Derivations Of Algebras Research Articles

Lie higher derivations of incidence algebras

Lie higher derivations of incidence algebras

A commutative algebra approach to multiplicative Hom-Lie algebras

Let be a finite-dimensional complex Lie algebra and be the affine variety of all multiplicative Hom-Lie algebras on . We use a method of computational ideal theory to describe , showing that consists of two 1-dimensional and one 3-dimensional irreducible components and for . We construct a new family of multiplicative Hom-Lie algebras on the Heisenberg Lie algebra and characterize the affine varieties and . We also study the derivation algebra of a multiplicative Hom-Lie algebra D on and, under some hypotheses on D, we prove that the Hilbert series is a rational function.

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.

Holomorphic anomalies, fourfolds and fluxes

We investigate holomorphic anomalies of partition functions underlying string compactifications on Calabi-Yau fourfolds with background fluxes. For elliptic fourfolds the partition functions have an alternative interpretation as elliptic genera of N = 1 supersymmetric string theories in four dimensions, or as generating functions for relative, genus zero Gromov-Witten invariants of fourfolds with fluxes. We derive the holomorphic anomaly equations by starting from the BCOV formalism of topological strings, and translating them into geometrical terms. The result can be recast into modular and elliptic anomaly equations. As a new feature, as compared to threefolds, we find an extra contribution which is given by a gravitational descendant invariant. This leads to linear terms in the anomaly equations, which support an algebra of derivatives mapping between partition functions of the various flux sectors. These geometric features are mirrored by certain properties of quasi-Jacobi forms. We also offer an interpretation of the physics from the viewpoint of the worldsheet theory, and comment on holomorphic anomalies at genus one.

On complete Leibniz algebras

This paper is devoted to the so-called complete Leibniz algebras. We construct some complete Leibniz algebras with complete radical and prove that the direct sum of complete Leibniz algebras is also complete. It is known that a Lie algebra with a complete ideal is split. We discuss the analogs of this result for the Leibniz algebras and show that it is true for some special classes of Leibniz algebras. Finally, we consider derivations of Leibniz algebras and present some classes of Leibniz algebras which are not complete, since they admit outer derivation.

The hom-associative Weyl algebras in prime characteristic

We introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed from associative algebras by a general ``twisting'' procedure. Then, with the help of these results, we determine the commuter, center, nuclei, and set of derivations of the first hom-associative Weyl algebras. We also classify them up to isomorphism, and show, among other things, that all nonzero endomorphisms on them are injective, but not surjective. Last, we show that they can be described as a multi-parameter formal hom-associative deformation of the first associative Weyl algebra, and that this deformation induces a multi-parameter formal hom-Lie deformation of the corresponding Lie algebra, when using the commutator as bracket.

Classical Radicals of Invariants of Algebraic Skew Derivations

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

Derivation of the Optimal Solution for the Economic Production Quantity Model with Planned Shortages without Derivatives

In this paper, we study a reformulation of the Economic Production Quantity (EPQ) problem. We study a more general version of the problem first and derive the conditions for an optimal solution, as well as the optimal solution itself, all without using derivatives. Then, we apply the approach to the reformulated EPQ problem. This version of the EPQ problem has been tackled by a number of researchers, wherein they have derived the conditions for the optimal solution and proposed algebraic derivations. However, their derivations for the conditions, as well as the optimal solution, have been shown to be questionable. Other than being questionable, the existing approaches are so complicated that they defeat the purpose of simplifying the optimization by using a derivative-free approach. We propose a correct and more succinct, much less complicated approach to derive the conditions and the optimal solution without using derivatives.

On special subalgebras of derivations of Leibniz algebras

Our aim in this work is to study the central derivations of Leibniz algebras and investigate the properties of Leibniz algebras by comparing the set of central derivations with the inner derivations. We prove that, the set of all central derivations of a Leibniz algebra with non-trivial center coincide with the set of all inner derivations if and only if the Leibniz algebra is metabelian. In addition, we will show, by examples, that some statements hold for arbitrary Lie algebras, but does not hold for some Leibniz algebras.

On a structure of the one-loop divergences in 4D harmonic superspace sigma-model

We study the quantum structure of four-dimensional {{mathcal {N}}}=2 superfield sigma-model formulated in harmonic superspace in terms of the omega-hypermultiplet superfield omega . The model is described by harmonic superfield sigma-model metric g_{ab}(omega ) and two potential-like superfields L^{++}_{a}(omega ) and L^{(+4)}(omega ). In bosonic component sector this model describes some hyper-Kähler manifold. The manifestly {{mathcal {N}}}=2 supersymmetric covariant background-quantum splitting is constructed and the superfield proper-time technique is developed to calculate the one-loop effective action. The one-loop divergences of the superfield effective action are found for arbitrary g_{ab}(omega ), L^{++}_{a}(omega ), L^{(+4)}(omega ), where some specific analogy between the algebra of covariant derivatives in the sigma-model and the corresponding algebra in the {{mathcal {N}}}=2 SYM theory is used. The component structure of divergences in the bosonic sector is discussed.

Inner derivations of finite dimensional dendriform algebras

Inner derivations of finite dimensional dendriform algebras

On n-Lie derivations of triangular algebras

On n-Lie derivations of triangular algebras

Stability of hyper homomorphisms and hyper derivations in complex Banach algebras

<abstract><p>In this paper, we introduce the concept of hyper homomorphisms and hyper derivations in Banach algebras and we establish the stability of hyper homomorphisms and hyper derivations in Banach algebras for the following 3-additive functional equation:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} g(x_1+x_2, y_1+y_2, z_1+z_2) = \sum\limits_{i, j, k = 1}^2 g(x_i, y_j, z_k). \end{align*} $\end{document} </tex-math></disp-formula></p> </abstract>

Jordan {g,h}-derivations of unital algebras

Jordan {g,h}-derivations of unital algebras

The algebra of derivations of quasi-modular forms from mirror symmetry

We study moduli spaces of mirror non-compact Calabi-Yau threefolds enhanced with choices of differential forms. The differential forms are elements of the middle dimensional cohomology whose variation is described by a variation of mixed Hodge structures which is equipped with a flat Gauss-Manin connection. We construct graded differential rings of special functions on these moduli spaces and show that they contain rings of quasi-modular forms. We show that the algebra of derivations of quasi-modular forms can be obtained from the Gauss--Manin connection contracted with vector fields on the enhanced moduli spaces. We provide examples for this construction given by the mirrors of the canonical bundles of $\mathbb{P}^2$ and $\mathbb{F}_2$.

Configuration analysis method and geometric interpretation of UUVs cooperative localization based on error ellipse

Unmanned underwater vehicles (UUVs) share information through underwater acoustic communication to realize cooperative localization, which can expand the sensing range and accomplish complex tasks, e.g., underwater exploration and reconnaissance, marine engineering, and underwater operations. However, due to the lack of underwater observation information, the configuration will directly affect the accuracy of UUVs cooperative localization. The former researches are based on algebraic derivation to analyze the observability of UUVs configuration, which is not intuitive and convenient in practical application. A new analysis method based on error ellipse for UUVs configuration and its geometric interpretation are proposed. On one hand, it proves the consistency of error ellipse index with traditional Fisher Information Matrix (FIM) and Cramer Rao lower bound (CRLB) index. On the other hand, it improves the intuitiveness and convenience for analyzing the observability of UUVs configuration. In order to validate the effectiveness of the proposed method, three UUVs are simulated, including one slave UUV and two master UUVs. Five different configurations of UUVs with separation angles 30°, 60°, 90°, 120°and 150° are selected. The experimental results show that the error ellipse index can characterize the accuracy of UUVs cooperative localization, and improve the intuitiveness of the observability of UUVs configurations.

On derivations of the Jordan triple product in classical Cartan factors

Let U be a classical Cartan factor of any of the types In, IIn, IIIn or IVn with dim⁡(U)=∞, and let δ:U→U be a map (neither linearity nor continuity of δ is assumed) that satisfies the functional equation δ{aba}={δ(a)ba}+{aδ(b)a}+{abδ(a)}, (a,b∈U), where {abc} stands for the Jordan triple product in U. We prove that δ is additive and homogeneous over the field Q of rational numbers. Moreover δ is real-linear if and only if it is continuous. This gives a new description of the real Banach Lie algebra of derivations of U.

Cohomology and deformations of BiHom-Lie conformal algebras

In this paper, we introduce the notion of BiHom-Lie conformal algebras and develop the cohomology theory.Furthermore, we discuss the deformations of regular BiHom-Lie conformal algebras and present some applications. Finally, we introduce derivations of BiHom-Lie conformal algebras and study their properties.

Jordan *-triple derivations on the exceptional Cartan factors

Let U be any of the two exceptional Cartan factors V and VI and let δ:U→U be a Jordan *-triple derivation of U, that is, a map (neither linearity nor continuity of δ is assumed) that satisfies the functional equation δ{ab⁎a}={δ(a)b⁎a}+{aδ(b)⁎a}+{ab⁎δ(a)}, (a,b∈U), where (a,b)↦{ab⁎a} stands for the Jordan triple product in U. We give an explicit representation of δ as certain multipliers on U and prove that δ automatically is a continuous real linear map on U. This gives a new description of the real Banach Lie algebra of Jordan triple derivations of U.

Isomorphisms and derivations of partial flag incidence algebras

Let [Formula: see text] and [Formula: see text] be finite posets and [Formula: see text] a commutative unital ring. In the case where [Formula: see text] is indecomposable, we prove that the [Formula: see text]-linear isomorphisms between partial flag incidence algebras [Formula: see text] and [Formula: see text] are exactly those induced by poset isomorphisms between [Formula: see text] and [Formula: see text]. We also show that the [Formula: see text]-linear derivations of [Formula: see text] are trivial.