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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.
References
- Abramov, V. (2017). Super 3-Lie algebras induced by super Lie algebras. Adv. Appl. Clifford Algebras 27(1):9–16. DOI: 10.1007/s00006-015-0604-3.
- Abramov, V. (2019). 3-Lie superalgebras induced by Lie superalgebras. Axioms. 8(1):21. DOI: 10.3390/axioms8010021.
- Abramov, V. (2020). Weil algebra, 3-Lie algebra and B.R.S. algebra. In: Silvestrov, S., Malyarenko, A., Rancic, M., eds. Algebraic Structures and Applications. Springer Proceedings in Mathematics and Statistics, Vol. 317, Ch 1. Cham: Springer, pp. 1–12.
- Abramov, V., Lätt, P. (2016). Classification of low dimensional 3-Lie superalgebras. In: Silvestrov S., Rancic M., eds. Engineering Mathematics II. Springer Proceedings in Mathematics and Statistics, Vol. 179. Cham: Springer, pp. 1–12.
- Abramov, V., Lätt, P. (2020). Ternary Lie superalgebras and Nambu-Hamilton equation in superspace. In: Silvestrov S., Malyarenko A., Rancić, M., eds. Algebraic Structures and Applications. Springer Proceedings in Mathematics and Statistics, Vol. 317, Ch 3. Springer, pp. 47–80.
- Abramov, V., Silvestrov, S. (2020). 3-Hom-Lie algebras based on σ-derivation and involution. Adv. Appl. Clifford Algebras 30(3):45. DOI: 10.1007/s00006-020-01068-6.
- Arnlind, J., Kitouni, A., Makhlouf, A., Silvestrov, S. (2014). Structure and cohomology of 3-Lie algebras induced by Lie algebras. In: Makhlouf, A., Paal, E., Silvestrov, S. D., Stolin, A., eds. Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics and Statistics, Vol. 85, Berlin, Heidelberg: Springer, pp. 123–144.
- Arnlind, J., Makhlouf, A., Silvestrov, S. (2010). Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras. J. Math. Phys. 51(4):043515. DOI: 10.1063/1.3359004.
- Arnlind, J., Makhlouf, A., Silvestrov, S. (2011). Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras. J. Math. Phys. 52(12):123502–123513. DOI: 10.1063/1.3653197.
- Albuquerque, H., Barreiro, E., Calderón, A. J., Sánchez, J. M. (2020) Split Lie-Rinehart algebras. J. Algebra Appl. DOI: 10.1142/S0219498821501644.
- Awata, H., Li, M., Minic, D., Yoneya, T. (2001). On the quantization of Nambu brackets. J. High Energy Phys. 2001(02):013–017. DOI: 10.1088/1126-6708/2001/02/013.
- Bai, R., Li, X., Wu, Y. (2019). 3-Lie-Rinehart algebras. arXiv:1903.12283v1.
- Bai, R., Li, X., Wu, Y. (2020). Hom 3-Lie-Rinehart algebras. arXiv:2001.07570.
- Bai, R., Bai, C., Wang, J. (2010). Realizations of 3-Lie algebras. J. Math. Phys. 51(6):063505. DOI: 10.1063/1.3436555.
- Bai, R., Wu, Y. (2015). Constructions of 3-Lie algebras. Lin. Multilin. Alg. 63(11):2171–2186. DOI: 10.1080/03081087.2014.986121.
- Bai, R., Li, Z., Wang, W. (2017). Infnite-dimensional 3-Lie algebras and their connections to Harish-Chandra module. Front. Math. China 12(3):515–530. DOI: 10.1007/s11464-017-0606-7.
- Ben Hassine, A., Mabrouk, S., Ncib, O. (2019). Some constructions of multiplicative n-ary Hom-Nambu algebras. Adv. Appl. Clifford Algebras 29(5):88. DOI: 10.1007/s00006-019-0996-6.
- Ben Hassine, A., Chtioui, T., Elhamdadi, M., Mabrouk, S. (2021). Extensions and crossed modules of n-Lie Rinehart algebras. arXiv:2103.15006.
- Bkouche, R. (1966). Structures (K, A)-linéaires. C. R. Acad. Sci. Paris Sér. A-B. 262:A373–A376. (in French)
- Calderon Martin, A. J., Forero Piulestan, M. (2011). Split 3-Lie algebras. J. Math. Phys. 52(12):123503.
- Cantarini, N., Kac, V. G. (2010). Classification of simple linearly compact n-Lie superalgebras. Commun. Math. Phys. 298(3):833–853. DOI: 10.1007/s00220-010-1049-0.
- Casas, J. M. (2011). Obstructions to Lie-Rinehart algebra extensions. Algebra Colloq. 18(01):83–104. DOI: 10.1142/S1005386711000046.
- Casas, J. M., Ladra, M., Pirashvili, T. (2004). Crossed modules for Lie-Rinehart algebras. J. Algebra 274(1):192–201. DOI: 10.1016/j.jalgebra.2003.10.001.
- Casas, J. M., Ladra, M., Pirashvili, T. (2005). Triple cohomology of Lie-Rinehart algebras and the canonical class of associative algebras J. Algebra 291(1):144–163. DOI: 10.1016/j.jalgebra.2005.05.018.
- Chemla, S. (1995). Operations for modules on Lie-Rinehart superalgebras. Manuscripta Math. 87(1):199–224. DOI: 10.1007/BF02570471.
- Chen, Z., Liu, Z., Zhong, D. (2011). Lie-Rinehart bialgebras for crossed products. J. Pure Appl. Algebra 215(6):1270–1283. DOI: 10.1016/j.jpaa.2010.08.011.
- Dokas, I. (2012). Cohomology of restricted Lie-Rinehart algebras. Adv. Math. 231(5):2573–2592. DOI: 10.1016/j.aim.2012.08.003.
- Daletskii, Y. L., Kushnirevich, V. A. (1996). Inclusion of the Nambu-Takhtajan algebra in the structure of formal differential geometry. Dopov. Akad. Nauk Ukr. 4:12–18.
- Filippov, V. (1985). n-Lie algebras. Sib. Mat. Zh. 26:126–140.
- Guan, B., Chen, L., Sun, B. (2017). 3-ary Hom-Lie superalgebras induced by Hom-Lie superalgebras. Adv. Appl. Clifford Algebras 27(4):3063–3082. DOI: 10.1007/s00006-017-0801-3.
- Guo, S., Zhang, X., Wang, S. (2019). On 3-Hom-Lie-Rinehart algebras. arXiv:1911.10992.
- Herz, J. (1953). Pseudo-algebras de Lie. I, II. C. R. Acad. Sci. Paris. 236:2289–2291, 1935-1937.
- Higgins, P. J., Mackenzie, K. (1990). Algebraic constructions in the category of Lie algebroids. J. Algebra 129(1):194–230. DOI: 10.1016/0021-8693(90)90246-K.
- Huebschmann, J. (1990). Poisson cohomology and quantization. J. Reine Angew. Math. 408:57–113.
- Huebschmann, J. (1998). Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. Ann. Inst. Fourier (Grenoble). 48(2):425–440. DOI: 10.5802/aif.1624.
- Huebschmann, J. (1999). Duality for Lie-Rinehart algebras and the modular class. J. Reine Angew. Math. 1999(510):103–159. DOI: 10.1515/crll.1999.043.
- Huebschmann, J. (2004). Lie-Rinehart algebras, descent, and quantization. In: Galois Theory, Hopf Algebras, and Semiabelian Categories, (Fields Institute Communications), Vol. 43. Providence, RI: American Mathematical Society, pp. 295–316.
- Huebschmann, J., Perlmutter, M., Ratiu, T. S. (2013). Extensions of Lie-Rinehart algebras and cotangent bundle reduction. Proc. Lond. Math. Soc. 107(5):1135–1172. DOI: 10.1112/plms/pdt030.
- Huebschmann, J. (2017). Multi derivation Maurer-Cartan algebras and sh Lie-Rinehart algebras. J. Algebra 472:437–479. DOI: 10.1016/j.jalgebra.2016.10.008.
- Krahmer, U., Rovi, A. (2015). A Lie-Rinehart algebra with no antipode. Commun. Algebra 43(10):4049–4053. DOI: 10.1080/00927872.2014.896375.
- Kac, V. G. (1977). A sketch of Lie superalgebra theory. Commun. Math. Phys. 52(1):31–64. DOI: 10.1007/BF01609166.
- Kitouni, A., Makhlouf, A., Silvestrov, S. (2016). On (n + 1)-Hom-Lie algebras induced by n-Hom-Lie algebras. Georgian Math. J. 23(1):75–95.
- Kitouni, A., Makhlouf, A., Silvestrov, S. (2020). On solvability and nilpotency for n-Hom-Lie algebras and (n + 1)-Hom-Lie algebras induced by n-Hom-Lie algebras. In: Silvestrov, S., Malyarenko, A., Rancic, M., eds. Algebraic Structures and Applications, Springer Proceedings in Mathematics and Statistics, Vol 317, Ch 6, Cham: Springer, pp. 127–157.
- Mackenzie, K. (1987). Lie groupoids and Lie algebroids in differential geometry. London Math. Soc. Lecture Note Series 124, Cambridge: Cambridge University Press.
- Mackenzie, K. C. H. (1995). Lie algebroids and Lie pseudoalgebras. Bull. London Math. Soc. 27(2):97–147. DOI: 10.1112/blms/27.2.97.
- Mandal, A., Mishra, S. K. (2018). Hom-Lie-Rinehart algebras. Commun. Algebra 46(9):3722–3744. DOI: 10.1080/00927872.2018.1424865.
- Mandal, A., Mishra, S. K. (2018). On hom-Gerstenhaber algebras, and hom-Lie algebroids. J. Geom. Phys. 133:287–302. DOI: 10.1016/j.geomphys.2018.07.018.
- Mishra, S. K., Silvestrov, S. (2020). A review on Hom-Gerstenhaber algebras and Hom-Lie algebroids. In: Silvestrov S., Malyarenko A., Rancić, M., eds. Algebraic Structures and Applications, Springer Proceedings in Mathematics and Statistics, Vol. 317, Ch 11. Cham: Springer, pp. 285–315.
- Palais, R. (1961). The cohomology of Lie rings. Proc. Symp. Pure Math., Amer. Math. Soc. Providence, R. I., pp. 130–137
- Rinehart, G. (1963). Differential forms on general commutative algebras. Trans. Amer. Math. Soc. 108(2):195–222. DOI: 10.1090/S0002-9947-1963-0154906-3.
- Rota, G. (1969). Baxter algebras and combinatorial identities I, II. Bull. Amer. Math. Soc. 75(2):330–334. DOI: 10.1090/S0002-9904-1969-12156-7.
- Takhtajan, L. (1994). On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160(2):295–315. DOI: 10.1007/BF02103278.