自同构，证明幂零指数为3和4时存在非导子的局部导子及非自同构的局部自同构，并给出存在性充分条件及一个局部导子空间与导子空间重合的非结合代数例子。

二维演化代数的Hopf代数与结合表示

Hopf algebras and associative representations of two-dimensional evolution algebras

本文通过自同构代数群建立二维演化代数与Hopf代数的联系，计算了相关联的p-代数，并给出了忠实交换结合表示存在性及同构条件的刻画。

On the hom-associative Weyl algebras

The first (associative) Weyl algebra is formally rigid in the classical sense. In this paper, we show that it can however be formally deformed in a nontrivial way when considered as a so-called hom-associative algebra, and that this deformation preserves properties such as the commuter, while deforming others, such as the center, power associativity, the set of derivations, and some commutation relations. We then show that this deformation induces a formal deformation of the corresponding Lie algebra into what is known as a hom-Lie algebra, when using the commutator as bracket. We also prove t…

The automorphisms of Petit's algebras

Let $σ$ be an automorphism of a field $K$ with fixed field $F$. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras $K[t;σ]/fK[t;σ]$ obtained when the twisted polynomial $f\in K[t;σ]$ is invariant, and were first defined by Petit. We compute all their automorphisms if $σ$ commutes with all automorphisms in ${\rm Aut}_F(K)$ and $n\geq m-1$, where $n$ is the order of $σ$ and $m$ the degree of $f$,and obtain partial results for $n<m-1$. In the case where $K/F$ is a finite Galois field extension, we obtain more detai…

On The Zariski Topology Of Automorphism Groups Of Affine Spaces And Algebras

We study the Zariski topology of the ind-groups of polynomial and free associative algebras $\Aut(K[x_1,...,x_n])$ (which is equivalent to the automorphism group of the affine space $\Aut(K^n))$) and $\Aut(K< x_1,..., x_n>$ via $\Ind$-schemes, toric varieties, approximations and singularities. We obtain some nice properties of $\Aut(\Aut(A))$, where $A$ is polynomial or free associative algebra over a field $K$. We prove that all $\Ind$-scheme automorphisms of $\Aut(K[x_1,...,x_n])$ are inner for $n\ge 3$, and all $\Ind$-scheme automorphisms of $\Aut(K< x_1,..., x_n>)$ are semi-inner. We also …

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 …