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

Hom-algebraic ‎structures ‎appeared ‎first ‎as a‎ ‎generalization ‎of ‎Lie ‎algebras ‎in [1,3],  ‎where ‎the ‎authors ‎studied ‎‎q-deformations ‎of ‎Witt ‎and ‎Virasoro ‎algebras. A‎ ‎general ‎study ‎and ‎construction ‎of ‎Hom-Lie ‎algebras ‎were ‎considered ‎in [7, 8]. ‎Since ‎then, ‎other ‎interesting ‎Hom- type ‎algebraic ‎structures ‎of ‎many ‎classical ‎structures ‎were ‎studied ‎Hom-associative ‎algebras, ‎Hom-Lie ‎admissible ‎algebras ‎and ‎Hom-Jordan ‎algebras. ‎Hom-algebraic ‎structures ‎were ‎extended ‎to ‎Hom-Lie ‎superalgebras ‎in .‎As a‎ ‎generalization ‎of ‎Lie ‎superalgebras ‎and ‎Jordan ‎Lie ‎algebras, ‎the ‎notion ‎of ‎‎ δ-Jordan ‎Lie ‎superalgebra ‎was ‎introduced ‎in [6, 12] which is intimately related to both Jordan-super and atiassociative algebras. The case of δ=1 ‎yields ‎the ‎Lie ‎superalgebra, ‎and ‎we ‎call ‎the ‎other ‎case ‎of δ=1 a‎ ‎Jordan ‎Lie ‎superalgebra,   ‎because ‎it ‎turns ‎out ‎to ‎be a‎ ‎Jordan ‎superalgebra. ‎It ‎is ‎often ‎convenient ‎to ‎consider ‎both ‎cases ‎of δ= 1, ‎and ‎call δ-Jordan ‎Lie ‎superalgebras.‎ ‎The ‎motivations ‎to ‎characterize ‎Hom-Lie ‎structurers ‎are ‎related ‎to ‎physics ‎and ‎to ‎deformations ‎of ‎Lie ‎algebras, ‎in ‎particular ‎Lie ‎algebras ‎of ‎vector ‎fields. ‎Hom-Lie superalgebras are a generalization of Hom-Lie algebras, where the classical super Jacobi  identity is twisted by a linear map. If the skew-super symmetric bracket of a Hom-Lie superalgebra is replaced by δ-Jordan-super ‎symmetric‎, it is called a   δ-Jordan-Hom-Lie ‎superalgebra ‎(see [11]).‎There are several notions of differential operators and differential calculus on‎ non-associative algebras (see [4, 5])‎. A ‎ ‎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 ‎Lie ‎algebras ‎and ‎Hom-Lie ‎algebras [9, 10]. ‎ According  ‎to ‎various ‎applications ‎in ‎both ‎mathematics ‎and ‎physics,‎‎‎‎‎ we will investigate a notion of differential operators and differential calculus on‎ ‎ multiplicative δ-Jordan-Hom-Lie ‎superalgebras.

A ‎key ‎point ‎is ‎that ‎the ‎multiplications ‎on ‎ multiplicative δ-Jordan-Hom-Lie ‎superalgebras are their derivations. Therefore, definition of differential operators on a ‎‎‎multiplicative δ-Jordan-Hom-Lie ‎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 ‎multiplicative δ-Jordan-Hom-Lie ‎superalgebra. We also consider a geometric aspect to the concept of differential calculus on ‎ multiplicative δ-Jordan-Hom-Lie ‎superalgebra by using the cohomology theory for this algebra.

‎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 ‎ multiplicative   δ-Jordan-Hom-Lie ‎superalgebras and their modules. We also study some property of differential operators on ‎ multiplicative   δ-Jordan-Hom-Lie ‎superalgebras, for examples, the brackets and composition of two differential operators of higher order on these algebras. Finally, by using theory of cohomology for ‎ multiplicative   δ-Jordan-Hom-Lie ‎superalgebras, we investigate a notion of differential calculus on these algebras. In other words, for a ‎multiplicative   δ-Jordan-Hom- Lie ‎superalgebra L  ‎with ‎center Z(L) ‎and ‎‎Der(L), ‎the ‎derivation ‎of ‎‎ L, ‎we ‎consider ‎the ‎cochain ‎complex ‎of  L ‎as ‎‎Der(L)-module ‎its ‎subcomplex ‎of ‎‎ Z(L)-multilinear  ‎morphism ‎is said ‎to ‎be a‎ ‎‎ differential calculus based on derivation of ‎ L. ‎Next, ‎we ‎compute ‎the‎ differential calculus based on derivation of Hom-Lie super algebra ‎‎‎osp(1, 2).‎

The following conclusions were drawn from this research.
• Definition of the differential operators of any order on ‎ multiplicative   δ-Jordan-Hom-Lie ‎superalgebras and prove several properties of it.‎
• Definition of the differential operators of any order on δ-modul ‎of‎ ‎ multiplicative δ-Jordan-Hom-Lie ‎superalgebras and state some properties of it.‎
• The study of ‎‎ differential calculus based on derivation of a ‎ multiplicative δ-Jordan-Hom-Lie ‎superalgebra.
• Compute the ‎‎ differential calculus based on derivation of Hom-Lie superalgebra ‎ osp (1, 2).‎