Local Derivative Research Articles

Local properties of Schrödinger-Virasoro type Lie algebras and a full diamond thin Lie algebra

Let S V ( z ) be a class of Schrödinger-Virasoro type Lie algebras, where z ∈ Z . In this paper, we show that S V ( z ) is 2-local complete (namely, every 2-local derivation is a derivation) for almost all integers z’s. We also introduce the notion of full diamond thin Lie algebras, and construct an example F from S V ( − 1 ) . We completely determine the derivations of F by combinatoric techniques, and show that F is not 2-local complete.

2-Local derivations of Heisenberg–Virasoro type Lie algebras

Let [Formula: see text] with [Formula: see text] be the Heisenberg–Virasoro type Lie algebras, and [Formula: see text] be a thin Lie algebra with only one diamond. In this paper, we study [Formula: see text]-local derivations on [Formula: see text] and [Formula: see text]. For [Formula: see text], we show that every [Formula: see text]-local derivation is a derivation for almost all parameter pairs [Formula: see text], generalizing some previous results, while for [Formula: see text], we show that it admits infinite many [Formula: see text]-local derivations which are not derivations. Along the way, we also prove that the space of outer derivations of [Formula: see text] is infinite-dimensional.

Local and 2-local derivations on filiform associative algebras

This paper is devoted to the study of local and 2-local derivations of null-filiform, filiform and naturally graded quasi-filiform associative algebras. We prove that these algebras as a rule admit local derivations which are not derivations. We show that filiform and naturally graded quasi-filiform associative algebras admit 2-local derivations which are not derivations and any 2-local derivation of null-filiform associative algebras is a derivation.

Local and 2-local derivations of 𝔰𝔩(2) on any finite-dimensional completely reducible module

This paper determines all local and 2-local derivations of the 3-dimensional complex simple Lie algebra [Formula: see text] on its any finite-dimensional completely reducible module. In particular, the quotient space of the one consisting of all local derivations by the one consisting of all derivations of [Formula: see text] on its [Formula: see text]-dimensional simple module has the dimension [Formula: see text] if [Formula: see text] is odd, and [Formula: see text] otherwise. On the other hand, each 2-local derivation of [Formula: see text] on any finite-dimensional completely reducible module is a derivation.

Local derivations and 2-local Lie derivations of triangular algebras

Local derivations and 2-local Lie derivations of triangular algebras

Local and 2-Local Derivations of Locally Simple Lie Algebras

Local and 2-Local Derivations of Locally Simple Lie Algebras

Description of 2-local derivations on Baer JB-algebras of type I$$_2$$

Description of 2-local derivations on Baer JB-algebras of type I$$_2$$

Biderivations, commuting mappings and 2-local derivations of -graded Lie algebras of maximal class

In Fialowski's classification for algebras of maximal class, there are three Lie algebras of maximal class with one-dimensional homogeneous components: m 0 , L 1 and m 2 . In this paper, we study their biderivations by considering the embedded mapping to derivation algebras. Then we determine commuting mappings on these algebras as an application of biderivations. Finally, local and 2-local derivations for these three algebras are characterized as the given gradings.

Local and 2-local derivations on small dimensional Zinbiel algebras

Local and 2-local derivations on small dimensional Zinbiel algebras

2-Local derivations on the planar Galilean conformal algebra

This paper is devoted to studying 2-local derivations on the planar Galilean conformal algebra. We prove that every 2-local derivation on the planar Galilean conformal algebra is a derivation.

2-Local derivations on the Schrödinger-Virasoro algebra

The present paper is devoted to the study of 2-local derivations on the Schrödinger-Virasoro Algebra, which is an infinite-dimensional Lie algebra with three outer derivations. We prove that all 2-local derivations on the Schrödinger-Virasoro Algebra are derivations.

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 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.

2-Local derivations on Schrödinger algebra in (2 + 1)- dimensional space-time

The present paper is devoted to study 2-Local derivations on the Schrödinger algebra in (2 + 1)-dimensional space-time of Schrödinger Lie groups. We prove that every 2-local derivation on this algebra is a derivation.

2-Local Derivations on the Schrödinger Algebra

2-Local Derivations on the Schrödinger Algebra

Local and 2-Local Derivations of Locally Simple Lie Algebras

In the present paper, we study local and 2-local derivations of the classical locally simple Lie algebras. Firstly, we prove that every local and 2-local derivations on classical locally simple Lie algebra is a derivation. Further, we show that every local derivation of Borel subalgebras of locally simple Lie algebras is a derivation.

Local and 2-Local Derivations of Locally Simple Lie Algebras

In this paper, we study a two-state Hard-Core (HC) model with activity λ0{\lambda >0} on a Cayley tree of order k ≥2{k \ge 2}. It is known that there are λcr{\lambda_{cr}}, λcr0{\lambda_{cr}^0} and λcr'{\lambda_{cr}^'} such that
 
 for λ≤λcr{\lambda \le \lambda_{cr}} this model has a unique Gibbs measure λ0{\lambda >0}, which is translation invariant. The measure λ0{\lambda >0} is extreme for λλcr0{\lambda < \lambda_{cr}^0} and not extreme for λλcr'{\lambda > \lambda_{cr}^'};
 for λλcr{\lambda > \lambda_{cr}} there exist exactly three 2-periodic Gibbs measures, one of which is μ*{\mu^*}, the other two are not translation-invariant and are always extreme.
 
 The extremity of these periodic measures was proved using the maximality and minimality of the corresponding solutions of some equation, which ensures the consistency of these measures. In this paper, we give a brief overview of the known Gibbs measures for the HC-model and an alternative proof of the extremity of 2-periodic measures for k=2,3{k = 2 , 3}. Our proof is based on the tree reconstruction method.

Local and 2-local derivations on octonion algebras

This paper is devoted to study of local and 2-local derivations on octonion algebras. We shall give a general form of local derivations on the real octonion algebra [Formula: see text]. This description implies that the space of all local derivations on [Formula: see text] when equipped with Lie bracket is isomorphic to the Lie algebra [Formula: see text] of all real skew-symmetric [Formula: see text]-matrices. We also consider [Formula: see text]-local derivations on an octonion algebra [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero and prove that every [Formula: see text]-local derivation on [Formula: see text] is a derivation. Further, we apply these results to similar problems for the simple seven-dimensional Malcev algebra. As a corollary, we obtain that the real octonion algebra [Formula: see text] and Malcev algebra [Formula: see text] are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation.

Local properties of Jacobson-Witt algebras

This paper studies local properties of Jacobson-Witt algebras over fields of prime characteristic, i.e., initiates the study on 2-local derivations of Lie algebras of prime characteristic. Let Wn be a simple Jacobson-Witt algebra over a field F of prime characteristic p with |F|≥pn. In this paper, it is shown that every 2-local derivation on Wn is a derivation.