عملگرها و حساب دیفرانسیل روی δ-هوم-ابرجبرهای ‌لی جردن

در این مقاله، به بررسی و مطالعه نوعی از عملگرهای دیفرانسیل روی هومابرجبرهای لی جردن می‌پردازیم. هم‌چنین، به تعریف نوعی از عملگرهای دیفرانسیل روی مدول‌های این دسته از جبرها می‌پردازیم. سرانجام، مفهوم نوعی از حساب دیفراسیل بر پایه مشتقات روی این دسته از جبرها را بررسی می‌کنیم و مثالی برای محقق‌سازی این مفاهیم می‌آوریم.

Differential Operators and Differential Calculus on $delta-$Hom-Jordan-Lie Superalgebras

IntroductionHomalgebraic lrm;structures lrm;appeared lrm;first lrm;as a lrm; lrm;generalization lrm;of lrm;Lie lrm;algebras lrm;in [1,3], lrm;where lrm;the lrm;authors lrm;studied lrm; lrm;qdeformations lrm;of lrm;Witt lrm;and lrm;Virasoro lrm;algebras. A lrm; lrm;general lrm;study lrm;and lrm;construction lrm;of lrm;HomLie lrm;algebras lrm;were lrm;considered lrm;in [7, 8]. lrm;Since lrm;then, lrm;other lrm;interesting lrm;Hom type lrm;algebraic lrm;structures lrm;of lrm;many lrm;classical lrm;structures lrm;were lrm;studied lrm;Homassociative lrm;algebras, lrm;HomLie lrm;admissible lrm;algebras lrm;and lrm;HomJordan lrm;algebras. lrm;Homalgebraic lrm;structures lrm;were lrm;extended lrm;to lrm;HomLie lrm;superalgebras lrm;in lrm;[2]. lrm;As a lrm; lrm;generalization lrm;of lrm;Lie lrm;superalgebras lrm;and lrm;Jordan lrm;Lie lrm;algebras, lrm;the lrm;notion lrm;of lrm; lrm; delta;Jordan lrm;Lie lrm;superalgebra lrm;was lrm;introduced lrm;in [6, 12] which is intimately related to both Jordansuper and atiassociative algebras. The case of delta;=1 lrm;yields lrm;the lrm;Lie lrm;superalgebra, lrm;and lrm;we lrm;call lrm;the lrm;other lrm;case lrm;of delta;=1 a lrm; lrm;Jordan lrm;Lie lrm;superalgebra, lrm;because lrm;it lrm;turns lrm;out lrm;to lrm;be a lrm; lrm;Jordan lrm;superalgebra. lrm;It lrm;is lrm;often lrm;convenient lrm;to lrm;consider lrm;both lrm;cases lrm;of delta;= 1, lrm;and lrm;call delta;Jordan lrm;Lie lrm;superalgebras. lrm; lrm;The lrm;motivations lrm;to lrm;characterize lrm;HomLie lrm;structurers lrm;are lrm;related lrm;to lrm;physics lrm;and lrm;to lrm;deformations lrm;of lrm;Lie lrm;algebras, lrm;in lrm;particular lrm;Lie lrm;algebras lrm;of lrm;vector lrm;fields. lrm;HomLie superalgebras are a generalization of HomLie algebras, where the classical super Jacobi identity is twisted by a linear map. If the skewsuper symmetric bracket of a HomLie superalgebra is replaced by delta;Jordansuper lrm;symmetric lrm;, it is called a delta;JordanHomLie lrm;superalgebra lrm;(see [11]). lrm;There are several notions of differential operators and differential calculus on lrm; nonassociative algebras (see [4, 5]) lrm;. A lrm; lrm;comprehensive definition of differential operators on nonassociative algebras fails to be formulated. But many authors was studied a notion of differential operators and differential calculus on lrm;Lie lrm;algebras lrm;and lrm;HomLie lrm;algebras [9, 10]. lrm; According lrm;to lrm;various lrm;applications lrm;in lrm;both lrm;mathematics lrm;and lrm;physics, lrm; lrm; lrm; lrm; lrm; we will investigate a notion of differential operators and differential calculus on lrm; lrm; multiplicative delta;JordanHomLie lrm;superalgebras. lrm;Material and methodsA lrm;key lrm;point lrm;is lrm;that lrm;the lrm;multiplications lrm;on lrm; multiplicative delta;JordanHomLie lrm;superalgebras are their derivations. Therefore, definition of differential operators on a lrm; lrm; lrm;multiplicative delta;JordanHomLie lrm;superalgebra must treat the derivations of this algebra as a firstorder differential operators too. By our considerations, we will define higher order differential operators as composition of the firstorder differential operators on a lrm;multiplicative delta;JordanHomLie lrm;superalgebra. We also consider a geometric aspect to the concept of differential calculus on lrm; multiplicative delta;JordanHomLie lrm;superalgebra by using the cohomology theory for this algebra. Results and discussion lrm;The theory of differential operators on associative algebras is not extended to the nonassociative algebras in a straightforward way. But, we provide a notion of differential operators of any order on lrm; multiplicative delta;JordanHomLie lrm;superalgebras and their modules. We also study some property of differential operators on lrm; multiplicative delta;JordanHomLie lrm;superalgebras, for examples, the brackets and composition of two differential operators of higher order on these algebras. Finally, by using theory of cohomology for lrm; multiplicative delta;JordanHomLie lrm;superalgebras, we investigate a notion of differential calculus on these algebras. In other words, for a lrm;multiplicative delta;JordanHom Lie lrm;superalgebra L lrm;with lrm;center Z(L) lrm;and lrm; lrm;Der(L), lrm;the lrm;derivation lrm;of lrm; lrm; L, lrm;we lrm;consider lrm;the lrm;cochain lrm;complex lrm;of L lrm;as lrm; lrm;Der(L)module lrm;its lrm;subcomplex lrm;of lrm; lrm; Z(L)multilinear lrm;morphism lrm;is said lrm;to lrm;be a lrm; lrm; lrm; differential calculus based on derivation of lrm; L. lrm;Next, lrm;we lrm;compute lrm;the lrm; differential calculus based on derivation of HomLie super algebra lrm; lrm; lrm;osp(1, 2). lrm;ConclusionThe following conclusions were drawn from this research. bull; Definition of the differential operators of any order on lrm; multiplicative delta;JordanHomLie lrm;superalgebras and prove several properties of it. lrm; bull; Definition of the differential operators of any order on delta;modul lrm;of lrm; lrm; multiplicative delta;JordanHomLie lrm;superalgebras and state some properties of it. lrm; bull; The study of lrm; lrm; differential calculus based on derivation of a lrm; multiplicative delta;JordanHomLie lrm;superalgebra. bull; Compute the lrm; lrm; differential calculus based on derivation of HomLie superalgebra lrm; osp (1, 2). lrm;./files/site1/files/62/5Abstract.pdf