Derivations Of Algebras Research Articles

Generalized derivations with nilpotent values in semiprime rings

Let R be a (semi-) prime ring with extended centroid C, let f(X 1, … , Xk ) be a multilinear polynomial over C in k noncommutative indeterminates which is not central-valued on R and let g be a generalized derivation of R. In this paper, we completely characterize the form of g and the structure of R such that (g(f(x 1, … , xk )) m − γf(x 1, … , xk ) n ) s = 0 for all x 1, … , xk ∈ R, where γ ∈ C and m, n, s are fixed positive integers. Our results naturally improve and generalize the theorems obtained by Huang and Davvaz in [Generalized derivations of rings and Banach algebras, Comm. Algebra (2013); 43, 1188–1194] and the theorems recently obtained by De Filippis et al. in [Generalized derivations with nilpotent, power-central and invertible values in prime and semiprime rings, Comm. Algebra (2019); 47, 3025–3039]. Moreover, we describe a revised version of the theorem obtained by Huang in [On generalized derivations of prime and semiprime rings, Taiwanese J. Math. (2012); 16, 771–776.]

Characterization of Lie-Type Higher Derivations of von Neumann Algebras with Local Actions

Let m and n be fixed positive integers. Suppose that A is a von Neumann algebra with no central summands of type I1, and Lm:A→A is a Lie-type higher derivation. In continuation of the rigorous and versatile framework for investigating the structure and properties of operators on Hilbert spaces, more facts are needed to characterize Lie-type higher derivations of von Neumann algebras with local actions. In the present paper, our main aim is to characterize Lie-type higher derivations on von Neumann algebras and prove that in cases of zero products, there exists an additive higher derivation ϕm:A→A and an additive higher map ζm:A→Z(A), which annihilates every (n−1)th commutator pn(S1,S2,⋯,Sn) with S1S2=0 such that Lm(S)=ϕm(S)+ζm(S)forallS∈A. We also demonstrate that the result holds true for the case of the projection product. Further, we discuss some more related results.

Skew derivations of incidence algebras

Skew derivations of incidence algebras

Anatomy of path integral Monte Carlo: Algebraic derivation of the harmonic oscillator's universal discrete imaginary-time propagator and its sequential optimization.

The direct integration of the harmonic oscillator path integral obscures the fundamental structure of its discrete, imaginary time propagator (density matrix). This work, by first proving an operator identity for contracting two free propagators into one in the presence of interaction, derives the discrete propagator by simple algebra without doing any integration. This discrete propagator is universal, having the same two hyperbolic coefficient functions for all short-time propagators. Individual short-time propagator only modifies the coefficient function's argument, its portal parameter, whose convergent order is the same as the thermodynamic energy. Moreover, the thermodynamic energy can be given in a closed form for any short-time propagator. Since the portal parameter can be systematically optimized by matching the expansion of the product of the two coefficients, any short-time propagator can be optimized sequentially, order by order, by matching the product coefficient's expansion alone, without computing the energy. Previous empirical findings on the convergence of fourth and sixth-order propagators can now be understood analytically. An eight-order convergent short-time propagator is also derived.

Geometric properties of special orthogonal representations associated to exceptional Lie superalgebras

From an octonion algebra [Formula: see text] over a field [Formula: see text] of characteristic not two or three, we show that the fundamental representation [Formula: see text] of the derivation algebra [Formula: see text] and the spinor representation [Formula: see text] of [Formula: see text] are special orthogonal representations. They have particular geometric properties coming from their similarities with binary cubics and we show that the covariants of these representations and their Mathews identities are related to the Fano plane and the affine space [Formula: see text]. This also permits to give constructions of exceptional Lie superalgebras.

Nonlinear Jordan higher derivations of incidence algebras

Let be a 2-torsion free commutative ring with unity and X be a locally finite preordered set. We prove in this paper that every nonlinear Jordan higher derivation on the incidence algebra is an additive higher derivation, provided that all connected components of X are nontrivial.

The derivation algebra and automorphism group of the n-th Schrödinger algebra

The n-th Schrödinger algebra is the semidirect product of the special linear Lie algebra and the Heisenberg algebra . In this paper, we describe explicitly the structure of the derivation algebra and automorphism group of the n-th Schrödinger algebra. Especially, the automorphism group of the Schrödinger algebra is obtained.

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.

The Implementation of Scientific Approach to Increase Mathematics Understanding Ability on Algebraic Derivative Materials

Referring to the objectives of learning mathematics and NCTM, one of the abilities that must be developed is the ability to understand mathematics. The importance of having the ability to understand mathematics includes this ability listed in the learning of mathematics in the 2006 KTSP mathematics curriculum and the 2013 curriculum. This study aims to examine the increased understanding  of the material concept of derivatives of algebraic functions in class XI high school students. This type of research is classroom action research. This research was conducted on class XI high school students for the 2020/2021 academic year, consisting of 16 students. The instrument used is a test of students' mathematical understanding abilities, in the form of 6 questions that focus more on students' understanding abilities regarding the derivative of algebraic functions; observation sheet for teachers and students to see the condition of the implementation of the action. This study shows that students' mathematical understanding abilities have increased as seen from the questions tested on each test with an increase in mastery from initially 31.25% to 56.25%. The results of this study indicate that the application of a scientific approach can improve the comprehension skills of class XI students on material derived from algebraic functions.

On the algebra of derivations of some nilpotent Leibniz algebras

We describe the algebra of derivations of some nilpotent Leibniz algebra, having dimensionality 3.

Descent study of the Lie algebra of derivations of certain infinite-dimensional Lie algebras

Descent study of the Lie algebra of derivations of certain infinite-dimensional Lie algebras

Derivations of two-step nilpotent algebras

In this paper we study the Lie algebras of derivations of two-step nilpotent algebras. We obtain a class of Lie algebras with trivial center and abelian ideal of inner derivations. Among these, the relations between the complex and the real case of the indecomposable Heisenberg Leibniz algebras are thoroughly described. Finally we show that every almost inner derivation of a complex nilpotent Leibniz algebra with one-dimensional commutator ideal, with three exceptions, is an inner derivation.

Genetic Algebras Associated with ξ(a)-Quadratic Stochastic Operators.

The present paper deals with a class of ξ(a)-quadratic stochastic operators, referred to as QSOs, on a two-dimensional simplex. It investigates the algebraic properties of the genetic algebras associated with ξ(a)-QSOs. Namely, the associativity, characters and derivations of genetic algebras are studied. Moreover, the dynamics of these operators are also explored. Specifically, we focus on a particular partition that results in nine classes, which are further reduced to three nonconjugate classes. Each class gives rise to a genetic algebra denoted as Ai, and it is shown that these algebras are isomorphic. The investigation then delves into analyzing various algebraic properties within these genetic algebras, such as associativity, characters, and derivations. The conditions for associativity and character behavior are provided. Furthermore, a comprehensive analysis of the dynamic behavior of these operators is conducted.

Conformal Triple Derivations and Triple Homomorphisms of Lie Conformal Algebras

Let [Formula: see text] be a finite Lie conformal algebra. We investigate the conformal derivation algebra [Formula: see text], conformal triple derivation algebra [Formula: see text] and generalized conformal triple derivation algebra [Formula: see text], focusing mainly on the connections among these derivation algebras. We also give a complete classification of (generalized) conformal triple derivation algebras on all finite simple Lie conformal algebras. In particular, [Formula: see text], where [Formula: see text] is a finite simple Lie conformal algebra. But for [Formula: see text], we obtain a conclusion that is closely related to [Formula: see text]. Finally, we introduce the definition of a triple homomorphism of Lie conformal algebras. Triple homomorphisms of all finite simple Lie conformal algebras are also characterized.

On Boolean elements and derivations in 2-dimension linguistic lattice implication algebras

A 2-dimension linguistic lattice implication algebra (2DL-LIA) can build a bridge between logical algebra and 2-dimension fuzzy linguistic information. In this paper, the notion of a Boolean element is proposed in a 2DL-LIA and some properties of Boolean elements are discussed. Then derivations on 2DL-LIAs are introduced and the related properties of derivations are investigated. Moreover, it proves that the derivations on 2DL-LIAs can be constructed by Boolean elements.

Derivations in group algebras and combinatorial invariants of groups

Derivations in group algebras and combinatorial invariants of groups

Derivations and loops of some evolution algebras

In this work we study the space of derivations of non-degenerate evolution algebras. We improve some results obtained recently in the literature and, as a consequence, we advance in the description of the derivations for n-dimensional Volterra evolution algebras. In addition, we introduce the notion of loop of an evolution algebra and we analyze under which conditions the set of loops is invariant under change of basis.

On the derivations of Leibniz algebras of low dimension

Let L be an algebra over a field F. Then L is called a left Leibniz algebra if its multiplication operations [×, ×] addition- ally satisfy the so-called left Leibniz identity: [[a,b],c] = [a,[b,c]] – [b,[a,c]] for all elements a, b, c Î L. In this paper, we begin the description of the algebra of derivations of Leibniz algebras having dimension 3. It is clear that the description of the algebra of derivations of all Leibniz algebras, having dimension 3, is quite large. Therefore, in this article, we will focus on the description of the nilpotent Leibniz algebra, whose nilpotency class is 3, and the nilpotent Leibniz algebra, whose center has dimension 2.

Derivations of Hilbert Algebras

In this paper, we introduce the notions of (l, r)-derivations, (r, l)-derivations, and derivations of Hilbert algebras and investigate some related properties. In addition, we define two subsets for a derivation d of a Hilbert algebra X, Ker d(X) and Fix d(X), and we also take a look at some of their characteristics.

Images of locally finite [formula omitted]-derivations of bivariate polynomial algebras

This paper presents an E-derivation analogue of a result on derivations due to van den Essen, Wright and Zhao. We prove that the image of a locally finite K-E-derivation of polynomial algebras in two variables over a field K of characteristic zero is a Mathieu subspace. This result together with that of van den Essen, Wright and Zhao confirms the LFED conjecture in the case of polynomial algebras in two variables.