Let $L_{n}$ be the free Lie algebra, $F_{n}$ be the free metabelian Lie algebra, and $L_{n,c}$ be the free metabelian nilpotent of class $c$ Lie algebra of rank $n$ generated by $x_1,\ldots,x_n$ over a field $K$ of characteristic zero. We call a polynomial $p(X_n)$ symmetric in these Lie algebras if $p(x_1,\ldots,x_n)=p(x_{π(1)},\ldots,x_{π(n)})$ for each element $π$ of the symmetric group $S_n$. The sets $L_n^{S_n}$, $F_n^{S_n}$, and $L_{n,c}^{S_n}$ of symmetric polynomials coincide with the algebras of invariants of the group $S_n$ in $L_{n}$, $F_{n}$, and $L_{n,c}$, respectively. We determine the groups $\text{Inn}(F_{n}^{S_n})$ and $\text{Inn}(L_{n,c}^{S_n})$ of inner automorphisms of the algebras $F_{n}^{S_n}$ and $L_{n,c}^{S_n}$, respectively. In particular, we obtain the descriptions of the groups $\text{Aut}(L_{2}^{S_2})$, $\text{Aut}(F_{2}^{S_2})$, and $\text{Aut}(L_{2,c}^{S_2})$ of all automorphisms of the algebras $L_{2}^{S_2}$, $F_{2}^{S_2}$, and $L_{2,c}^{S_2}$, respectively.
We describe the groups of inner and outer automorphisms of the free metabelian nilpotent Lie algebra of finite rank over a field of characteristic 0. To obtain this result we first describe the groups of inner and continuous outer automorphisms of the completion with respect to the formal power series topology of the free metabelian Lie algebra finite rank.
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.
In this paper, we study the structure and representability of the automorphism group functor of the N=4 Lie conformal superalgebra over an algebraically closed field k of characteristic zero.
Let ${\mathfrak g}$ be a finite dimensional simple Lie algebra over an algebraically closed field $K$ of characteristic $0$. A linear map $φ:{\mathfrak g}\to {\mathfrak g}$ is called a local automorphism if for every $x$ in ${\mathfrak g}$ there is an automorphism $φ_x$ of ${\mathfrak g}$ such that $φ(x)=φ_x(x)$. We prove that a linear map $φ:{\mathfrak g}\to {\mathfrak g}$ is local automorphism if and only if it is an automorphism or an anti-automorphism.
Let F_m=F_m(var(sl(2,K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl(2,K) over a field K of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion of F_m with respect to the formal power series topology. Our results are more precise for m=2 when F_2 is isomorphic to the Lie algebra L generated by two generic traceless 2 x 2 matrices. We give a complete description of the group of inner automorphisms of the completion of L. As a consequence we obtain si…
In this paper, a simple Lie algebra, referred to as the completed Witt Lie algebra, is introduced. Its derivation algebra and automorphism group are completely described. As a byproduct, it is obtained that the first cohomology group of this Lie algebra with coefficients in its adjoint module is trivial. Furthermore, we completely determine the conjugate classes of this Lie algebra under its automorphism group, and also obtain that this Lie algebra does not contain any nonzero ad-locally finite element.
Let L be the free m-generated metabelian nilpotent of class c Lie algebra over a field of characteristic 0. An automorphism f of L is called normal if f(I)=I for every ideal I of the algebra L. Such automorphisms form a normal subgroup N(L) of Aut(L) containing the group of inner automorphisms. We describe the group of normal automorphisms of L and the quotient group of Aut(L) modulo N(L).
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.
For a Lie algebra $\mathfrak g$ related to a quantum torus, we compute its automorphisms, derivations and universal central extension. This Lie algebra $\mathfrak g$ is isomorphic to a subalgebra of the Lie algebra of derivations over the quantum torus, and moreover contains a subalgebra isomorphic to the centerless higher rank Virasoro algebra.
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 …
Let $0 \to A \to L \to B \to 0$ be a short exact sequence of Lie algebras over a field $F$, where $A$ is abelian. We show that the obstruction for a pair of automorphisms in $\Aut(A) \times \Aut(B)$ to be induced by an automorphism in $\Aut(L)$ lies in the Lie algebra cohomology $\Ha^2(B;A)$. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in $\Aut\big(L_{n,2}^{(1)}\big) \times \Aut\big(L_{n,2}^{ab}\big)$ to be induced by an au…
Let $X$ be a finite connected poset and $K$ a field. We give a full description of the Lie automorphisms of the incidence algebra $I(X,K)$. In particular, we show that they are in general not proper.
Suppose that a Lie algebra $L$ admits a finite Frobenius group of automorphisms $FH$ with cyclic kernel $F$ and complement $H$ of order 2, such that the fixed-point subalgebra of $F$ is trivial and the fixed-point subalgebra of $H$ is metabelian. Then the derived length of $L$ is bounded by a constant.
This paper addresses several structural aspects of the insertion-elimination algebra, a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras, the automorphism group, the derivation group, and a generating set for the insertion-elimination Lie algebra. Many parts of the results are stated for a more general class of Lie algebras and reproduce results for the generalized Virasoro algebras.
Let $\BB$ be the Lie algebra of Block type with basis $\{L_{\a,i}|\,\a,i\in\Z, i\geq0\}$ and relations $[L_{\a,i},L_{\b,j}]=\left((\a-1)(j+1)-(\b-1)(i+1)\right)L_{\a+\b,i+j}$. In the present paper, the derivation algebra and the automorphism group of $\BB$ are explicitly described. In particular, it is shown that the outer derivation space is 1-dimensional and the inner automorphism group of $\BB$ is trivial.
We implement two algorithms in MATHEMATICA for classifying automorphisms of lower-dimensional non-commutative Lie algebras. The first algorithm is a brute-force approach whereas the second is an evolutionary strategy. These algorithms are delivered as the MATHEMATICA package cwsAutoClass. In order to facilitate the application of this package to symmetry Lie algebras of differential equations, we also provide a package, cwsLieSymTools, for manipulating finite-dimensional Lie algebras of vector fields. In particular, this package allows the computations of Lie brackets, structure constants, and…
Suppose that a Lie type algebra L over a field K admits a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H such that the fixed-point subalgebra of F is trivial and the fixed-point subalgebra of H is nilpotent of class c. If the ground field K contains a primitive n-th root of unity, then L is nilpotent and the nilpotency class of L is bounded in terms of |H| and c. The result extends the known theorem of Khukhro, Makarenko and Shumyatsky on Lie algebras with metacyclic Frobenius group of automorphisms.
Let R be a Lie nilpotent algebra of index k over a field K of characteristic zero. If G is an n-element subgroup of Aut(R) of the K-automorphisms, then we prove that R is right integral over Fix(G) of degree n^k. In the presence of a primitive n-th root of unity e in K, for a K-automorphism d in Aut(R) with d^n=id, we prove that the skew polynomial algebra R[w,d] is right integral of degree n^k over Fix(d)[w^n].
Let $(\mathfrak{g},ω)$ be a finite-dimensional non-Lie complex $ω$-Lie algebra. We study the derivation algebra $Der(\mathfrak{g})$ and the automorphism group $Aut(\mathfrak{g})$ of $(\mathfrak{g},ω)$. We introduce the notions of $ω$-derivations and $ω$-automorphisms of $(\mathfrak{g},ω)$ which naturally preserve the bilinear form $ω$. We show that the set $Der_ω(\mathfrak{g})$ of all $ω$-derivations is a Lie subalgebra of $Der(\mathfrak{g})$ and the set $Aut_ω(\mathfrak{g})$ of all $ω$-automorphisms is a subgroup of $Aut(\mathfrak{g})$. For any 3-dimensional and 4-dimensional nontrivial $ω$-L…
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 …
Many important quantum algebras such as quantum symplectic space, quantum Euclidean space, quantum matrices, $q$-analogs of the Heisenberg algebra and the quantum Weyl algebra are semi-commutative. In addition, enveloping algebras $U(L_+)$ of even Lie color algebras are also semi-commutative. In this paper, we generalize work of Montgomery and examine the $X$-inner automorphisms of such algebras. The theorems and examples in our paper show that for algebras $R$ of this type, the non-identity $X$-inner automorphisms of $R$ tend to have infinite order. Thus if $G$ is a finite group of automorphi…
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 prove that every $2$-local automorphism on a finite-dimensional semi-simple Lie algebra $\mathcal{L}$ over an algebraically closed field of characteristic zero is an automorphism. We also show that each finite-dimensional nilpotent Lie algebra $\mathcal{L}$ with $\dim \mathcal{L}\geq 2$ admits a $2$-local automorphism which is not an automorphism.
For a real semisimple Lie algebra, we consider its automorphism group quotient by its identity component. This is known as the outer automorphism group. In this article, we compute the outer automorphism groups of all real semisimple Lie algebras.
We introduce the concept of a double automorphism of an A-graded Lie algebra L. Roughly, this is an automorphism of L which also induces an automorphism of the group A. It is clear that the set of all double automorphisms of L forms a subgroup in Aut(L). In the present paper we prove several nilpotency criteria for a graded Lie algebra admitting a finite group of double automorphisms. We also give an application of our results to groups admitting a Frobenius group of automorphisms.
In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras -- subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and spin groups. We study the corresponding Lie algebras. Some of the results can be reformulated for the case of more general algebras -- graded central simple algebras or graded central simple alge…
We prove that every monomorphism of the Lie algebra $\ggu_n$ of unitriangular derivations of the polynomial algebra $P_n=K[x_1,..., x_n]$ is an automorphism.
Each matrix A in GL_n(Z) naturally defines an automorphism f of the free r-step nilpotent Lie algebra on n generators. We study the relationship between the matrix A and the eigenvalues and rational invariant subspaces for f. We give applications to the study of Anosov automorphisms.
We prove that the group of automorphisms of the category of free modules over Noterian rings is semi-inner and present also a short proof of a similar result for the variery of Lie algebras.
Let $L_n$ be the free metabelian Leibniz algebra generated by the set $X_n=\{x_1,\ldots,x_n\}$ over a field $K$ of characteristic zero. This is the free algebra of rank $n$ in the variety of solvable of class $2$ Leibniz algebras. We call an element $s(X_n)\in L_n$ symmetric if $s(x_{σ(1)},\ldots,x_{σ(n)})=s(x_1,\ldots,x_n)$ for each permutation $σ$ of $\{1,\ldots,n\}$. The set $L_n^{S_n}$ of symmetric polynomials of $L_n$ is the algebra of invariants of the symmetric group $S_n$. Let $K[X_n]$ be the usual polynomial algebra with indeterminates from $X_n$. The description of the algebra $K[X_n…
We discuss the group of automorphisms of a general MR-algebra. We develop several functors between implication algebras and cubic algebras. These allow us to generalize the notion of inner automorphism. We then show that this group is always isomorphic to the group of inner automorphisms of a filter algebra.
We give a detailed description of the algebraic group Aut(g) of automorphisms of a simple finite dimensional Lie superalgebra g over an algebraically closed field k of characteristic 0. We also give a description of the group of automorphisms of the k-Lie superalgebra $g \otimes_k R$ whenever R is a Noetherian domain with trivial Picard group.
We prove that the group of automorphisms of the Lie algebra $\Der_K (P_n)$ of derivations of a polynomial algebra $P_n=K[x_1,..., x_n]$ over a field of characteristic zero is canonically isomorphic to the the group of automorphisms of the polynomial algebra $P_n$.
We prove that a linear mapping on the algebra \(\mathfrak{sl}_n\) of all trace zero complex matrices is a local automorphism if and only if it is an automorphism or an anti-automorphism. We also show that a linear mapping on a simple Leibniz algebra of the form \(\mathfrak{sl}_n\dot +\mathcal{I}\) is a local automorphism if and only if it is an automorphism. We give examples of finite-dimensional nilpotent Lie algebras \(\mathcal{L}\) with \(\dim \mathcal{L} \geq 3\) which admit local automorphisms which are not automorphisms.
In this paper, we study the derivations, central extensions and the automorphisms of the infinite-dimensional Lie algebra W which appeared in [8] and Dong-Zhang's recent work [22] on the classification of some simple vertex operator algebras.
We describe the structure of the algebraic group of automorphisms of all simple finite dimensional Lie superalgebras. Using this and étale cohomology considerations, we list all different isomorphism classes of the corresponding twisted loop superalgebras.
In algebraic geometry over a variety of universal algebras $Θ$, the group $Aut(Θ^{0})$ of automorphisms of the category $Θ^{0}$ of finitely generated free algebras of $Θ$ is of great importance. In this paper, semi-inner automorphisms are defined for the categories of free (semi)modules and free Lie modules; then, under natural conditions on a (semi)ring, it is shown that all automorphisms of those categories are semi-inner. We thus prove that for a variety $_{R}\mathcal{M}$ of semimodules over an IBN-semiring $R$ (an IBN-semiring is a semiring analog of a ring with IBN), all automorphisms of …
We prove that the group of automorphisms of the Lie algebra $\Der_K (Q_n)$ of derivations of the field of rational functions $Q_n=K(x_1,..., x_n)$ over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the $K$-algebra $Q_n$.
