Characterizations of additive -Lie derivations on unital algebras

UDC 512.5 Let be a commutative ring with unity and be a unital algebra over (or field ).An -linear map is called a Lie derivation on if holds for all For scalar an additive map is called an additive -Lie derivation on if where holds for all In the present paper, under certain assumptions on it is shown that every Lie derivation (resp., additive -Lie derivation) on is of standard form, i.e., where is an additive derivation on and is a mapping vanishing at with in Moreover, we also characterize the additive -Lie derivation for by its action at zero product in a unital algebra over

Lie Derivatives and Ricci Tensor on Real Hypersurfaces in Complex Two-plane Grassmannians

On a real hypersurface M in a complex two-plane Grassmannian G2() we have the Lie derivation and a differential operator of order one associated with the generalized Tanaka–Webster connection . We give a classification of real hypersurfaces M on G2() satisfying , where ξ is the Reeb vector field on M and S the Ricci tensor of M.

Lie Derivations on Trivial Extension Algebras

In this paper we provide some conditions under which a Lie derivation on a trivial extension algebra is proper, that is, it can be expressed as a sum of a derivation and a center valued map vanishing at commutators. We then apply our results for triangular algebras. Some illuminating examples are also included.

N-Derivations for Finitely Generated Graded Lie Algebras

The N-derivation is a natural generalization of derivation and triple derivation. Let [Formula: see text] be a finitely generated Lie algebra graded by a finite-dimensional Cartan subalgebra. In this paper, a sufficient condition for the Lie N-derivation algebra of [Formula: see text] coinciding with the Lie derivation algebra of [Formula: see text] is given. As applications, any N-derivation of the Schrödinger-Virasoro algebra, generalized Witt algebra, Kac-Moody algebra or their Borel subalgebra is a derivation.

Synchronous Control of Hyper-Chaotic System Based on Inverse-System Design

Hyper-chaotic synchronization can be applied in vast areas of physics and engineering science, and especially in secure communication. A hyper-chaotic synchronization method based on inverse-system theory is studied. The reversibility and relative degree of the synchronization error function are determined through Lie derivation calculation. A close-loop controller is obtained by synthesizing and designing a pseudo-linear feedback system, which used for estimating output variable synchronization error. Hype-chaotic synchronization of a four dimensional oscillatory system is selected as typical example and the simulation results demonstrate the validity of the method. The research results provide a useful reference for realizing hyper-chaotic synchronization.

Characterizations of Nonlinear Lie Derivations ofB(X)

LetXbe an infinite dimensional Banach space, andΦ:B(X)→B(X)is a nonlinear Lie derivation. Then Φ is the formδ+τwhereδis an additive derivation ofB(X)andτis a map fromB(X)into its centerZB(X), which maps commutators into the zero.

Lie Triple Derivations on Triangular Matrices

Let [Formula: see text] be the algebra of all n × n upper triangular matrices over a commutative unital ring [Formula: see text], and let [Formula: see text] be a 2-torsion free unital [Formula: see text]-bimodule. We show that every Lie triple derivation [Formula: see text] is a sum of a standard Lie derivation and an antiderivation.