Hom-Lie superalgebras in characteristic 2

The main goal of this paper is to develop the structure theory of Hom-Lie superalgebras in characteristic 2. We discuss their representations, semidirect product, $\alpha^k$-derivations and provide a classification in low dimension. We introduce another notion of restrictedness on Hom-Lie algebras in characteristic 2, different from one given by Guan and Chen. This definition is inspired by the process of queerification of restricted Lie algebras in characteristic 2. We also show that any restricted Hom-Lie algebra in characteristic 2 can be queerified to give rise to a Hom-Lie superalgebra. Moreover, we develop a cohomology theory of Hom-Lie superalgebras in characteristic 2, which provides a cohomology of ordinary Lie superalgebras. Furthermore, we establish a deformation theory of Hom-Lie superalgebras in characteristic 2 based on this cohomology.


Introduction
Throughout the text, K stands for an arbitrary field of characteristic 2. In almost all our constructions, K is arbitrary.There are a few instances where K is required to be infinite.We will point out at these instances.
1.1.Lie superalgebras in characteristic 2. Roughly speaking, a Lie superalgebra in characteristic 2 is a Z/2-graded vector space that has a Lie algebra structure on the even part, and endowed with a squaring on the odd part that satisfies a modified Jacobi identity, see §2.1 for a precise definition.Because we are in characteristic 2, those Lie superalgebras are sometimes being confused with Z/2-graded Lie algebras, though they are totally different algebras due to the presence of the squaring.They can, however, be considered as Z/2-graded Lie algebra by forgetting the super structure.The other way round is not always true in general.
The classification of simple Lie superalgebras in characteristic 2 is still an open and wide problem.Nevertheless, Lie superalgebras in characteristic 2 admitting a Cartan matrix have been classified in [BGL], with the following assumption: each Lie superalgebra posses a Dynkin diagram with only one odd node.The list of non-equivalent Cartan matrices for each Lie superalgebra is also listed in [BGL].Moreover, it has recently been showed in [BLLSq] that each finite-dimensional simple Lie superalgebra in characteristic 2 can be obtained from a simple finite-dimensional Lie algebra in characteristic 2 by one of two methods, hence reducing the classification to the classification of simple Lie algebras which in its own is a very tough problem.As a matter of fact, there are plenty of (vectorial and non-vectorial) Lie superalgebras in characteristic 2 that have no analogue in other characteristics, see [BeBou, BGLLS, BLLSq, BLS, LeD].
It is worth mentioning that the characteristic 2 case is a very tricky case, due to the presence of the squaring.It does require new ideas and techniques.
1.2.Hom-Lie superalgebras in characteristic 2. The first instances of Hom-type algebras appeared in physics literature, see for example [CKL], where q-deformations of some Lie algebras of vector fields led to a structure in which Jacobi identity is no longer satisfied.This class of algebras were formalized and studied in [HLS, LS], where they were called Hom-Lie algebras since the Jacobi identity is twisted by a homomorphism.The super case were considered in [AM], where Hom-Lie superalgebras were introduced as a Z/2-graded generalization of the Hom-Lie algebras.The authors of [AM] characterized Hom-Lie admissible superalgebras and proved a Z/2-graded version of a Hartwig-Larsson-Silvestrov Theorem which led to a construction of a q-deformed Witt superalgebra using σ-derivations.Moreover, they derived a one parameter family of Hom-Lie superalgebras deforming the orthosymplectic Lie superalgebra osp(1|2).The cohomology of Hom-Lie superalgebras was defined in [AMS].Notice that all these studies and results were performed over a field of characteristic 0.
1.3.The main results.The main purpose of this this paper is to tackle the positive characteristic and provide a study of Hom-Lie superalgebras in characteristic 2. We introduce the main definitions and some key constructions as well as a cohomology theory fitting with a deformation theory.In Section 2, we recall some basic definitions, and introduce Hom-Lie algebras and Hom-Lie superalgebras over fields of characteristic 2 and some related structures.We show that a Lie superalgebra in characteristic 2 and an even Lie algebra morphism give rise to a Hom-Lie superalgebra in characteristic 2.Moreover, we provide a classification of Hom-Lie superalgebras in characteristic 2 in low dimensions.In Section 3, we consider representations and semidirect product of Hom-Lie superalgebras in characteristic 2. The structure map defining a Hom-Lie superalgebra in characteristic 2 allows a new type of derivations called α k -derivations discussed in Section 4. In Section 5, we introduce the notion of p-structure and discuss queerification of restricted Hom-Lie algebras in characteristic 2. Section 6 is dedicated to cohomology theory.We construct a cohomology complex of a Hom-Lie superalgebra g in characteristic 2 with values in a g-module.This cohomology complex has no analogue in characteristic p = 2.In the last section, we provide a deformation theory of Hom-Lie superalgebras in characteristic 2 using the cohomology we constructed.

Backgrounds and main definitions
Let V and W be two vector spaces over K.A map s : V → W is called a squaring if (1) s(λx) = λ 2 s(x) for all λ ∈ K and for all x ∈ V .
2.1.Lie superalgebras in characteristic 2. Following [LeD], a Lie superalgebra in characteristic 2 is a superspace g = g0 ⊕ g1 over K such that g0 is an ordinary Lie algebra, g1 is a g0-module made two-sided by symmetry, and on g1 a squaring, denoted by s g : g1 → g0, is given.The bracket on g0, as well as the action of g0 on g1, is denoted by the same symbol [•, •].For any x, y ∈ g1, their bracket is then defined by The bracket is extended to non-homogeneous elements by bilinearity.The Jacobi identity involving the squaring reads as follows: , [x, y]] for any x ∈ g1 and y ∈ g.
For any Lie superalgebra g in characteristic 2, its derived algebras are defined to be (for i ≥ 0) We denote the space of all derivations of g by der(g).
Therefore, morphisms in the category of Lie superalgebras in characteristic 2 preserve not only the bracket but the squaring as well.In particular, subalgebras and ideals have to be stable under the bracket and the squaring.
Remark 2.1.Associative superalgebras in characteristic 2 leads to Lie superalgebras in characteristic 2. The bracket is standard and the squaring is defined by s(x) = x • x, for every odd element x.

2.2.
Hom-Lie algebras in characteristic 2. A Hom-Lie algebra in characteristic 2 is a vector space g over K and a map α ∈ End(g) together with a bracket satisfying the following conditions: Such a Hom-Lie algebra will be denoted by (g, [•, •], α).
A representation of a Hom-Lie algebra where V is a vector space, β ∈ gl(V ), and [•, •] V is the action of g on V such that (for all x, y ∈ g and for all v ∈ V ): Writing Eq. ( 5) using the notation of Eq. ( 4), we put ρ β := [•, •] V and obtain (for all x, y ∈ g): 2.3.Hom-Lie superalgebras in characteristic 2. Our main definition is given below.Due to the presence of the squaring, our approach to define Hom-Lie superalgebras in characteristic 2 will differ from that used in characteristics p = 2, see [AM].
•], a squaring s : g1 → g0, and an even map α ∈ End(g) such that (i) (g0, [•, •], α| g0 ) is an ordinary Hom-Lie algebra, (ii) g1 is a g0-module made two-sided by symmetry, where the action is still denoted by the bracket is bilinear, and induces the bracket on odd elements; namely, for any x, y ∈ g1: (iv) the following three conditions hold The Jacobi identity on triples in {g 0 , g 1 , g 1 } and {g 1 , g 1 , g 1 } follow from condition 8. We, therefore, recover the usual definition of Hom-Lie superalgebras [AM].
(iii) We may want to consider Hom-Lie superalgebras in characteristic 2 without conditions ( 9) and (10), which corresponds to the multiplicativity of the structure map α.
s, α) be a Hom-Lie superalgebra in characteristic 2. Let I be a subset of g.The set I is called an ideal of g if and only if I is closed under addition and scalar multiplication, together with [I, g] ⊆ I, α(I) ⊆ I and s(x) ∈ I whenever x ∈ I ∩ g1.
In particular, if the ideal I is homogeneous; namely I = I ∩ g0 ⊕ I ∩ g1 = I0 ⊕ I1 then the condition involving the squaring reads s(x) ∈ I0 for all x ∈ I1.In addition, the superspace g/I is also a Hom-Lie superalgebra in characteristic 2. The bracket and the squaring are defined as follows: [x + I, y + I] := [x, y] + I for all x, y ∈ g, s(x + I) := s(x) + I for all x ∈ g1, while the twist map α on g/I is defined by α(x + I) = α(x) + I for all x ∈ g.
We will only show that the squaring is well-defined.Suppose that x − x = i ∈ I1 we have, In the following proposition, we will show that an ordinary Lie superalgebra together with a morphism give rise to a Hom-Lie superalgebra structure on the underlying vector space.
Proposition 2.4.Let (g, [•, •], s) be a Lie superalgebra in characteristic 2, and let α : g → g be an even superalgebra morphism.Then Proof.The first part of the proof is given in [AM].We have to check Eqns.( 1) and ( 8).Indeed, let λ ∈ K and let x ∈ g1.We have On the other hand, for any x ∈ g1 and y ∈ g, we have More generally, let (g, [•, •], s, α) be a Hom-Lie superalgebra in characteristic 2 and let β : g → g be an even weak superalgebra morphism (the third condition of (11) is not necessary satisfied).Then (g, [•, •] The proof is similar to that of Proposition 2.4.
Let us define the map α on the vector space underlying oo (1) A direct computation shows that the map α is a morphism of Lie superalgebras if and only if (where we have put for simplicity T := 1 + δ 2 ε 1 + δ 1 ε 2 ): together with ( 12) The only solutions to Eqns.(12) that do not produce the zero map are given by T = 0. We can, therefore, construct a Hom-Lie superalgebra by means of the map α, depending on three parameters, as in Proposition 2.4.So, we have In particular, we have the following Hom-Lie superalgebra in characteristic 2, which we denote by oo (1) IΠ (1|2) α , defined by the brackets: with the corresponding squaring: and the twist map where ε is a parameter in K.We recover the Lie superalgebra oo 2.4.The classification in low dimensions.Let us assume here that the field K is infinite (for instance, algebraically closed).For the classification of Hom-Lie algebras and superalgebras in low dimensions, see [MS,GSS,GSSc,LL,ORS1,ORS2,R,WZW].
2.4.1.The case sdim(g) = 1|1.Assume that g0 = Span{e} and g1 = Span{f }.We set By straightforward computations on the conditions, one gets that the only non-trivial case is given by λ 1 = 1 and γ = ρλ 2 .Therefore, any (1|1)-dimensional Hom-Lie superalgebra in characteristic 2 is isomorphic to the Hom-Lie superalgebra given, with respect to basis {e, f }, by where λ and ρ are non-zero parameters.As the field K is infinite, we have a family of Hom-Lie superalgebras that depends on parameters λ and ρ.
2.4.2.The case sdim(g) = 1|2.Assume that g0 = Span{e} and g1 = Span{f 1 , f 2 }.We define the brackets as (where and finally the squaring as (where ρ i ∈ K for i = 1, 2, 3): Let us consider a linear map α by which we will construct the Hom-structure.As α preserves the Z/2Z-grading, and by using the Jordan decomposition we distinguish two cases: Case 1: Suppose that α is given by (where s, t 1 , r 2 ∈ K): A direct computation shows that there are only the following sub-cases: Sub-case 1a): We have Here are the two possible cases: We can disregard this case, because it produces a Lie algebra instead of a Lie superalgebra.

Subcase 2c):
We have ρ 1 = 0 but ρ 2 = ρ 3 = 0, together with Subcase 2d): We have ρ 1 = 0, ρ 2 = 0 but ρ 3 arbitrary, together with The tables below summarize our finding.We find it convenient to order the Hom-Lie superalgebras into two groups: (i) of type I are those for which the g 0 -module structure on g1 is trivial; (ii) of type II are those for which the g 0 -module structure on g1 is not trivial.
The HLSA The squaring s The conditions The HLSA The squaring s [g0, g1] The conditions The HLSA The squaring s The conditions The HLSA The squaring s [g0, g1] The conditions We say that V is a g-module.
Sometimes it is more convenient to use the notation ρ β = [•, •] V and write: (14) With the above notation, we define a Hom-Lie superalgebra structure on the superspace g ⊕ V = (g0 + V0) ⊕ (g1 + V1), where the bracket is defined by the squaring s g+V : g1 + V1 → g0 + V0 is defined by and the structure map α g⊕V : g ⊕ V → g ⊕ V defined by The Hom-Lie superalgebra Proof.Checking Axioms (i) and (ii) of Definition 2.2 is a routine; we can refer to [AM].
We should check the conditions relative to the squaring.Let us first check that the map s g ⊕ V is indeed a squaring.For all x + v ∈ g1 ⊕ V1 and for all λ ∈ K, we have Now, for all x + v ∈ g1 ⊕ V1 and for all y + w ∈ g ⊕ V , we have On the other hand, Therefore, Eq. ( 8) is satisfied.Now, Therefore, Eq. ( 10) is satisfied.
In the following proposition, we show how to twist a Lie superalgebra and its representation into a Hom-Lie superalgebra together with a representation in characteristic 2.
Proposition 3.3.Let (g, [•, •] g , s g ) be a Lie superalgebra and (V, ρ) a representation.Let α : g → g be an even superalgebra morphism and β ∈ gl(V ) be a linear map such that ρ(α(x)) Proof.We have already proved in Proposition 2.4 that (g, [•, •] g,α , s g,α , α) is a Hom-Lie superalgebra.Let us check that (V, ρ β , β) is a representation with respect to (g, [•, •] g,α , s g,α , α).Indeed, the first condition is provided by the hypothesis while the second and the third ones are straightforward.Let us check the last one.For any x ∈ g1 and v ∈ V , we have and The equality follows from the fact that Example 3.4.The classification of irreducible modules over oo (1) IΠ (1|2) having highest weight vectors has been carried out in [BGKL].We will borrow here the simplest example.Consider the Hom-Lie superalgebra oo (1) IΠ (1|2) with the twist α given as in Example (2.5).We consider the oo (1) The vector m 1 is a highest weight vector with weight (m 1 ) = (1).The map β is given as follows: , where the coefficients δ 1 , δ 2 , ε 1 , ε 2 are given as in Example 2.5.
Here we will introduce another point of view concerning the representations of Hom-Lie superalgebras in characteristic 2, inspired by [Sh].
Let V = V0 ⊕ V1 be a vector superspace, and let β ∈ GL(V ) be even map.We will define a bracket on gl(V ) as well as a product as follows: (where β −1 is the inverse of β): is obviously bilinear on gl(V )1 as well.

Denote by Ad
Proposition 3.5.The brackets and the squaring defined in Eqns.( 15) and ( 16) make Proof.The map Ad β is invertible with inverse Ad β −1 .Let us check the multiplicativity conditions: Similarly, For the Jacobi identity, let us just deal with the squaring.The LHS of the JI reads (for all f ∈ gl(V )1 and for all g ∈ gl(V )) Theorem 3.6.Let (g, [•, •] g , s g , α) be a Hom-Lie superalgebra in characteristic 2. Let V be a vector superspace, and let β ∈ GL(V ) be even.Then, the map Proof.Let us only proof one direction.Suppose that For the squaring, we have It follows that ρ β is a homomorphism of Hom-Lie superalgebras in characteristic 2.

α k -Derivations
Let (g, [•, •], s, α) be a Hom-Lie superalgebra in characteristic 2. We denote by α k the k-times composition of α, where α 0 is the identity map.We will be needing the following linear map  17)) is an α k -derivation.Let us just check the condition related to the squaring.Indeed, Let us denote the space of α k -derivations by der α (g).We have the following proposition.
Proposition 4.3.The space der α (g) can be endowed with a Lie superalgebra structure in characteristic 2. The bracket is the usual commutator, and the squaring is given by s der α (g) (D) := D 2 for all D ∈ der α 1 (g).Proof.As we did before, we only prove the requirements when the squaring is involved.Let us first show that D 2 is an α 2k -derivation.Checking the bracket is a routine.For the squaring, we have (for all x ∈ g1): Before we proceed with the proof, let us re-denote the space der α (g) by h for simplicity.
Now, for all D ∈ h1 and for all E ∈ h1, we have (for all x ∈ g): On the other hand, The space der α (g) is actually graded as der α (g) = ⊕der α k (g) where der α k (g) is the space of α k -derivations where k is fixed.Indeed, we have [der α k (g), der α l (g)] ⊆ der α k+l (g) and s(der α k (g)1) ⊆ der α 2k (g).

p-structures and queerification of Hom-Lie algebras in characteristic 2
We will first introduce the concept of p-structures on Hom-Lie algebras.In the case of Lie algebras, the definition is due to Jacobson [J].Roughly speaking, one requires the existence of an endomorphism on the modular Lie algebra that resembles the pth power mapping x → x p in associative algebras.In the case of Hom-Lie algebra, there is a definition proposed in [GC] but it turns out that this definition is not appropriate to queerify a restricted Hom-Lie algebras in characteristic two, as done in [BLLSq] in the case of ordinary restricted Lie algebras.Here, we will give an alternative definition and justify the construction.Definition 5.1.Let g be a Hom-Lie algebra in characteristic p with a twist α.A mapping [p] α : g → g, a → a [p]α is called a p-structure of g and g is said to be restricted if α for all x ∈ g and for all λ ∈ K; Let us exhibit this p-structure in the case where p = 2.The conditions (R2) and (R3) read, respectively, as Proposition 5.2.Twisting with a morphism α an ordinary Lie algebra with a p-structure gives rise to a Hom-Lie algebra with a p-structure.More precisely, given an ordinary Lie algebra (g, [•, •]) and a Lie algebra morphism α.

Proof. It has been shown in
, is a Hom-Lie algebra.Now, let us show that the map [p] α defines a p-structure on the Hom-Lie algebra (g, [•, •] g ).Indeed, let us check Axiom (R1).The LHS reads [p] , y]).

The RHS reads
Axiom (R2) is obviously satisfied.Let us check Axiom (R3).Indeed, The proof is now complete.
Proposition 5.3.Let g be a restricted Hom-Lie algebra in characteristic 2 with a twist map α.On the superspace h := g ⊕ Π(g) there exists a Hom-Lie superalgebra structure defined as follows (for all x, y ∈ g): Proof.Let us check that the map s h is indeed a squaring on h.The condition s h (λΠ(x)) = λ 2 s h (Π(x)), for all λ ∈ K and for all x ∈ g, is an immediate consequence of condition (R2).Moreover, the map is obviously bilinear because it coincides with the Lie bracket on g.
Let us check the Jacobi identity involving the squaring.Indeed, for all y ∈ h0 and for all Π(x) ∈ h1, we have On the other hand For all Π(y) ∈ h1 and for all Π(x) ∈ h1, we have On the other hand Proposition 5.4.Let g be a restricted Lie algebra in characteristic 2 and h := g ⊕ Π(g) be its queerification, see [BLLSq], defined as follows (for all x, y ∈ g): Let α : g → g be a Lie algebra morphism.Let us extend it to α on h by declaring α(Π(x)) := Π(α(x)) for all x ∈ g.Then twisting the Lie superalgebra h along α is exactly the queerification of the Hom-Lie algebra g α obtained by twisting g along α.Namely, Proof.Let x, y ∈ g.We have On the other hand, Similarly, one can easily prove that Let us only prove that their squarings coincide.Indeed, for all x ∈ g we have 2] ).
6. Cohomology and Deformations of finite dimensional Hom-Lie superalgebras 6.1.Cohomology of ordinary Lie superalgebras in characteristic 2. In this section we define a cohomology theory of Lie superalgebras in characteristic 2. The first instances can be found in [BGLL1].Let g be a Lie superalgebra in characteristic 2 and M be a g-module.Let us introduce a map with the following properties: (i) p(λx, z) = λ 2 p(x, z) for all x ∈ g1, for all z ∈ ∧ n g and for all λ ∈ K; (ii) For all x ∈ g1, the map z → p(x, z) is multi-linear.
For n = 0, the map p should be understood as a quadratic form on g1 with values in M.
We are now ready to define the space of cochains on g with values in M. We set (n > 1) is a map as in ( 21) such that p(x + y, z) + p(x, z) + p(y, z) = c(x, y, z) for all x, y ∈ g1 and z ∈ ∧ n−2 g}.
Theorem 6.1.The maps d n is well defined.Moreover, for all integers n Hence, the pair (XC * (g, M), d * ) defines a cohomology complex for Lie superalgebras in characteristic 2.
The proof of the theorem will be given next when considering the cohomology of Hom-Lie superalgebras that reduce to ordinary Lie superalgebras when the structure map is the identity.6.2.Elucidation for n = 2, 3. Let us first exhibit the sets of cochains in the case where n = 2, 3.
If v ∈ M and a, b ∈ g * 1, we can define the cochain (v ⊗ a ∧ b, q) ∈ XC 2 (g, M) such that the quadratic form is q(x) = a(x)b(x) v for all x ∈ g1.The polar form1 associated to q is B q (x, y) = (a(x)b(y) + a(y)b(x)) v for all x, y ∈ g1.
In particular, we can define the cochain c = v ⊗ a ∧ a, where q(x) = v(a(x)) 2 for all x ∈ g1 and c(x, y) = 0 for all x, y ∈ g.
A 1-cocycle c on g with values in a g-module M must satisfy the following conditions: A 2-cocycle (c, q) on g with values in M must satisfy the following conditions: for all x ∈ g1 and for all z ∈ g, 6.3.Cohomology of Hom-Lie superalgebras in characteristic 2. Let (g, [•, •], s, α) be a Hom-Lie superalgebra in characteristic 2 and (M, β) be a g-module, see Definition 3.1.The space of n-cochains are defined similarly to ( 22) with a slight difference with respect to degree 0 space and an extra condition that is where Note that these definitions are consistent as showed by the following Lemma.
In particular for n = 2, the differential is given by where for all x ∈ g1, and for all z 1 ∈ g.
A 2-cocycle is 2-tuple (c, p) satisfying the following conditions: for all x ∈ g1 and for all z 1 ∈ g, The first step here is to show that the map d n α is well defined, for every twist α.By doing so, we give a proof to the first part of Theorem 6.1 in the case where α = id.Proposition 6.3.The map d n α is well-defined; namely, Im (d n α ) ⊆ XC n+1 α (g, M).
In order to prove this theorem, we will need the following Lemma.Lemma 6.5.
There are five terms in the expression above.We will compute each term separately.
Now, using Lemma 6.5 a direct computation shows that Part 1+Part 2+Part 3+Part 4+Part 5 = 0. Now, we are ready to define a cohomology of Hom-Lie superalgebras in characteristic 2. The kernel of the map d n α , denoted by Z n α (g; M), is the space of n-cocycles.The range of the map d n−1 α , denoted by B n α (g; M), is the space of coboundaries.We define the n th cohomology space as H n α (g; M) := Z n α (g; M)/B n α (g; M).Remark 6.6.The cohomology defined above coincides when α = id g and β = id M , with the cohomology of Lie superalgebras in characteristic 2 defined in the previous section.
Example 6.7.We compute the second cohomology of the Hom-Lie superalgebra oo (1) IΠ (1|2) α defined in Example 2.5.We will assume here that the field K is infinite.
Let us first show that cocycles of the form (0, p) are necessarily trivial.In fact, the condition p = p • α and ε = 0 implie that p(x 1 ) = 0 and p(y 1 ) = m (arbitrary).
Choose B m = m y * 2 , where m ∈ K.A direct computation shows that d 2 α B m = 0. Let us compute the corresponding q m .Indeed, ) and hence its cohomlogy class is trivial.Let us now describe 2-cocycles of the form (c, p).A direct computation shows that Let us describe the corresponding p's.We have We then get that d IΠ (1|2) α ; K) is trivial.(ii) Let us now compute the cohomology space: H 2 α (oo IΠ (1|2) α ).Recall that in the case where α = Id, this cohomology space has only two non-trivial 2-cocycles.
6.4.Deformations of Hom-Lie superalgebras.The deformation theory of Hom-Lie superalgebras in characteristic 2 will be discussed here.As a result, we also cover the Lie case, namely α = Id g .Over a field of characteristic zero, the study has been carried out in [AMS, TR].
Let (g, [•, •], s, α) be a Hom-Lie superalgebra over a field K of characteristic 2. A deformation of g is a family of Hom-Lie superalgebras g t specializing in g when the parameter t equals 0 and where the Hom-Lie superalgebra structure is defined on the tensor product g⊗K[[t]] when g is finite dimensional.The bracket in the deformed Hom-Lie superalgebra g t is a K[[t]]-bilinear map of the form (for all x, y ∈ g): [x, y] t = [x, y] + i≥1 c i (x, y)t i , while the squaring s t , with respect to K[[t]], on the Hom-Lie superalgebra g t is given by (for all x ∈ g1): where (c i , p i ) ∈ XC 2 α, 0(g; g) for all i ≥ 1.We will assume that c 0 (•, •) = [•, •] and p 0 (•) = s(•).
According to deformation theory, we call a deformation infinitesimal if the bracket [•, •] t and the squaring s t (•) define a Hom-Lie superalgebra structure mod (t 2 ) (degree 1), that is [•, •] t = [•, •]+c 1 (•, •)t and s t (•) = s(•)+p 1 (•)t.A deformation is said to be of order n if the bracket [•, •] t and the squaring s t (•) define a Hom-Lie superalgebra structure mod (t n+1 ), that is Afterwards, one tries to extend a deformation of order n to a deformation of order n + 1.All obstructions are cohomological, as we will see.
A deformation of order n − 1 can be extended to a deformation of order n if and only there exists Proof.(i) Checking that c 1 satisfies the condition (34) is a routine, see [AMS].Let us deal with the squaring s t .We have Collecting the coefficient of t in the condition [s t (x), α(y)] t = [α(x), [x, y] t ] t , we get c 1 (s(x), α(y)) + [α(y), p i (x)] + c 1 (α(x), [x, y]) + [α(x), c 1 (x, y)] = 0, which corresponds to Condition (35).Therefore, (c 1 , p 1 ) is a 2-cocycle on g with values in the adjoint representation.