On classification and deformations of Lie-Rinehart superalgebras

The purpose of this paper is to study Lie-Rinehart superalgebras over characteristic zero fields, which are consisting of a supercommutative associative superalgebra $A$ and a Lie superalgebra $L$ that are compatible in a certain way. We discuss their structure and provide a classification in small dimensions. We describe all possible pairs defining a Lie-Rinehart superalgebra for $\dim(A)\leq 2$ and $\dim(L)\leq 4$. Moreover, we construct a cohomology complex and develop a theory of formal deformations based on formal power series and this cohomology.


Introduction
Lie-Rinehart algebras are algebraic analogs of Lie algebroids.They first appeared in the work of Rinehart ([19]) and Palais ([17]) and have been studied by Huebschmann ( [11] and [12]).A Lie-Rinehart algebra is a pair (A, L), with A an associative K-algebra, K being a commutative ring, and L a Lie K-algebra.They must be endowed with an action of A on L, making the latter an A-module, and with a Lie algebra map ρ : L −→ Der(A) such that L acts on A by derivations.Some authors put the emphasis on L by saying that L is a K-A Lie-Rinehart algebra.An example comes from Differential Geometry ( [20]): if V is a differential manifold, let A := O V be the algebra of smooth functions on V , and L := Vect(V ) be the Lie algebra of vector fields on V .Then the pair (A, L) carries a Lie-Rinehart algebra structure.According to Claude Roger, Lie-Rinehart algebras give direct methods to deal with the relations between differential operators on manifolds and the enveloping algebra of the Lie algebra of vector fields of a manifold ( [20]).More details can be found in ( [19]).Definition 2.1.Let K be an arbitrary field.A Lie-Rinehart superalgebra is a pair (A, L), where • L is a Lie superalgebra over K, endowed with a bracket [•, •]; • A is an associative and supercommutative K-superalgebra, such that, for all x, y ∈ L and a, b ∈ A: 1. there is an action A × L −→ L, (a, x) −→ a • x, making L an A-module; 2. there is an action of L on A by superderivations: L → Der(A), x → (ρ x : a → ρ x (a)), such that ρ is an even morphism of Lie superalgebras; We sometimes write ρ(x)(a) instead of ρ x (a).
Remark 2.2.If A = K, the one dimensional even superalgebra, then any Lie-Rinehart superalgebra reduces in a certain sense to the ordinary Lie superalgebra over K.
Example 2.3.Let A be a supercommutative unital associative superalgebra and L be a Lie superalgebra.Then the pair (A, L) can always be endowed with a Lie-Rinehart superalgebra structure with the trivial action (the neutral element e 0 for the multiplication of A acts by e 0 • x = x for x ∈ L, and all the other elements of A act by 0) and the zero anchor (ρ(x) = 0 ∀x ∈ L) (see Proposition 3.1).
Example 2.4 (Lie superalgebra of superderivations).Let A be a supercommutative unital associative superalgebra, and L = Der(A) be its superalgebra of superderivations.Then one can check that the pair (A, Der(A)) is a Lie-Rinehart superalgebra, with the action A Der(A) being given by (a•δ)(b) = aδ(b) and the trivial anchor being given by ρ(δ) = δ, for all δ ∈ Der(A), a, b ∈ A.
Example 2.5 (Crossed product).We give now another example of Lie-Rinehart superalgebra, following Chemla ([8]).Let g be a Lie superalgebra, with bracket [•, •], and A a supercommutative associative superalgebra.We suppose that g is endowed with a Lie superalgebra morphism σ : Then, σ x is a superderivation of A. We can define a new Lie superalgebra by setting L := A ⊗ g endowed with the bracket with homogeneous a, b ∈ A and homogeneous x, y ∈ g.
One can check that this formula gives a Lie superalgebra structure on L. We then extend σ to a A-module morphism on L = A ⊗ g, denoted by σ ∈ Der(A), in the following way: define σ(a ⊗ x) := aσ(x) = aσ x .We define a Lie-Rinehart structure on (A, L), with a, b, c ∈ A and x, y ∈ g:

Classification in low dimensions
In this section, we fix K = C, the complex field.We describe all Lie-Rinehart superstructures on a pair (A, L) when dim(A) ≤ 2 and dim(L) ≤ 4.This means, if we are given a supercommutative associative superalgebra (with unit) A and a Lie superalgebra L, we give all the pairs action-anchor which are compatible in the sense of Definition 2.1.We will denote basis elements of A by e s i ,

Supercommutative associative superalgebras
We list the supercommutative associative superalgebras with unit, of dimension up to 2. A list of all supercommutative associative superalgebras of dimension up to 4 can be found in Appendix 5.1.As above, we denote basis elements of A 0 by e 0 i and those of A 1 by e 1 j .The unit is e 0 1 .We only write the non zero products that have to be completed by supercommutativity and multilinearity.
• The purely odd superalgebras A 0|p always have a zero product.

Lie superalgebras
We provide in the sequel a list of Lie superalgebras.As above, we denote the basis elements of L 0 by f 0 i and those of L 1 by f 1 j .We only write the non zero brackets, the other brackets are obtained by super-skewsymmetry and bilinearity.
• The purely odd Lie superalgebras L 0|q always have a zero bracket.

Superderivations of associative superalgebras
If A is a superalgebra, we have given the definition of superderivations of A in Definition 1.4.Let (A, L) be a Lie-Rinehart superalgebra.As an anchor ρ : L −→ L corresponds to a family {ρ(x)} x∈L of superderivations of A, it seems quite natural to study and describe superderivations spaces for all the superalgebras we deal with.We recall that in this case, Der(A) has a Lie superalgebra structure, the bracket being given by the supercommutator.
For every associative superalgebra listed above, we give the general form of the superderivations D. We describe D with respect to the basis e 0 1 , e 0 2 , . . ., e 0 n , e 1 1 , e 1 2 , . . ., e 1 p of the algebra A k n|p .We only write the non-zero values.All the parameters are complex and independent in each column.

Superalgebra
Superderivations of degree 0 Superderivations of degree 1 We provide in this section a classification of Lie-Rinehart superalgebras in low dimensions, using the following general results and the computer algebra system Mathematica.We write only non trivial and non zero relations.If (A, L) is a Lie-Rinehart superalgebra, we denote its dimension by a tuple (n|p, m|q), where n = dim A 0 , p = dim A 1 , m = dim L 0 , q = dim L 1 .We say that an action is trivial if e s i • f t j = f t j if i = 1, s = 0, and 0 otherwise.Using properties of the degrees and basic calculations, we obtain some general results.3. (1|p, 0|q) case: L 0 = {0}, so the action is trivial.Let 1 ≤ k ≤ q and 1 ≤ l ≤ p.We have ρ(f 1 k )(e 1 l ) = r 1 (k,1)(l,1) e 0 1 , r 1 (k,1)(l,1) ∈ C.Then, using the fourth condition of the Definition 2.1, So r 1 (k,1)(l,1) = 0 and the anchor vanishes.There are also some exceptional cases, found by computer calculations, where the only suitable pair is the trivial action and the zero anchor, given by the following proposition: Proposition 3.3.For the following Lie-Rinehart superalgebras, the only compatible action/anchor pair is the trivial action and the zero anchor: One notices that all supercommutative associative superalgebras in the above list have a trivial multiplication, i.e. all the products vanishes, except the products involving the unit.One may conjecture that if the only suitable pair of a Lie-Rinehart superalgebra (A, L) is the trivial action and the zero anchor, then A must have a trivial multiplication.This result is to be proven yet.Proposition 3.4.
• If dim(A, L) = (n|p, m|0) or (A, L) = (n|p, 0|q), the elements of A 1 are acting by 0; • If dim(A, L) = (n|0, m|q), the elements of L 1 are acting (via the anchor) by 0. Now, we list the Lie-Rinehart superalgebras which are not already classified by the above results.They are arranged in lexicographic order of the tuple (n|p, m|q).For each tuple, a table gives all the possible pairs, with all compatible actions and anchors.Every row of a table gives a different Lie-Rinehart superstructure.We give here all the tables with dim(A) ≤ 2 and dim(L) ≤ 2. The tables with dim(L) > 2 are given in Appendix 5.2.
(1|1, 1|1)-type: (λ ∈ C) ), endowed with the null anchor and the action e 0 (2|0, 0|2)-type: Here L = L 1 0|2 and the anchor is always null.We list the possible compatible actions for each supercommutative associative (2|0)-type superalgebra: On classification and deformations of Lie-Rinehart superalgebras We see here that for A = A 2 2|0 , all the anchors are null.It is a straightforward consequence of the fact that the super derivations space of A 2 2|0 is {0}.
Proposition 3.7.Let (A, L) be a Lie-Rinehart superalgebra, with dim(A) ≤ 2 and L abelian.Then either the action is trivial, or the anchor is null.
Proof.Since L is abelian, the compatibility condition becomes: ρ(x)(a) • y = 0 for all x, y ∈ L and a ∈ A.
• A = A 1 (1|0) .We have already seen that the only compatible pair is the one with the trivial action and the null anchor.
. f a = e 0 1 , ρ(x)(e 0 1 ) = 0, so the compatibility condition is satisfied.If a = e 1  1 , we have two cases, depending on the degree of x.
. The superderivations space is reduced to {0}, so all the anchors are null.

Deformation theory of Lie-Rinehart superalgebras
In this section, we provide a deformation theory of Lie-Rinehart superalgebras.The following results are strongly inspired by [14], where the authors discussed a deformation theory of Hom-Lie-Rinehart algebras, including Lie-Rinehart algebras.We also use results from [22] and [5].One needs a cohomology complex constructed below and that controls deformations.
We could deform four different operations: the multiplication of A, the Lie bracket of L, the action A L and the anchor map ρ : L −→ Der(A).Here, we restrict ourselves to deforming only the bracket of L and the anchor map ρ.It follows that the multiplication of A and the action A L remain undeformed in the following theory.All the superalgebras are now K-superalgebras, K being a characteristic zero field.

Super-multiderivations
Let A be a supercommutative associative K-superalgebra, M an A-module and L an A-Lie-Rinehart superalgebra with bracket [•, •] and anchor map ρ.We recall that both maps are even.Definition 4.1 (Super-multiderivations space).We define Der n (M, M ) as the space of multilinear maps f : such that there exists σ f : M ⊗n −→ Der(A) (called symbol map), 1.For all i ∈ {1, . . ., n}: 2. For all a ∈ A: With this definition, we can check that the bracket [•, •] on L belongs to Der 1 (L, L), with symbol map given by the anchor ρ.
Every space Der n (M, M ) has a natural Z 2 -graduation, given by We have Next, we provide a bracket on Der * (M, M ).We adapt the formula of the Nijenhuis-Richardson bracket ( [16]) to the super case.As explained in [22], the space of supermultiderivations Der * (M, M ) is a Z-graded Lie-algebra, but not a bigraded Lie algebra.For f ∈ Der p (M, M ) and g ∈ Der q (M, M ), we define where Sh(q + 1, p) denotes the set of all permutations τ of {0, 1, The sign ε(τ, x 1 , • • • , x p+q+1 ) is implicitly defined by the permutation τ with respect to the parity of the homogeneous elements x 1 , • • • , x p+q+1 ∈ L. For example, if τ = (i, i + 1) is an elementary transposition, then Some computations for τ ∈ S 3 can be found in [2].An explicit expression of ε is given in [5].Proposition 4.3 ([22]).For f ∈ Der p (M, M ) and g ∈ Der q (M, M ), we define a bracket by With this bracket, Der * (M, M ) has a Z-graded Lie algebra structure.
Remark 4.4.The reader should be aware that in [VL15], different conventions are adopted, but the result remains unchanged.

Deformation complex
Let (A, L) be a Lie-Rinehart superalgebra.We construct a deformation complex.
).Or, by using computations on S 3 done in [2] for the signs, we get with an explicit formula given by where ε i and ε j i denote the signs associated to the permutations with respect to the parity of the homogeneous elements x 1 , • • • , x n+1 ∈ L and D ∈ Der * (L, L).Proposition 4.6.The operator δ is a differential, i.e. δ 2 = 0.
This enables us to define a cohomology complex, which will be used next in the deformation theory of Lie-Rinehart superalgebras.Therefore we call it deformation cohomology.For p, q ∈ Z, we have the usual definitions of p-cocycles and q-coboundaries, denoted respectively by Z p def (L) and B q def (L).Finally we set H p def (L) := Z p def (L)/B p def (L) to be the p-th cohomology group.We have for all x, y, z ∈ L.

Formal deformations
In this section, we discuss deformation theory of Lie-Rinehart superalgebras and show that the deformations are controlled by the cohomology defined above.Notice that in the sequel, we aim to deform the Lie bracket and the anchor while we keep fixed the multiplication of the associative superalgebra and its action.We denote by K[[t]] (resp.L[[t]]) the formal power series ring in t with coefficients in K (resp.the formal space in t with coefficients in the vector superspace L).
such that m 0 = m and m i ∈ Der This equation is called deformation equation and is equivalent to an infinite system by identifying the coefficients of t.
Theorem 4.10.Let m t be a deformation of a Lie-Rinehart superalgebra (A, L).Then the infinitesimal of the deformation m 1 is a 2-cocycle with respect to the deformation cohomology.

Supercommutative associative superalgebras
We list supercommutative associative superalgebras with unit.We denote basis elements of A 0 by e 0 i and those of A 1 by e 1 j .The unit is e 0 1 .
• The purely odd superalgebras A 0|p always have a zero product.
On classification and deformations of Lie-Rinehart superalgebras 83 Hence, we can identify Lie-Rinehart superstructures on (A, L) and the corresponding element m ∈ Der 1 (L, L).We setC n def (L, L) := Der n−1 (L, L) and C * def (L, L) := n≥0 C n def (L, L),which we endow with an operator

Definition 4 . 7 .
Let (A, L, [•,•], ρ) be a Lie-Rinehart superalgebra over a field K of characteristic zero, and let m ∈ Der 1 (L, L) be the corresponding element obtained by Proposition 4.5.A deformation of the Lie-Rinehart superalgebra is given by a non-trivial product is zero.5.2 Lie-Rinehart superalgebras with dim(L) > 2 with symbol map given by σ mt = i t i σ m i .As a consequence, m t gives rise to a Lie-Rinehart superstructure on(A[[t]], L[[t]]), with bracket [•, •] t := m t and anchor ρ t := σ mt .Remark 4.9.The first non-zero element m i of the deformation is called the infinitesimal of the deformation.Since m t satisfies [m t , m t ] = 0, we have 1(L, L) with symbol map denoted by σ m i for i ≥ 1, satisfying [m t , m t ] = 0, the bracket being the Z-graded bracket on Der * (L[[t]], L[[t]]).Remark 4.8.The map m t defined on L × L can be extended to a map on L[[t]] × L[[t]] using the K[[t]]-bilinearity.We check that m t is a 1-degree super-multiderivation of L[[t]],