fa عملگرها و حساب دیفرانسیل روی δ-هوم-ابرجبرهای ‌لی جردن جبر alg علمی پژوهشی بنیادی S <span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">در این مقاله، به بررسی و مطالعه نوعی از عملگرهای دیفرانسیل روی</span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"> <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.png" > </span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">-هوم-ابرجبرهای 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</span></span><span dir="LTR"><span style="font-size:10.0pt;">.</span></span><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;"></span></span></span><br> <span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;"></span></span></span><br> &nbsp;<span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:12.0pt;"></span></span></span> <strong>Introduction</strong><br> Hom-algebraic &lrm;structures &lrm;appeared &lrm;first &lrm;as a&lrm; &lrm;generalization &lrm;of &lrm;Lie &lrm;algebras &lrm;in [1,3],&nbsp; &lrm;where &lrm;the &lrm;authors &lrm;studied &lrm;&lrm;q-deformations &lrm;of &lrm;Witt &lrm;and &lrm;Virasoro &lrm;algebras. A&lrm; &lrm;general &lrm;study &lrm;and &lrm;construction &lrm;of &lrm;Hom-Lie &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;Hom-associative &lrm;algebras, &lrm;Hom-Lie &lrm;admissible &lrm;algebras &lrm;and &lrm;Hom-Jordan &lrm;algebras. &lrm;Hom-algebraic &lrm;structures &lrm;were &lrm;extended &lrm;to &lrm;Hom-Lie &lrm;superalgebras &lrm;in &lrm;[2].&lrm;<br> 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 Jordan-super 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,&nbsp;&nbsp; &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;Hom-Lie &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;Hom-Lie superalgebras are a generalization of Hom-Lie algebras, where the classical super Jacobi&nbsp; identity is twisted by a linear map. If the skew-super symmetric bracket of a Hom-Lie superalgebra is replaced by &delta;-Jordan-super &lrm;symmetric&lrm;, it is called a&nbsp;&nbsp; &delta;-Jordan-Hom-Lie &lrm;superalgebra &lrm;(see [11]).&lrm;<br> There are several notions of differential operators and differential calculus on&lrm; non-associative algebras (see [4, 5])&lrm;. A &lrm; &lrm;comprehensive definition of differential operators on non-associative 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;Hom-Lie &lrm;algebras [9, 10]. &lrm; According&nbsp; &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;-Jordan-Hom-Lie &lrm;superalgebras.<br> <strong>&lrm;</strong><strong>Material and methods</strong><br> A &lrm;key &lrm;point &lrm;is &lrm;that &lrm;the &lrm;multiplications &lrm;on &lrm; multiplicative &delta;-Jordan-Hom-Lie &lrm;superalgebras are their derivations. Therefore, definition of differential operators on a &lrm;&lrm;&lrm;multiplicative &delta;-Jordan-Hom-Lie &lrm;superalgebra must treat the derivations of this algebra as a first-order differential operators too. By our considerations, we will define higher order differential operators as composition of the first-order differential operators on a &lrm;multiplicative &delta;-Jordan-Hom-Lie &lrm;superalgebra. We also consider a geometric aspect to the concept of differential calculus on &lrm; multiplicative &delta;-Jordan-Hom-Lie &lrm;superalgebra by using the cohomology theory for this algebra.<br> &nbsp;<br> <strong>Results and discussion</strong><br> &lrm;The theory of differential operators on associative algebras is not extended to the non-associative algebras in a straightforward way. But, we provide a notion of differential operators of any order on &lrm; multiplicative&nbsp;&nbsp; &delta;-Jordan-Hom-Lie &lrm;superalgebras and their modules. We also study some property of differential operators on &lrm; multiplicative&nbsp;&nbsp; &delta;-Jordan-Hom-Lie &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&nbsp;&nbsp; &delta;-Jordan-Hom-Lie &lrm;superalgebras, we investigate a notion of differential calculus on these algebras. In other words, for a &lrm;multiplicative&nbsp;&nbsp; &delta;-Jordan-Hom- Lie &lrm;superalgebra L&nbsp; &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&nbsp; L &lrm;as &lrm;&lrm;Der(L)-module &lrm;its &lrm;subcomplex &lrm;of &lrm;&lrm; Z(L)-multilinear&nbsp; &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 Hom-Lie super algebra &lrm;&lrm;&lrm;osp(1, 2).&lrm;<br> <strong>Conclusion</strong><br> The following conclusions were drawn from this research.<br> &bull; Definition of the differential operators of any order on &lrm; multiplicative&nbsp;&nbsp; &delta;-Jordan-Hom-Lie &lrm;superalgebras and prove several properties of it.&lrm;<span dir="RTL"></span><br> &bull; Definition of the differential operators of any order on &delta;-modul &lrm;of&lrm; &lrm; multiplicative &delta;-Jordan-Hom-Lie &lrm;superalgebras and state some properties of it.&lrm;<br> &bull; The study of &lrm;&lrm; differential calculus based on derivation of a &lrm; multiplicative &delta;-Jordan-Hom-Lie &lrm;superalgebra.<br> &bull; Compute the &lrm;&lrm; differential calculus based on derivation of Hom-Lie superalgebra &lrm; osp (1, 2).&lrm;<a href="./files/site1/files/62/5Abstract.pdf">./files/site1/files/62/5Abstract.pdf</a><span dir="RTL"></span><br> <span dir="RTL"></span>  هوم-جبرهای لی, هوم-ابرجبرهای لی, مشتقات و نظریه کوهمولوژی روی هوم-ابرجبرهای لی. Hom-Lie algebras, Hom-Lie superalgebras, Derivation and cohomology on Hom-Lie superalgebras 191 206 http://mmr.khu.ac.ir/browse.php?a_code=A-10-627-1&slc_lang=fa&sid=1 Valiollah Khalili ولی الله خلیلی V-Khalili@araku.ac.ir 10031947532846003742 10031947532846003742 Yes Arak University دانشگاه اراک، دانشکدۀ علوم پایه، گروه ریاضی