Abstract
We introduce a concept of 3-Lie-Rinehart superalgebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we study the relationships between a Lie-Rinehart superalgebra and its induced 3-Lie-Rinehart superalgebra. The deformations of 3-Lie-Rinehart superalgebra are considered via a cohomology theory.
1. Introduction
The notion of Lie-Rinehart algebras was introduced by J. Herz in [Citation32] and mainly developed in [Citation49, Citation50]. A Lie-Rinehart algebra can be thought as a Lie -algebra, which is also an A-module, where A is an associative and commutative -algebra, in such a way that both structures are related by a compatibility condition. A first approach to this class of algebras can be found in [Citation34, Citation36].
G. Rinehart developed in [Citation50] a formalism of differential forms for general commutative algebras which relies on the notion of -Lie algebra where is a commutative ring with unit and A is a commutative -algebra. The notion of Lie-Rinehart algebra includes an abstract algebraic characterization of the algebraic structure which underlies a Lie algebroid [Citation33, Citation34, Citation44, Citation45]. Thus a Lie-Rinehart algebra is an algebraic generalization of the notion of a Lie algebroid: the space of sections of a vector bundle is replaced by a module over a ring, a vector field by a derivation of the ring. For further details and a history of the notion of Lie-Rinehart algebra, we refer to [Citation37]. Lie-Rinehart algebras have been investigated further in many papers [Citation19, Citation22, Citation26, Citation34–36, Citation38–40, Citation44].
Some generalizations of Lie-Rinehart algebras, such as Lie-Rinehart superalgebras [Citation25] or restricted Lie-Rinehart algebras [Citation27], have been recently studied. The cohomology of a Lie-Rinehart algebra L with coefficients in a Lie-Rinehart module M was first defined by Rinehart [Citation50] and further developed in [Citation34].
In [Citation22–24], the authors introduced a notion of cross modules of Lie-Rinehart algebras. Hom-Lie-Rinehart algebras and their extensions in the small dimension cohomology space was introduced and studied in [Citation46]. Hom-Rinehart algebras have close relations with Hom-Gerstenhaber Algebras and Hom-Lie Algebroids [Citation47, Citation48].
The study of 3-Lie algebras [Citation29] gets a lot of attention since it has close relationships with Lie algebras, Hom-Lie algebras, commutative associative algebras, and cubic matrices [Citation14–16, Citation20]. For example, it is applied to the study of Nambu mechanics and the study of supersymmetry and gauge symmetry transformations of the world-volume theory of multiple coincident M2-branes [Citation10, Citation46, Citation51, Citation52]. A notion of n-Lie Rinehart algebras was introduced recently in [Citation18] and extensions and crossed modules were developed for such algebras.
In 1996, the concept of n-Lie superalgebras was firstly introduced by Daletskii and Kushnirevich in [Citation28]. Moreover, Cantarini and Kac gave a more general concept of n-Lie superalgebras again in 2010 (see [Citation21]). n-Lie superalgebras are more general structures including n-Lie algebras, n-ary Nambu-Lie superalgebras, and Lie superalgebras. The construction of (n + 1)-Lie algebras induced by n-Lie algebras using a combination of bracket multiplication with a trace, motivated by the works on the quantization of the Nambu brackets [Citation11], was generalized using the brackets of general Hom-Lie algebra or n-Hom-Lie algebra and trace-like linear forms satisfying some conditions depending on the linear maps defining the Hom-Lie or n-Hom-Lie algebras in [Citation8, Citation9]. The structure of 3-Lie algebras induced by Lie algebras, classification of 3-Lie algebras and application to constructions of B.R.S. algebras have been considered in [Citation1, Citation3, Citation4]. Interesting constructions of ternary Lie superalgebras in connection to superspace extension of Nambu-Hamilton equation is considered in [Citation5]. In [Citation17], a method was demonstrated of how to construct n-ary multiplications from the binary multiplication of a Hom-Lie algebra and a -linear function satisfying certain compatibility conditions. Solvability and Nilpotency for n-Hom-Lie algebras and (n + 1)-Hom-Lie algebras induced by n-Hom-Lie algebras have been considered in [Citation43]. In [Citation1, Citation2, Citation5], 3-Lie superalgebras were constructed starting with a Lie superalgebras and various properties of such algebras considered. Related constructions for n-ary hom-Lie algebras and for n-ary hom-Lie superalgebras can be found in [Citation6–9, Citation17, Citation30, Citation42]. The ternary case of (Hom-)Lie Rinehart algebras was developed in [Citation12, Citation13, Citation31].
This paper is organized as follows. In Section 1, we recall the notion of Lie-Rinehart superalgebras and define their cohomology groups. We generalize the Lie-Rinehart superalgebra to the ternary case. Section 2 is devoted to some construction results. We begin by constructing 3-Lie-Rinehart superalgebras starting with a Lie-Rinehart superalgebras. We construct some new 3-Lie-Rinehart superalgebras from a given 3-Lie-Rinehart superalgebra. The notion of a module for a 3-Lie-Rinehart superalgebras appears in Section 3, and subsequently we introduce a cochain complex and cohomology of a 3-Lie-Rinehart superalgebras with coefficients in a module. Then we study relations between 1-cocycles and 2-cocycles of a Lie-Rinehart superalgebra and the induced 3-Lie-Rinehart superalgebra. At the end of this section, we study the deformation of 3-Lie-Rinehart superalgebras.
2. Definitions and notations
In this section, we review basic definitions of Lie superalgebras, 3-Lie superalgebras, Lie-Rinehart superalgebras and generalize the notion of 3-Lie-Rinhehart algebras to the super case.
Let be a -graded vector space. If is a homogenous element, then its degree will be denoted by where and Let End(V) be the -graded vector space of endomorphisms of a -graded vector space Denoted by the set of homogenous elements of V. For the composition of endomorphisms the graded commutator, defined by for homogeneous induces the structure of Lie superalgebra on the -graded vector space End(V).
2.1. Lie-Rinehart superalgebras
Definition 1.1
([Citation41]). A Lie superalgebra is a pair consisting of a -graded vector space an even bilinear map satisfying the following identities: (1.1) (1.1) (1.2) (1.2) where x, y and z are homogeneous elements in
Definition 1.2.
A representation of a Lie superalgebra on a -graded vector space is an even linear map such that the following identity is satisfied for all homogeneous (1.3) (1.3)
Now, we recall the definition of Lie-Rinehart superalgebra [Citation25].
Definition 1.3.
A Lie-Rinehart superalgebra L over (an associative supercommutative superalgebra) A is a Lie superalgebra over with an A-module structure and an even linear map such that that following conditions hold:
μ is a representation of on A.
for all
The compatibility condition: (1.4) (1.4)
Example 1.4.
Let us observe that Lie-Rinehart superalgebras over A with trivial map are exactly Lie superalgebras. If then Der(A) = 0 and there is no difference between Lie and Lie-Rinehart superalgebras. Therefore, the concept of Lie-Rinehart superalgebras generalizes the concept of Lie superalgebras.
Example 1.5.
Let A an associative supercommutative superalgebra. Then, the A-module is a Lie-Rinehart superalgebra over A with the bracket and an even map the projection onto the first factor.
Now, we define the cohomology of a Lie-Rinehart superalgebra. First we introduce the notion of left module over a Lie-Rinehart superalgebra.
Definition 1.6.
Let M be an A-module. Then M is a left module over a Lie-Rinehart superalgebra L if there exits an even map such that:
θ is a representation of the Lie superalgebra on M.
for all
for all
Let be a Lie-Rinehart superalgebra, and let M be a left module over L. For Lie-Rinehart superalgebra L with coefficients in M, consider the -graded space of -modules where consisting of elements satisfying for all and
Define the even linear map given by for all where
With the above notation, the map δLR gives rise to a coboundary map.
Proposition 1.7.
If , then and
By the above proposition, is a cochain complex. The resulting cohomology of the cochain complex we define to be the cohomology space of Lie-Rinehart superalgebra with coefficients in M, and we denote this cohomology as
2.2. 3-Lie-Rinehart superalgebras
Definition 1.8
([Citation21]). A -graded vector space is said to be a 3-Lie superalgebra, if it is endowed with an even trilinear map (bracket) satisfying the following conditions: (1.5) (1.5) (1.6) (1.6) where are homogeneous elements.
Proposition 1.9.
Let be a -graded vector space together with a super skew-symmetric even linear map . Then is a 3-Lie superalgebra if and only if the following identities hold: (1.7) (1.7) (1.8) (1.8) for all
Proof.
If is a 3-Lie superalgebra, then applying Equation(1.6)(1.6) (1.6) to the last term in Equation(1.6)(1.6) (1.6) for and using Equation(1.5)(1.5) (1.5) , yields Equation(1.7)(1.7) (1.7) .
Similarly, applying Equation(1.6)(1.6) (1.6) to the first and second terms on the right hand side of Equation(1.6)(1.6) (1.6) for and using Equation(1.5)(1.5) (1.5) , yields Equation(1.8)(1.6) (1.6) .
Conversely, suppose that Equation(1.7)(1.6) (1.6) and Equation(1.8)(1.6) (1.6) hold. First Equation(1.7)(1.6) (1.6) gives
by Equation(1.8)(1.6) (1.6) . Thus, is a 3-Lie superalgebra. □
In the following we recall that given a Lie superalgebra analogue of supertrace one can construct a 3-Lie superalgebra [Citation1]. Let be a Lie superalgebra and an even linear form. We say that τ is a supertrace of if For any we define the 3-ary bracket by (1.9) (1.9)
Theorem 1.10.
For any Lie superalgebra and supertrace τ, the pair is a 3-Lie superalgebra.
Definition 1.11.
Let be a 3-Lie superalgebra, V be a graded vector space and be an even linear mapping. If ρ satisfies (1.10) (1.10) (1.11) (1.11) for all then is called a representation of or is an -module.
Define (1.12) (1.12)
Using Equation(1.6)(1.6) (1.6) , we can see that is a representation of the 3-Lie superalgebra and it is called the adjoint representation of
Proposition 1.12.
Let be a representation of a Lie superalgebra and τ be a supertrace of . Then is a representation of the 3-Lie superalgebra where is defined by (1.13) (1.13)
Proof.
For all the left hand of Equation(1.10)(1.10) (1.10) becomes
Using Equation(1.9)(1.6) (1.6) and Equation(1.13)(1.6) (1.6) , the right hand of Equation(1.10)(1.6) (1.6) can be written as follows:
By direct identification, we proof that
Similarly, we can proof Equation(1.11)(1.11) (1.11) . □
We generalize the Lie-Rinehart superalgebra to the ternary case.
Definition 1.13.
A 3-Lie-Rinehart superalgebra over A is a tuple where A is an associative supercommutative superalgebra, L is an module, is an even super skew-symmetric trilinear map, and the -map such that the following conditions hold:
is a 3-Lie superalgebra.
ρ is a representation of on A.
For all (1.14) (1.14)
The compatibility condition: (1.15) (1.15)
Remark 1.14.
If ρ = 0, then is called a 3-Lie A-superalgebra.
Remark 1.15.
If the condition Equation(1.14)(1.6) (1.6) is not satisfied, then we call a weak 3-Lie-Rinehart superalgebra over A.
Definition 1.16.
Let and be two 3-Lie-Rinehart superalgebras, then a 3-Lie-Rinehart superalgebra homomorphism is defined as a pair of maps (g, f), where and are two -algebra homomorphisms such that
for all
for all
3. Some constructions of 3-Lie-Rinehart superalgebras
In this section, we give some construction results. We begin by constructing 3-Lie-Rinehart superalgebras starting with a Lie-Rinehart superalgebras. We also construct Lie-Rinehart superalgebras starting from 3-Lie-Rinehart superalgebras. At the end of this section, we construct some new 3-Lie-Rinehart superalgebras from a given 3-Lie-Rinehart superalgebra.
The following Theorem generalizes results in [Citation2] to Lie-Rinehart case.
Theorem 2.1.
Let be a Lie-Rinehart superalgebra and τ is a supertrace. If the condition (2.1) (2.1) is satisfied for any , then is a 3-Lie-Rinehart superalgebra, where and are defined in Equation(1.9)(1.6) (1.6) and Equation(1.12)(1.6) (1.6) respectively. We say that is induced by and denote it by
Proof.
Proposition 1.12 gives that is a representation of the 3-Lie superalgebra and For all and
So, we obtain Equation(1.15)(1.5) (1.5) . Since Equation(2.1)(1.6) (1.6) is satisfied and the condition Equation(1.14)(1.4) (1.4) holds. □
Conversely, we can construct a Lie-Rinehart superalgebra structure from a given 3-Lie-Rinehart superalgebra.
Proposition 2.2.
Let be a 3-Lie-Rinehart superalgebra. Let . Define the bracket and . Then is a Lie-Rinehart superalgebra.
Proof.
It is easy to check that the bracket is super skew-symmetric. For any we have
It is obvious to see that is a representation of L on A and It remains to prove Equation(1.4)(1.4) (1.4) . Using Equation(1.15)(1.5) (1.5) , we obtain which completes the proof. □
Definition 2.3.
Let be a 3-Lie-Rinehart superalgebra.
If S is a subalgebra of the 3-Lie superalgebra satisfying then is a 3-Lie-Rinehart superalgebra, which is called a subalgebra of the 3-Lie-Rinehart superalgebra
If I is an ideal of the 3-Lie superalgebra and satisfies and then is a 3-Lie-Rinehart superalgebra, which is called an ideal of the 3-Lie-Rinehart superalgebra
If a 3-Lie-Rinehart superalgebra cannot be decomposed into the direct sum of two nonzero ideals, then L is called an indecomposable 3-Lie-Rinehart superalgebra.
Proposition 2.4.
If is a 3-Lie-Rinehart superalgebra. Then is an ideal, which is called the kernel of the representation ρ.
Proof.
By Equation(1.10)(1.10) (1.10) and Equation(1.11)(1.11) (1.11) , for all homogeneous elements
Therefore, By Equation(1.14)(1.14) (1.14) , for all
We get that is,
Therefore, is an ideal of the 3-Lie-Rinehart superalgebra □
Theorem 2.5.
Let be a 3-Lie-Rinehart superalgebra. Then the following identities hold, for all (2.2) (2.2) (2.3) (2.3) (2.4) (2.4) (2.5) (2.5)
Proof.
Using Equation(1.15)(1.6) (1.6) we have
Thanks to Proposition 1.9, we have Using Equation(1.15)(1.15) (1.15) , Equation(1.10)(1.10) (1.10) and Equation(1.11)(1.11) (1.11) , we obtain Equation(2.2)(2.2) (2.2) . Similarly, Equation(2.3)(2.3) (2.3) -Equation(2.5)(2.5) (2.5) can be verified by a direct computation according to Equation(2.2)(2.2) (2.2) and Definition 1.13. □
Now we construct some new 3-Lie-Rinehart superalgebra from a given 3-Lie-Rinehart superalgebra
Theorem 2.6.
Let be a 3-Lie-Rinehart superalgebra and . Then is a 3-Lie-Rinehart superalgebra, where the multiplication is defined by, for all (2.6) (2.6) and is defined by
Note that where and
Proof.
For the super skew-symmetry of the bracket we have
Similarly, we obtain
Using Equation(2.2)(2.2) (2.2) and Equation(2.3)(2.3) (2.3) , we can deduce that is a 3-Lie superalgebra and B is an A-module. Thanks to Equation(1.10)(1.10) (1.10) and Equation(1.11)(1.11) (1.11) , for all and it is easy to show that is a representation of B. Indeed,
To prove the compatibility condition Equation(1.15)(1.15) (1.15) , we compute as follows
It is obvious to show that
Therefore, is a 3-Lie-Rinehart superalgebra. □
Theorem 2.7.
Let be a 3-Lie-Rinehart superalgebra and
Then is a 3-Lie-Rinehart superalgebra, where for any and (2.7) (2.7) (2.8) (2.8) (2.9) (2.9)
Note that , where and
Proof.
Let Thanks to Equation(2.7)(1.12) (1.12) , E is an module, and the 3-ary linear multiplication defined by Equation(2.8)(2.8) (2.8) is super skew-symmetric. We have
Similarly, we obtain
Then Equation(1.6)(1.6) (1.6) holds thanks to Equation(1.10)(1.13) (1.13) . Therefore, is a 3-Lie superalgebra.
Now we prove that ρ1 is a representation of E over A. By Equation(2.9)(1.7) (1.7) , we have
Therefore, Equation(1.10)(1.10) (1.10) holds. Similarly, we can prove Equation(1.11)(1.11) (1.11) . Then ρ1 is a representation of E over A. For all we have
Moreover, we have
Similarly, we obtain Then is a 3-Lie-Rinehart superalgebra. □
4. Cohomology and deformations of 3-Lie-Rinehart superalgebras
In this section, we study the notion of a module for a 3-Lie-Rinehart superalgebras and subsequently we introduce a cochain complex and cohomology of a 3-Lie-Rinehart superalgebras with coefficients in a module, then we study relations between 1 and 2 cocycles of a Lie-Rinehart superalgebra and the induced 3-Lie-Rinehart superalgebra. At the end of this section, we study the deformation of 3-Lie-Rinehart superalgebras
4.1. Representations and cohomology of 3-Lie-Rinehart superalgebras
Definition 3.1.
Let M be an A-module. and be an even linear map. The pair is called a left module of the 3-Lie-Rinehart superalgebra if the following conditions hold:
ψ is a representation of on M,
for all and
for all and
Example 3.2.
A is a left module over L since ρ is a representation of over A and the other conditions are satisfied automatically by definition of the map ρ.
Example 3.3.
The pair (L, ad) is a left module over L, which is called the adjoint representation of
Proposition 3.4.
Let be a 3-Lie-Rinehart superalgebra. Then is a left module over if and only if is a 3-Lie-Rinehart superalgebra with the multiplication: for any and . Note that , implying that if , then
Proof.
Since L and M are A-modules, then is an A-module via
If is a left module over Then is a 3-Lie superalgebra. It is obvious that is a representation of the 3-Lie superalgebra over A.
For any and
Moreover,
Similarly, we obtain Therefore, is a 3-Lie-Rinehart superalgebra. The sufficient condition can be done in the same way. □
Let be a left module of the 3-Lie-Rinehart superalgebra and the space of all linear maps satisfying the following conditions:
for all
for all and
Next we consider the -graded space of -modules
Define the -linear maps by
Proposition 3.5.
If , then and
Proof.
Let f be a homogenous element in it is obvious that is skew-symmetric. For all and for
Using Definition 3.1 and Equation(1.15)(1.5) (1.5) , we obtain
Similarly, we can proof the same result if Then is well-defined. Further, follows from the direct but a long calculation. □
By the above proposition, is a cochain complex. The resulting cohomology of the cochain complex can be defined as the cohomology space of 3-Lie-Rinehart superalgebra with coefficients in and we denote this cohomology as
Definition 3.6.
Let be a 3-Lie-Rinehart superalgebra and be a left module over L. If satisfies for any then ν is called a 1-cocycle associated with ψ.
Definition 3.7.
Let be a 3-Lie-Rinehart superalgebra and be a left module over L. If satisfies for any then ω is called a 2-cocycle associated with ψ.
Theorem 3.8.
Let be a Lie-Rinehart superalgebra, τ be a supertrace and such that
Define the linear map by
Then is a 2-cocycle of the induced 3-Lie-Rinehart superalgebra
Proof.
Let It is obvious that is skew-symmetric and If then
Similarly, and On the other hand, if then
Since for all we get □
Corollary 3.9.
Let . Then is a 2-cocycle of the induced 3-Lie-Rinehart superalgebra.
Theorem 3.10.
Every 1-cocycle for the scalar cohomology of the Lie-Rinehart superalgebra is a 1-cocycle for the scalar cohomology of the induced 3-Lie-Rinehart superalgebra
Proof.
Let Then, which is equivalent to It is obvious to prove that and then that is which means that ω is a 1-cocycle for the scalar cohomology of □
Lemma 3.11.
Let . Then, for all
Proof.
Let Then, completes the proof. □
Proposition 3.12.
Let . If are in the same cohomology class then defined by are in the same cohomology class.
Proof.
If are two cocycles in the same cohomology class, that is then which means that ψ1 and ψ2 are in the same cohomology class. □
4.2. Deformations of 3-Lie-Rinehart superalgebras
Let be a 3-Lie-Rinehart superalgebra. Denote by the space of formal power series ring with parameter t.
Definition 3.13.
A deformation of a 3-Lie-Rinehart superalgebra is a -trilinear map where and mi are even trilinear maps for satisfying Equation(1.5)(1.5) (1.5) , Equation(1.6)(1.6) (1.6) , Equation(1.14)(1.14) (1.14) and Equation(1.15)(1.15) (1.15) .
Let be a deformation of Then
Comparing the coefficients of tn, we get the following equation:
Definition 3.14.
The 3-cochain m1 is called the infinitesimal of the deformation More generally, if mi = 0 for and mn is non zero cochain, then mn is called the n-infinitesimal of the deformation
Definition 3.15.
Two deformations and are said to be equivalent if there exists a formal automorphism where is an even -linear maps such that
Definition 3.16.
A deformation is called trivial if it is equivalent to the deformation
Definition 3.17.
A 3-Lie-Rinehart superalgebra is said to be rigid if every deformation of it is trivial.
Theorem 3.18.
The cohomology class of the infinitesimal of a deformation is determined by the equivalence class of
Proof.
Let represents an equivalence of deformation given by and Then
Comparing the coefficients of t from both sides of the above equation we have or equivalently, So, cohomology class of infinitesimal of the deformation is determined by the equivalence class of deformation of □
Theorem 3.19.
A non-trivial deformation of a 3-Lie-Rinehart superalgebra is equivalent to a deformation whose n-infinitesimal cochain is not a coboundary for some
Proof.
Let be a deformation of 3-Lie-Rinehart superalgebra with n-infinitesimal mn for some Assume that there exists a 3-cochain with Set and Comparing the coefficients of tn, we get the following equation:
So Then, one has the deformation whose n-infinitesimal is not a coboundary for some □
Acknowlegments
Dr. Sami Mabrouk is grateful to the research environment in Mathematics and Applied Mathematics MAM, Division of Applied Mathematics and Physics, School of Education, Culture and Communication, Mälardalen University for hospitality and excellent and inspiring environment for research cooperation in Mathematics during his visit in 2019. Partial support from Swedish Royal Academy of Sciences foundations is also gratefully acknowledged.
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