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}.
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
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.
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.
Let $\mathcal{C}$ be a finite braided multitensor category. Let $B$ be Majid's automorphism braided group of $\mathcal{C}$, then $B$ is a cocommutative Hopf algebra in $\mathcal{C}$. We show that the center of $\mathcal{C}$ is isomorphic to the category of left $B$-comodules in $\mathcal{C}$, and the decomposition of $B$ into a direct sum of indecomposable $\mathcal{C}$-subcoalgebras leads to a decomposition of $B$-$\operatorname*{Comod}_{\mathcal{C}}$ into a direct sum of indecomposable $\mathcal{C}$-module subcategories. As an application, we present an explicit characterization of the struc…
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…
The relationship between local derivations and local automorphisms of some associative algebras
V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences（乌兹别克斯坦科学院 V.I.罗曼诺夫斯基数学研究所）；Andijan State University（安集延国立大学）；Namangan State University（纳曼干国立大学）
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.
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…
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…
Let $ Aut_{mHH}(H)$ denote the set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel and Goyvaerts \cite{CG2011}. The main aim of this paper is to provide new examples of braided $T$-category in the sense of Turaev \cite{T2008}. For this, first we construct a monoidal Hom-Hopf $T$-coalgebra $\mathcal{MHD}(H)$ and prove that the $T$-category $Rep(\mathcal{MHD}(H))$ of representation of $\mathcal{MHD}(H)$ is isomorphic to $\mathcal {MHYD}(H)$ as braided $T$-categories, if $H$ is finite-dimensional. Then we construct a new braided $T$-category $…
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 $\text{Aut}(A)$.
We study formal deformations of a crossed product $S(V)#_f G$, of a polynomial algebra with a group, induced from a universal deformation formula introduced by Witherspoon. These deformations arise from braided actions of Hopf algebras generated by automorphisms and skew derivations. We show that they are nontrivial in the characteristic free context, even if $G$ is infinite, by showing that their infinitesimals are not coboundaries. For this we construct a new complex which computes the Hochschild cohomology of $S(V)#_f G$.
Let $ Aut_{mHH}(H)$ denote a set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel S. and Goyvaerts I. (Commun. Algebra 39, 2216-2240, 2011) and let $G$ be a crossed product group $ Aut_{mHH}(H)\times Aut_{mHH}(H)$. The main aim of this paper is to provide further examples of braided $T$-category in the sense of Turaev (1994, 2008). For this purpose, we first introduce a class of new categories $_{H}\mathcal {MHYD}^{H}(A, B)$ of monoidal Hom $(A, B)$-Yetter-Drinfeld modules with $A, B \in Aut_{mHH}(H)$. Then we show that the category ${\cal M…
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…
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…
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.
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.
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.
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…
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.
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 $…
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.
In this paper, we introduce the notions of crossed module of associative conformal algebras, 2-term strongly homotopy associative conformal algebras, and discuss the relationship between them and the 3-th Hochschild cohomology of associative conformal algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. We show that non-abelian extensions of an associative conformal algebra can be viewed as Maurer-Cartan elements of a suitable differential graded Lie algebra, and prove that the Deligne groupoid of this differential graded Lie algebra corresponds one to on…
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.
Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra K<x,y,z> over a field K of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of K<x,y,z>. In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a similar result for several large classes of automorphisms of K<x,y,z>. We also find a…
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.
In this paper we construct "structural" pre-braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these pre-braidings seem original. On the other hand, we propose a general homology theory for pre-braided vector spaces and braided modules, based on the quantum co-shuffle comultiplication. Applied to the structural pre-braidings above, it gives a generalization and a unification of many known homology theories. All the constructions are categorified, resulting in particular in their super- and co-versions. L…
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.
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.
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 …
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 …
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…
We prove the Andruskiewitsch-Dumas conjecture that the automorphism group of the positive part of the quantized universal enveloping algebra $U_q({\mathfrak{g}})$ of an arbitrary finite dimensional simple Lie algebra g is isomorphic to the semidirect product of the automorphism group of the Dynkin diagram of g and a torus of rank equal to the rank of g. The key step in our proof is a rigidity theorem for quantum tori. It has a broad range of applications. It allows one to control the (full) automorphism groups of large classes of associative algebras, for instance quantum cluster algebras.
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 ${\text{Aut}_Q(Q*_FF<x,y>)}$.
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…
In this paper we study the existence of gradings on finite dimensional associative algebras. We prove that a connected algebra $A$ does not have a non-trivial grading if and only if $A$ is basic, its quiver has one vertex, and its group of outer automorphisms is unipotent. We apply this result to prove that up to graded Morita equivalence there do not exist non-trivial gradings on the blocks of group algebras with quaternion defect groups and one isomorphism class of simple modules.
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…
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 …
An algebra $L$ over a field $\Bbb F$, in which product is denoted by $[\,,\,]$, is said to be \textit{ Lie type algebra} if for all elements $a,b,c\in L$ there exist $α, β\in \Bbb F$ such that $α\neq 0$ and $[[a,b],c]=α[a,[b,c]]+β[[a,c],b]$. Examples of Lie type algebras are associative algebras, Lie algebras, Leibniz algebras, etc. It is proved that if a Lie type algebra $L$ admits an automorphism of finite order $n$ with finite-dimensional fixed-point subalgebra of dimension $m$, then $L$ has a soluble ideal of finite codimension bounded in terms of $n$ and $m$ and of derived length bounded …
We consider homogeneous varieties of linear algebras over an associative-commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F be a free algebra of some variety A of linear algebras over K freely generated by a finite set X, EndF be the semigroup of endomorphisms of F, and AutEndF be the group of automorphisms of the semigroup EndF. We investigate structure of the group AutEndF and its relation to the algebraical and categorical equivalence of algebras from A. We define a wide class of R1MF-domains containing, in particular, Bezout domains, unique factorizatio…
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 …
