Local Automorphisms Research Articles

Local and Linear 2-local Automorphisms of Heisenberg Lie (Super)algebras

Local and Linear 2-local Automorphisms of Heisenberg Lie (Super)algebras

Local Automorphisms and Local Superderivations of Model Filiform Lie Superalgebras

In this paper, we give the forms of local automorphisms (resp. superderivations) of model filiform Lie superalgebra Ln,m in the matrix version. Linear 2-local automorphisms (resp. superderivations) of Ln,m are also characterized. We prove that each linear 2-local automorphism of Ln,m is an automorphism.

Local derivations and local automorphisms on the super Virasoro algebras

This paper aims to study the local derivations, 2-local automorphisms and local automorphisms on the super-Virasoro algebras. The primary focus is to establish that every local derivation of the super-Virasoro algebras is indeed a derivation, and to demonstrate that every local or 2-local automorphism of the super-Virasoro algebras is an automorphism.

Local and 2-local automorphisms of solvable Leibniz algebras with abelian and model nilradicals

In the paper, we describe the automorphisms on the solvable Leibniz algebras with the model or abelian nilradicals, whose complementary spaces are equal to two and one, respectively. We show that any local automorphism on a solvable Leibniz algebra with model nilradical of maximal codimension is an automorphism. We show that solvable Leibniz algebras with abelian nilradicals, which have 1-dimension complementary space, admit local automorphisms which are not automorphisms. Moreover, similar problems concerning 2-local automorphisms of such algebras are investigated. We give examples of solvable Leibniz algebras in which 2-local automorphisms are automorphisms and examples of such algebras which admit 2-local automorphisms which are not automorphisms.

Local automorphisms of complex solvable Lie algebras of maximal rank

This paper is devoted to the descriptions of automorphisms and local automorphisms on complex solvable Lie algebras of maximal rank. First, it is established that any automorphism on a solvable Lie algebra of maximal rank can be represented as a product (composition) of inner, diagonal and graph automorphisms. We apply the description of automorphism to the specification of automorphisms on solvable Lie algebras of maximal rank with abelian nilradical, and to the description of automorphisms of standard Borel subalgebras of complex simple Lie algebras. Based on the representation of an automorphism, it is proved that all local automorphisms on a solvable Lie algebra of maximal rank are global automorphisms. We also present two examples of solvable Lie algebras which are not of maximal rank and have different behaviours of local automorphisms. Namely, the first algebra does not admit pure local automorphisms, while the second algebra admits a local automorphism which is not an automorphism.

Local properties of Virasoro-like algebra

This paper studies local properties of the Virasoro-like algebra. Namely, the 2-local derivations and 2-local automorphisms of the Virasoro-like algebra are considered. It is proved that every 2-local derivation of this Lie algebra is a derivation, and every 2-local automorphism of this Lie algebra is an automorphism.

A CHARACTERIZATION OF DERIVATIONS AND AUTOMORPHISMS ON SOME SIMPLE ALGEBRAS

In the present paper, we study simple algebras, which do not belong to the well-known classes of algebras (associative algebras, alternative algebras, Lie algebras, Jordan algebras, etc.). The simple finite-dimensional algebras over a field of characteristic 0 without finite basis of identities, constructed by Kislitsin, are such algebras. In the present paper, we consider two such algebras: the simple seven-dimensional anticommutative algebra \(\mathcal{D}\) and the seven-dimensional central simple commutative algebra \(\mathcal{C}\). We prove that every local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is a derivation, and every 2-local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also a derivation. We also prove that every local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is an automorphism, and every 2-local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also an automorphism.

Local Derivations and Automorphisms of Direct Sum Null-Filiform Leibniz Algebras

Local Derivations and Automorphisms of Direct Sum Null-Filiform Leibniz Algebras

Local derivations and automorphisms of Cayley algebras

The present paper is devoted to the description of local and 2-local derivations and automorphisms on Cayley algebras over an arbitrary field F. Given a Cayley algebra C with norm n, let C0 be its subspace of trace 0 elements. We prove that the space of all local derivations of C coincides with the Lie algebra {d∈so(C,n)|d(1)=0} which is isomorphic to the orthogonal Lie algebra so(C0,n). Surprisingly, the behavior of 2-local derivations depends on the Cayley algebra being split or division. Every 2-local derivation on the split Cayley algebra is a derivation, so they are isomorphic to the exceptional Lie algebra g2(F) if charF≠2,3. On the other hand, on division Cayley algebras over a field F, the sets of 2-local derivations and local derivations coincide. As a corollary we obtain descriptions of local and 2-local derivations of the seven-dimensional simple non-Lie Malcev algebras over fields of characteristic ≠2,3. Further, we prove that the group of all local automorphisms of C coincides with the group {φ∈O(C,n)|φ(1)=1}. As in the case of 2-local derivations, the behavior of 2-local automorphisms depends on the Cayley algebra being split or division. Every 2-local automorphism on the split Cayley algebra is an automorphism, so they form the exceptional Lie group G2(F) if charF≠2,3. On the other hand, on division Cayley algebras over a field F, the groups of 2-local automorphisms and local automorphisms coincide.

Local and 2-local automorphisms on octonion algebras

Local and 2-local automorphisms on octonion algebras

On 2-local automorphisms of symmetric groups

Let phi be a self-map of the symmetric group S_Omega acting on a countable set Omega . We show in an elementary fashion that if phi has the property that sigma tau is conjugate to phi (sigma )phi (tau ) for every choice of sigma ,tau in S_Omega , then it is necessarily an automorphism or an antiautomorphism of S_Omega . As a corollary to this result, the so-called 2-local automorphisms of S_Omega are also described.

Local and 2-local automorphisms of some solvable Leibniz algebras

In this paper we prove that any local automorphism on the solvable Leibniz algebras with null-filiform and naturally graded non-Lie filiform nilradicals, whose dimension of complementary space is maximal is an automorphism. Furthermore, the same problem concerning 2-local automorphisms of such algebras is investigated and we obtain the analogously results for 2-local automorphisms.

Local finiteness and automorphism groups of low complexity subshifts

AbstractWe prove that for any transitive subshift X with word complexity function $c_n(X)$ , if $\liminf ({\log (c_n(X)/n)}/({\log \log \log n})) = 0$ , then the quotient group ${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$ of the automorphism group of X by the subgroup generated by the shift $\sigma $ is locally finite. We prove that significantly weaker upper bounds on $c_n(X)$ imply the same conclusion if the gap conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of X of range n in terms of word complexity, which may be of independent interest. As an application, we are also able to prove that for any subshift X, if ${c_n(X)}/{n^2 (\log n)^{-1}} \rightarrow 0$ , then $\mathrm {Aut}(X,\sigma )$ is amenable, improving a result of Cyr and Kra. In the opposite direction, we show that for any countable infinite locally finite group G and any unbounded increasing $f: \mathbb {N} \rightarrow \mathbb {N}$ , there exists a minimal subshift X with ${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$ isomorphic to G and ${c_n(X)}/{nf(n)} \rightarrow 0$ .

The Split and Approximate Split Property in 2D Systems: Stability and Absence of Superselection Sectors

The split property of a pure state for a certain cut of a quantum spin system can be understood as the entanglement between the two subsystems being weak. From this point of view, we may say that if it is not possible to transform a state omega via sufficiently local automorphisms (in a sense that we will make precise) into a state satisfying the split property, then the state omega has a long-range entanglement. It is well known that in 1D, gapped ground states have the split property with respect to cutting the system into left and right half-chains. In 2D, however, the split property fails to hold for interesting models such as Kitaev’s toric code. Here we show that this failure is the reason that anyons can exist in that model. There is a folklore saying that the existence of anyons, like in the toric code model, implies long-range entanglement of the state. In this paper, we prove this folklore in an infinite dimensional setting. More precisely, we show that long-range entanglement, in a way that we will define precisely, is a necessary condition to have non-trivial superselection sectors. Anyons in particular give rise to such non-trivial sectors. States with the split property for cones, on the other hand, do not admit non-trivial sectors. A key technical ingredient of our proof is that under suitable assumptions on locality, the automorphisms generated by local interactions can be “approximately factorized.” That is, they can be written as the tensor product of automorphisms localized in a cone and its complement respectively, followed by an automorphism acting near the “boundary” of Lambda , and conjugation with a unitary. This result may be of independent interest. This technique also allows us to prove that the approximate split property, a weaker version of the split property that is satisfied in e.g. the toric code, is stable under applying such automorphisms.

Automorphisms and derivations of finite-dimensional algebras

Let A be a finite-dimensional algebra over a field F with char(F)≠2. We show that a linear map D:A→A satisfying xD(x)x∈[A,A] for every x∈A is the sum of an inner derivation and a linear map whose image lies in the radical of A. Assuming additionally that A is semisimple and char(F)≠3, we show that a linear map T:A→A satisfies T(x)3−x3∈[A,A] for every x∈A if and only if there exist a Jordan automorphism J of A lying in the multiplication algebra of A and a central element α satisfying α3=1 such that T(x)=αJ(x) for all x∈A. These two results are applied to the study of local derivations and local (Jordan) automorphisms. In particular, the second result is used to prove that every local Jordan automorphism of a finite-dimensional simple algebra A (over a field F with char(F)≠2,3) is a Jordan automorphism.

Modification of Poincaré's construction and its application in $CR$-geometry of hypersurfaces in $\mathbf{C}^4$

The modified Poincaré construction (a generalization of Poincaré's homological operator) was earlier used to estimate the dimension of the local automorphism group for an arbitrary germ of a real-analytic hypersurface in $\mathbf{C}^3$. In the present paper we prove the following alternative. For every hypersurface in $\mathbf{C}^4$, this dimension is either infinite or does not exceed $24$. Moreover, $24$ occurs only for a non-degenerate hyperquadric (one of the two). If the hypersurface is $2$-nondegenerate (resp. $3$-non-degenerate) at a generic point, the bound can be improved to $17$ (resp. $20$).

Infinitesimal symmetries of weakly pseudoconvex manifolds

We consider weakly pseudoconvex hypersurfaces with polynomial models in \({\mathbb {C}}^N\) and their symmetry algebras. In the most prominent case of special models, given by sums of squares of polynomials, we give their complete classification. In particular, we prove that such manifolds do not admit any nonlinear symmetries, depending only on complex tangential variables, nor do they admit real or nilpotent linear symmetries. This leads to a sharp 2-jet determination result for local automorphisms. We also give partial results in the general case and a more detailed description of the graded components in complex dimension three. The results also provide an important necessary step for solving the local equivalence problem on such manifolds.

VAUGHT’S CONJECTURE FOR ALMOST CHAINABLE THEORIES

AbstractA structure ${\mathbb Y}$ of a relational language L is called almost chainable iff there are a finite set $F \subset Y$ and a linear order $\,<$ on the set $Y\setminus F$ such that for each partial automorphism $\varphi $ (i.e., local automorphism, in Fraïssé’s terminology) of the linear order $\langle Y\setminus F, <\rangle $ the mapping $\mathop {\mathrm {id}}\nolimits _F \cup \varphi $ is a partial automorphism of ${\mathbb Y}$ . By theorems of Fraïssé and Pouzet, an infinite structure ${\mathbb Y}$ is almost chainable iff the profile of ${\mathbb Y}$ is bounded; namely, iff there is a positive integer m such that ${\mathbb Y}$ has $\leq m$ non-isomorphic substructures of size n, for each positive integer n. A complete first order L-theory ${\mathcal T}$ having infinite models is called almost chainable iff all models of ${\mathcal T}$ are almost chainable and it is shown that the last condition is equivalent to the existence of one countable almost chainable model of ${\mathcal T}$ . In addition, it is proved that an almost chainable theory has either one or continuum many non-isomorphic countable models and, thus, the Vaught conjecture is confirmed for almost chainable theories.

The characterization of 2-local Lie automorphisms of some operator algebras

The characterization of 2-local Lie automorphisms of some operator algebras

Conservative algebras of 2-dimensional algebras, III

Abstract In the present paper we prove that every local and 2-local derivation on conservative algebras of 2-dimensional algebras are derivations. Also, we prove that every local and 2-local automorphism on conservative algebras of 2-dimensional algebras are automorphisms.