Let $A$ be an algebra over a field $K$ of characteristic zero, let $\d_1, >..., \d_s\in \Der_K(A)$ be {\em commuting locally nilpotent}
$K$-derivations such that $\d_i(x_j)=\d_{ij}$, the Kronecker delta, for some elements $x_1,..., x_s\in A$. A set of algebra generators for the algebra $A^\d:= \cap_{i=1}^s\ker (\d_i)$ is found {\em explicitly} and a set of {\em defining relations} for the algebra $A^\d$ is described. Similarly, given a set $\s_1, ..., \s_s\in \Aut_K(A)$ of {\em commuting}
$K$-automorphisms of the algebra $A$ such that the maps $\s_i-{\rm id_A}$ are {\em locally nilpotent} and $\s_i (x_j)=x_j+\d_{ij}$, for some elements $x_1,..., x_s\in A$. A set of algebra generators for the algebra $A^\s:=\{a\in A | \s_1(a)=... =\s_s(a)=a\}$ is found {\em explicitly} and a set of defining relations for the algebra $A^\s$ is described. In general, even for a {\em finitely generated noncommutative} algebra $A$ the algebras of invariants $A^\d $ and $A^\s $ are {\em not} finitely generated, {\em not} (left or right) Noetherian and {\em does not} satisfy finitely many defining relations (see examples). Though, for a {\em finitely generated commutative} algebra $A$ {\em always} the {\em opposite} is true. The derivations (or automorphisms) just described appear often in may different situations after (possibly) a localization of the algebra $A$.
Subsets of a matrix algebra over a field that are invariant under conjugation and contain the linear span of each two of their commuting elements are described. They obviously include the subsets of diagonalizable and nilpotent matrices. In the paper, the case of an algebraically closed field is considered. The problem is easily reduced to description of subsets of diagonalizable matrices and subsets of nilpotent matrices with the given properties. So, among diagonalizable matrices, there are four of such subsets. As for the nilpotent case, it is proved that the subset should be defined by the…
Let $\mathfrak{g}$ be the $p$-dimensional Witt algebra over an algebraically closed field $k$ of characteristic $p>3$. Let $\mathscr{N}={x\in\ggg\mid x^{[p]}=0}$ be the nilpotent variety of $\mathfrak{g}$, and $\mathscr{C}(\mathscr{N}):=\{(x,y)\in \mathscr{N}\times\mathscr{N}\mid [x,y]=0\}$ the nilpotent commuting variety of $\mathfrak{g}$. As an analogue of Premet's result in the case of classical Lie algebras [A. Premet, Nilpotent commuting varieties of reductive Lie algebras. Invent. Math., 154, 653-683, 2003.], we show that the variety $\mathscr{C}(\mathscr{N})$ is reducible and equidimens…
In this paper is devoted to nilpotent finite-dimensional evolution algebras E with $dimE^2 = dimE-1$. We described Lie algebras associated with evolution algebras whose nilindex is maximal. Moreover, in terms of this Lie algebra we fully construct nilpotent evolution algebra with maximal index of nilpotency. Furthermore, this result allowed us fully characterize all local and 2-local derivations of the considered evolution algebras. All automorphisms and local automorphisms of the nilpotent evolution algebras with maximal nilindex are found.
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.
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).
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.
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].
We consider a family of 2-step nilpotent Lie algebras associated to uniform complete graphs on odd number of vertices. We prove that the symmetry group of such a graph is the holomorph of the additive cyclic group $\Z_n$. Moreover, we prove that the (Lie) automorphism group of the corresponding nilpotent Lie algebra contains the dihedral group of order $2n$ as a subgroup.
In this paper we study rank two commuting ordinary differential operators with polynomial coefficients and the orbit space of the automorphisms group of the first Weyl algebra on such operators. We prove that for arbitrary fixed spectral curve of genus one the space of orbits is infinite. Moreover, we prove in this case that for for any $n\ge 1$ there is a pair of self-adjoint commuting ordinary differential operators of rank two $L_4=(\partial_x^2+V(x))^2+W(x)$, $L_{6}$, where $W(x),V(x)$ are polynomials of degree $n$ and $n+2$. We also prove that there are hyperelliptic spectral curves with …
The Jordan type of a nilpotent matrix is the partition giving the sizes of its Jordan blocks. We study pairs of partitions $(P,Q)$, where $Q={\mathcal Q}(P)$ is the Jordan type of a generic nilpotent matrix A commuting with a nilpotent matrix B of Jordan type $ P$. T. Košir and P. Oblak have shown that $Q$ has parts that differ pairwise by at least two. Such partitions, which are also known as "super distinct" or "Rogers-Ramanujan", are exactly those that are stable or "self-large" in the sense that ${\mathcal Q}(Q)=Q$. In 2012 P. Oblak formulated a conjecture concerning the cardinality of the…
Let $D$ be a division algebra such that $D\t D^o$ is a Noetherian algebra, then any division subalgebra of $D$ is a {\em finitely generated} division algebra. Let $\D $ be a finite set of commuting derivations or automorphisms of the division algebra $D$, then the group $\Ev (\D)$ of common eigenvalues (i.e. {\em weights}) is a {\em finitely generated abelian} group. Typical examples of $D$ are the quotient division algebra ${\rm Frac} (\CD (X))$ of the ring of differential operators $\CD (X)$ on a smooth irreducible affine variety $X$ over a field $K$ of characteristic zero, and the quotient …
Automorphisms of algebras $R$ from a very large axiomatic class of quantum nilpotent algebras are studied using techniques from noncommutative unique factorization domains and quantum cluster algebras. First, the Nakayama automorphism of $R$ (associated to its structure as a twisted Calabi-Yau algebra) is determined and shown to be given by conjugation by a normal element, namely, the product of the homogeneous prime elements of $R$ (there are finitely many up to associates). Second, in the case when $R$ is connected graded, the unipotent automorphisms of $R$ are classified up to minor excepti…
We show that the set of all commuting probabilities in finite rings is a subset of the set of all commuting probabilities in finite nilpotent groups of class $\le2$. We believe that these two sets are equal; we prove they are equal, when restricted to groups and rings with odd number of elements.
We establish a Primitive Element Theorem for fields equipped with several commuting operators such that each of the operators is either a derivation or an automorphism. More precisely, we show that for every extension $F \subset E$ of such fields of zero characteristic such that $\bullet$ $E$ is generated over $F$ by finitely many elements using the field operations and the operators, $\bullet$ every element of $E$ satisfies a nontrivial equation with coefficient in $F$ involving the field operations and the operators, $\bullet$ the action of the operators on $E$ is irredundant there exists an…
We investigate the automorphism group of the quantised enveloping algebra U of the positive nilpotent part of certain simple complex Lie algebras g in the case where the deformation parameter q \in \mathbb{C}^* is not a root of unity. Studying its action on the set of minimal primitive ideals of U we compute this group in the cases where g=sl_3 and g=so_5 confirming a Conjecture of Andruskiewitsch and Dumas regarding the automorphism group of U. In the case where g=sl_3, we retrieve the description of the automorphism group of the quantum Heisenberg algebra that was obtained independently by A…
The principal filtration of the infinite-dimensional odd Contact Lie superalgebra over a field of characteristic $p>2$ is proved to be invariant under the automorphism group by investigating ad-nilpotent elements and determining certain invariants such as subalgebras generated by some ad-nilpotent elements. Then, it is proved that two automorphisms coincide if and only if they coincide on the -1 component with respect to the principal grading. Finally, all the odd Contact superalgebras are classified up to isomorphisms.
Let $\A$ be an algebra and $σ$ an automorphism of $\A$. A linear map $d$ of $\A$ is called a $σ$-derivation of $\A$ if $d(xy) = d(x)y + σ(x)d(y)$, for all $x, y \in \A$. A bilinear map $D: \A \times \A \to \A$ is said to be a $σ$-biderivation of $\A$ if it is a $σ$-derivation in each component. An additive map $Θ$ of $\A$ is $σ$-commuting if it satisfies $Θ(x)x - σ(x)Θ(x) = 0$, for all $x \in \A$. In this paper, we introduce the notions of inner and extremal $σ$-biderivations and of proper $σ$-commuting maps. One of our main results states that, under certain assumptions, every $σ$-biderivatio…
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.
The type and several invariant subspaces related to the upper annihilating series of finite-dimensional nilpotent evolution algebras are introduced. These invariants can be easily computed from any natural basis. Some families of nilpotent evolution algebras, defined in terms of a nondegenerate symmetric bilinear form and some commuting symmetric diagonalizable endomorphisms relative to the form, are explicitly constructed. Both the invariants and these families are used to review and complete the classification of nilpotent evolution algebras up to dimension five over algebraically closed fie…
In this work we find necessary and sufficient conditions for a free nilpotent or a free metabelian nilpotent Lie algebra to be endowed with an ad-invariant metric. For such nilpotent Lie algebras admitting an ad-invariant metric the corresponding automorphisms groups are studied.
The present paper is devoted to local and 2-local derivations and automorphism of complex finite-dimensional simple Leibniz algebras. We prove that all local derivations and 2-local derivations on a finite-dimensional complex simple Leibniz algebra are automatically derivations. We show that nilpotent Leibniz algebras as a rule admit local derivations and 2-local derivations which are not derivations. Further we consider automorphisms of simple Leibniz algebras. We prove that every 2-local automorphism on a complex finite-dimensional simple Leibniz algebra is an automorphism and show that nilp…
We determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field $\mathbb K$ of characteristic zero having an analogous family of flags of subalgebras as the four-dimensional non-Lie binary Lie algebras, and hence can be considered as the closest relatives of binary Lie algebras. These algebras are extensions of $\mathbb K$ by the 3-dimensional nilpotent Lie algebra and at the same time extensions of a two-dimensional Lie algebra by a two-dimensional abelian algebra. We describe their groups of automorphisms as extensions of a subgroup of the group of …
We consider the structure of groups and algebras that can be represented as automorphisms or derivations of distributive products -- which includes nonassociative rings, modules, forms, and commutation of groups and nonassociative loops. In many ways this is too broad to be possible, for instance even naive structure, such as inner automorphisms and derivations, are undefined for general bilinear maps. The appeal for working in this generality is broad applicability. We introduce fundamental structures and prove several results akin to ones in ring theory, including theorems of Morita and of S…
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.
We describe two algorithms for finding representatives of the nilpotent orbits of a theta-group. The algorithms have been implemented in the computer algebra system GAP (inside the package SLA). We comment on their performance. We apply the algorithms to study the nilpotent orbits of theta-groups, where theta is an N-regular automorphism of a simple Lie algebra of exceptional type.
In the present paper, we determine the group of automorphisms of pseudo $H$-type Lie algebras, which are two-step nilpotent Lie algebras closely related to the Clifford algebras $\Cl(\mathbb R^{r,s})$.
The group of automorphisms of the first Weyl algebra acts on commuting ordinary differential operators with polynomial coefficient. In this paper we prove that for fixed generic spectral curve of genus two the set of orbits is infinite.
Let P be a free Poisson algebra in two variables over a field of characteristic zero. We prove that the automorphisms of P are tame and that the locally nilpotent derivations of P are triangulable.
We obtain a recurrent and monotone method for constructing and classifying nilpotent Lie algebras by means of successive central extensions. It consists in calculating the second cohomology of an extendable nilpotent Lie algebra with the subsequent study of the orbit space geometry of the automorphism group action on Grassmannians defined in terms of the second cohomology of the extendable nilpotent Lie algebra. Such a geometric method allows to classify some classes of nilpotent Lie algebras. The concept of a rigid central extension is introduced.
In the present paper, every evolution algebra is endowed with Banach algebra norm. This together with the description of derivations and automorphisms of nilpotent evolution algebras, allows to investigated the set $\exp(Der(E))$. Moreover, it is proved that $\exp(Der(E))$ is a normal subgroup of $Aut(E)$, and its corresponding index is calculated.
