Lie Supergroups Obtained from 3-Dimensional Lie Superalgebras Associated to the Adjoint Representation and Having a 2-Dimensional Derived Ideal

Abstract

We give the explicit multiplication law of the Lie supergroups for which the base manifold is a 3-dimensional Lie group and whose underlying Lie superalgebra 𝔤=𝔤₀⊕𝔤₁ which satisfies 𝔤₁=𝔤₀, 𝔤₀ acts on 𝔤₁ via the adjoint representation and 𝔤₀ has a 2-dimensional derived ideal.

1. Introduction

A possible notion of superspace associated to a given 3-dimensional Lie algebra 𝔤₀ might be a Lie superalgebra 𝔤=𝔤₀⊕𝔤₁. If one is to introduce a minimal set of assumptions, it seems quite natural to consider those superalgebras for which 𝔤₁=𝔤₀ and for which the 𝔤₀ action on 𝔤₁ is the adjoint representation. A complete classification of the real and complex (3,3)-dimensional Lie superalgebras having precisely this restriction was recently obtained in [1]. The classification was given in terms of the dimension of the derived ideal 𝔤^′₀=[𝔤₀,𝔤₀]. The most trivial case was that corresponding to dim𝔤^′₀=3 whereas the most difficult case was that of dim𝔤^′₀=0. Our aim in this paper is to explicitly produce the real and complex Lie supergroups associated to the Lie superalgebras classified in [1] having a 2-dimensional derived ideal. This is just one step forward in our understanding of the real and complex (3,3)-dimensional Lie supergroups.

In order to maintain our exposition as self-contained as possible, we will summarize the basic statements from [1]. Once we fix the adjoint representation, it is well known (see [1, 2]) that there are as many Lie superalgebras as bilinear symmetric maps Γ∶𝔤₀×𝔤₀→𝔤₀ satisfyingΓ([𝑥,𝑦],𝑧)+Γ(𝑦,[𝑥,𝑧])=[𝑥,Γ(𝑦,𝑧)]. (1.1)

Furthermore, the isomorphism class of the Lie superalgebra defined by such a given Γ is completely determined by its orbit under the action of the group 𝐺⊂Aut(𝔤₀)×GL(𝔤₀) of pairs (𝑇,𝑆) satisfying 𝑆∘𝑇⁻¹∘ad(⋅)=ad(⋅)∘𝑆∘𝑇⁻¹ given by Γ→(𝑇,𝑆)⋅Γ=𝑇(Γ(𝑆⁻¹(⋅),𝑆⁻¹(⋅))). (1.2)

Now, let 𝔽 be either the real or the complex number field, and let 𝔤₀ be a fixed 3-dimensional Lie algebra over 𝔽. Let {𝑒₁,𝑒₂,𝑒₃} be a basis for 𝔤₀, and let Sym_ad(𝔤₀) be the real or complex vector space of symmetric, bilinear maps Γ∶𝔤₀×𝔤₀→𝔤₀ satisfying (1.1). Thus, we can identify the space Sym_ad(𝔤₀) with the set of triples (Γ¹,Γ²,Γ³) of symmetric bilinear forms, for which Γ(𝑢,𝑣)=∑³_𝑖=1Γ^𝑖(𝑢,𝑣)𝑒_𝑖 satisfies (1.1).

Let 𝔤^′₀=[𝔤₀,𝔤₀] be the derived ideal of 𝔤₀. Lie algebras having dim𝔤^′₀=2 can be classified by choosing 𝑒₁∉𝔤^′₀ and bringing ad(𝑒₁) into a convenient canonical form (see [3, 4] or [5] for details). Thus, [𝑒₁,𝑒₂]=𝑎𝑒₂+𝑐𝑒₃,[𝑒₁,𝑒₃]=𝑏𝑒₂+𝑑𝑒₃,sothatad(𝑒₁)|_𝔤′0⟷𝐴=,𝑎𝑏𝑐𝑑[𝑒₂,𝑒₃]=0, (1.3) and, since dim𝔤^′₀=2, then 𝐴 must be an invertible matrix. Now, the classification up to isomorphism of the Lie algebras 𝔤₀ having dim𝔤^′₀=2 can be written as in Table 1, where we followed the notation introduced in [1].

Table 1 

Full set of representatives of the isomorphism classes of the 3-dimensional Lie algebras 𝔤₀ that have 2-dimensional derived ideal. The classes were obtained from the canonical form of the 2×2 invertible matrix 𝐴. In the first two cases, the field 𝔽 can be either ℝ or ℂ.

It is easy to see that the matrices of ad(𝑒_𝑖) take the forms ad(𝑒₁)=000𝐴,ad(𝑒₂)=−00𝐴𝛿₁0,ad(𝑒₃)=−00𝐴𝛿₂0, (1.4) where 𝛿₁=(¹₀) and 𝛿₂=(⁰₁). It is well known that the associated Lie groups for the Lie algebras we are dealing with here are the so-called unimodular groups 𝐸(2) and 𝐸(1,1) for 𝔮₁(𝔽) and 𝔮₋₁(𝔽), respectively, and for the other Lie algebras, the associated Lie groups are called nonunimodular (see [6]). We shall denote by 𝐺₀(𝐴) any of those Lie groups.

It was proved in [1] that when 𝐴 is invertible, any triple (Γ¹,Γ²,Γ³), for which Γ(𝑢,𝑣)=∑³_𝑖=1Γ^𝑖(𝑢,𝑣)𝑒_𝑖 satisfies (1.1), is given, up to isomorphism, by Γ¹=𝑝100000000,Γ²=𝑝2010100000,Γ³=𝑝2001000100, (1.5) where 𝑝 is an arbitrary element of the ground field 𝔽 for the three classes 𝔭(𝔽),𝔮_𝜆(𝔽), and 𝔮¹_𝜆(ℝ), with no relation between the parameters 𝑝 and 𝜆.

What we do in this paper is to describe explicitly all the Lie supergroups whose underlying 3-dimensional Lie group is 𝐺₀(𝐴) and having as Lie superalgebras of left-invariant supervector fields the Lie superalgebras given by the triple (Γ¹,Γ²,Γ³) above. The method is the one given in [7] and is essentially the one used in Lie's classical theory, that is, we give first a faithful representation of the Lie superalgebra into the Lie superalgebra of vector fields of some supermanifold, and obtain the local coordinate version of the supergroup multiplication law through composition of their integral flows depending on the integration parameters (see [7]).

In more detail, we aim to describe the multiplication law in terms of tetrads (𝑠,𝑣;𝜎,𝜃), where (𝑠,𝑣)∈𝔽×𝔽² are the local coordinates on the 3-dimensional Lie group 𝐺₀(𝐴), and 𝜎 and 𝜃 are the odd coordinates on the supergroup (𝐺₀(𝐴),Λ(𝐸)), where Λ(𝐸) stands for the sheaf of sections of the exterior algebra bundle associated to the rank-3 vector bundle 𝐸→𝐺₀(𝐴), whose typical fiber 𝔤₀ can be decomposed as 𝑆⊕𝔤^′₀, with 𝑆≃𝔤₀/𝔤^′₀. Thus, 𝜎 is a local section of 𝐺₀(𝐴)×𝑆→𝐺₀(𝐴), and 𝜃 is a local section of 𝐺₀(𝐴)×𝔤^′₀→𝐺₀(𝐴). Then, the product (𝑠^′,𝑣^′;𝜎^′,𝜃^′)∗(𝑠,𝑣;𝜎,𝜃) is given by (see Theorem 3.1) (𝑠^′𝑝+𝑠−2𝜎^′𝜎,𝑣+𝑒^{(−𝑠+(𝑝/2)𝜎′𝜎)𝐴}𝑣^′+𝑝2𝜎𝑒^−𝑠𝐴𝜃^′;𝜎^′+𝜎,𝜃+𝑒^{(−𝑠+(𝑝/2)𝜎′𝜎)𝐴}𝜃^′+𝜎^′𝐴𝑣). (1.6)

The point of giving such an explicit expression is to actually see the way odd coordinates are combined, within the supergroup composition law, to produce even sections. It was this interaction between even and odd coordinates what apparently had some physical and geometrical ideas that were worth studying but there are only a few explicit examples in the literature; some of them are incomplete, and some are relatively trivial.

Once we obtain the multiplication law for the different Lie supergroups, we give in Proposition 3.2 the supermorphisms that define the Lie supergroups within the spirit of [7, 8]. Finally, in Proposition 3.3 we compute the left-invariant supervector fields associated to the Lie supergroups we have built, bringing us back to the Lie superalgebras we started with.

2. Lie Superalgebra Representations

Let us write𝐴¹=000𝐴,𝐴²=𝛿00₁0,𝐴³=𝛿00₂0, (2.1) where 𝛿₁ and 𝛿₂ are defined as above. Let 𝐵^𝑟 and 𝐶^𝑟 be 3×3 matrices (𝑟=1,2,3).

Proposition 2.1. Let 𝔤=𝔤₀⊕𝔤₁ be a (3,3)-dimensional Lie superalgebra, where 𝔤₁=𝔤₀, 𝔤₀ acts on 𝔤₁ via the adjoint representation, and 𝑑𝑖𝑚𝔤^′₀=2. Let {𝑒₁,𝑒₂,𝑒₃} and {𝑓₁,𝑓₂,𝑓₃} be bases for 𝔤₀ and 𝔤₁, respectively. Let 𝑉=𝑉₀⊕𝑉₁ be a (3,3)-dimensional supervector space and let 𝜌∶𝔤→𝔤𝔩(𝑉) be a linear map such that𝜌(𝑒_𝑖𝐴)=^𝑖00𝐴^𝑖,𝜌(𝑓_𝑖)=0𝐵^𝑖𝐶^𝑖0. (2.2)Then, the choices𝐵^𝑖=𝑝2𝐴^𝑖,𝐶^𝑖=𝐾𝐴^𝑖, (2.3) where 𝐾=(⁰⁰_0𝐴−1), turn 𝜌∶𝔤→𝔤𝔩(𝑉) into a faithful representation of the Lie superalgebra 𝔤.

Proof. Let us write𝐵^𝑟=𝛽_𝑟𝑢^𝑇_𝑟𝑣_𝑟𝐵_𝑟,𝐶^𝑟=𝜖_𝑟𝑥^𝑇_𝑟𝑦_𝑟𝐶_𝑟, (2.4)where 𝛽_𝑟,𝜖_𝑟∈𝔽, 𝑢_𝑟,𝑣_𝑟,𝑥_𝑟,𝑦_𝑟∈𝔽² and 𝐵_𝑟 and 𝐶_𝑟 are 2×2 matrices.
From the definition of 𝜌(𝑒_𝑖), it is straightforward to check that {𝜌(𝑒_𝑖)} defines a Lie algebra isomorphic to that described in (1.3).
In order to have 𝔤₀ acting on 𝔤₁ via the adjoint representation, we must have [𝜌(𝑒_𝑖),𝜌(𝑓_𝑗)]=𝑎𝜌(𝑓₂)+𝑐𝜌(𝑓₃),𝑖=1,𝑗=2,𝑏𝜌(𝑓₂)+𝑑𝜌(𝑓₃),𝑖=1,𝑗=3,−𝑎𝜌(𝑓₂)−𝑐𝜌(𝑓₃),𝑖=2,𝑗=1,−𝑏𝜌(𝑓₂)−𝑑𝜌(𝑓₃),𝑖=3,𝑗=1, (2.5) but [𝜌(𝑒_𝑖),𝜌(𝑓_𝑗)]=(^{0𝐴𝑖𝐵𝑗−𝐵𝑗𝐴𝑖}_{𝐴𝑖𝐶𝑗−𝐶𝑗𝐴𝑖0}), so we obtain 𝐵¹=𝛽₁00𝐵₁,𝐵²=𝑣00₂0,𝐵³=𝑣00₃0,𝐶¹=𝜖₁00𝐶₁,𝐶²=𝑦00₂0,𝐶³=𝑦00₃0, (2.6) where 𝐵₁=𝐴(𝑣₂𝑣₃)+𝛽₁𝟏,𝐶₁=𝐴(𝑦₂𝑦₃)+𝜖₁𝟏,𝐴𝐵₁=𝐵₁𝐴,𝐴𝐶₁=𝐶₁𝐴,𝐴𝑣₂=𝑎𝑣₂+𝑐𝑣₃,𝐴𝑣₃=𝑏𝑣₂+𝑑𝑣₃,𝐴𝑦₂=𝑎𝑦₂+𝑐𝑦₃,𝐴𝑦₃=𝑏𝑦₂+𝑑𝑦₃ (2.7) and 𝟏 is the 2×2 identity matrix. Now, we have to satisfy the condition [𝜌(𝑓_𝑖),𝜌(𝑓_𝑗)]=∑_𝑘Γ^𝑘_𝑖𝑗𝜌(𝑒_𝑘), but [𝜌(𝑓_𝑖),𝜌(𝑓_𝑗)]=(^{𝐵𝑖𝐶𝑗+𝐵𝑗𝐶𝑖0}_{0𝐶𝑖𝐵𝑗+𝐶𝑗𝐵𝑖}). Then, 𝛽₁𝜖₁=0,2𝐵₁𝐶₁=𝑝𝐴=2𝐶₁𝐵₁,𝑣₂𝜖₁+𝐵₁𝑦₂=𝑝2𝛿₁=𝑦₂𝛽₁+𝐶₁𝑣₂,𝑣₃𝜖₁+𝐵₁𝑦₃=𝑝2𝛿₂=𝑦₃𝛽₁+𝐶₁𝑣₃. (2.8)
With no loss of generality, we can choose 𝜖₁=0 and 𝑦₂=𝐴⁻¹𝛿₁,𝑦₃=𝐴⁻¹𝛿₂, so that 𝐶₁=𝟏 and 𝐵₁=(𝑝/2)𝐴. Finally, by choosing 𝑣₂=(𝑝/2)𝛿₁−𝛽₁𝑦₂ and 𝑣₃=(𝑝/2)𝛿₂−𝛽₁𝑦₃ we find that all equations are satisfied. Actually, we can always choose 𝛽₁=0, obtaining the expressions given in the statement.

The choices we have made in the proof of Proposition 2.1 produce easier exponential matrices (see Theorem 3.1 below). Since the representation 𝜌 is to be faithful, different choices made in the proof would have produced Lie superalgebras isomorphic to 𝔤 inside 𝔤𝔩(𝑉). Therefore, the corresponding supergroups obtained via the constructive method used in Theorem 3.1 would have been isomorphic at the end. This is so because Lie's theory looks for faithfully realizing 𝔤 in terms of vector fields whose integral flows will eventually define the supergroup multiplication law via composition of (local) diffeomorphisms.

3. Lie Supergroups for Which 𝐴 Is Invertible

Once we have the different Lie superalgebras represented in 𝔤𝔩(𝑉) for some (3,3)- dimensional supervector space 𝑉, we proceed to find a supermanifold that actually carries a Lie supergroup structure following essentially the same steps followed in the classical theory of Lie. In fact, we can always obtain explicitly a Lie group structure for 𝐺₀(𝐴) from its Lie algebra 𝔤₀, where Lie(𝐺₀(𝐴))=𝔤₀. So, let us write (𝑠,𝑣) as the local coordinates described in the introduction.

Theorem 3.1. Let 𝔤=𝔤₀⊕𝔤₁ be a Lie superalgebra satisfying 𝔤₁=𝔤₀, 𝔤₀ acting on 𝔤₁ via the adjoint representation, and having 2-dimensional derived ideal 𝔤^′₀=[𝔤₀,𝔤₀]. The Lie supergroups whose underlying Lie superalgebras are 𝔤 have the following multiplication law for the products of (𝑠^′,𝑣^′;𝜎^′,𝜃^′) and (𝑠,𝑣;𝜎,𝜃): (𝑠^′𝑝+𝑠−2𝜎^′𝜎,𝑣+𝑒^{(−𝑠+(𝑝/2)𝜎′𝜎)𝐴}𝑣^′+𝑝2𝜎𝑒^−𝑠𝐴𝜃^′;𝜎^′+𝜎,𝜃+𝑒^{(−𝑠+(𝑝/2)𝜎′𝜎)𝐴}𝜃^′+𝜎^′𝐴𝑣). (3.1)

Proof. According to [7, 8] we only have to compute the exponential of the matrices 𝜌(𝑒_𝑖) and 𝜌(𝑓_𝑗) given in Proposition 2.1, and the supergroup composition law will be obtained from first principles using the ODE theory in supermanifolds and following Lie's original techniques as described before (see [7]). If we denote by (𝑡₁,𝑡₂,𝑡₃;𝜏₁,𝜏₂,𝜏₃) the composition Exp(𝑡₁𝜌(𝑒₁))∘Exp(𝑡₂𝜌(𝑒₂))∘Exp(𝑡₃𝜌(𝑒₃))∘Exp(𝜏₁𝜌(𝑓₁))∘Exp(𝜏₂𝜌(𝑓₂))∘Exp(𝜏₃𝜌(𝑓₃)), (3.2) we notice that (𝑡₁,𝑡₂,𝑡₃;𝜏₁,𝜏₂,𝜏₃)=(^𝑃𝑄_−𝑅𝑃), where 𝑒𝑃=10_𝑡1𝐴(𝑡₂𝑡₃−𝑝2𝜏₁𝜏₂𝜏₃)𝑒^𝑡1𝐴,𝑄=𝑝002𝑒^𝑡1𝐴𝜏₂𝜏₃𝑝2𝜏₁𝑒^𝑡1𝐴𝐴,𝑒𝑅=00_𝑡1𝐴𝐴⁻¹𝜏₂𝜏₃𝜏₁𝑒^𝑡1𝐴. (3.3)
In order to find the multiplication law, we have to find when the following identity holds: (𝑡₁^′′,𝑡₂^′′,𝑡₃^′′;𝜏₁^′′,𝜏₂^′′,𝜏₃^′′)=(𝑡^′₁,𝑡^′₂,𝑡^′₃;𝜏^′₁,𝜏^′₂,𝜏^′₃)⋅(𝑡₁,𝑡₂,𝑡₃;𝜏₁,𝜏₂,𝜏₃), (3.4) that is, we have to solve𝑃^′′𝑄^′′−𝑅^′′𝑃^′′=𝑃^′𝑄^′−𝑅^′𝑃^′=𝑃𝑄𝑃−𝑅𝑃^′𝑃−𝑄^′𝑅𝑃^′𝑄+𝑄^′𝑃−(𝑅^′𝑃+𝑃^′𝑅)𝑃^′𝑃−𝑅^′𝑄. (3.5)
From 𝑃^′′=𝑃^′𝑃−𝑄^′𝑅, we obtain𝑒^{𝑡1′′𝐴}=𝑒^{(𝑡′1+𝑡1)𝐴}𝑝(𝟏−2𝜏^′₁𝜏₁𝐴)=𝑒^{(𝑡′1+𝑡1−(𝑝/2)𝜏′1𝜏1)𝐴},𝑒^{𝑡1′′𝐴}(𝑡₂^′′𝑡₃^′′−𝑝2𝜏₁^′′𝜏₂^′′𝜏₃^′′)=𝑒^𝑡′1𝐴(𝑡^′₂𝑡^′₃−𝑝2𝜏^′₁𝜏^′₂𝜏^′₃)+𝑒^{(𝑡′1+𝑡1)𝐴}(𝑡₂𝑡₃−𝑝2𝜏₁𝜏₂𝜏₃−𝑝2𝜏^′₁𝜏₂𝜏₃); (3.6) and from 𝑅^′′=𝑅^′𝑃+𝑃^′𝑅, we obtain𝜏₁^′′𝑒^{𝑡1′′𝐴}=(𝜏^′₁+𝜏₁)𝑒^{(𝑡′1+𝑡1)𝐴},𝑒^{𝑡1′′𝐴}𝐴⁻¹𝜏₂^′′𝜏₃^′′=𝑒^𝑡′1𝐴𝐴⁻¹𝜏^′₂𝜏^′₃+𝑒^{(𝑡′1+𝑡1)𝐴}⋅𝐴⁻¹𝜏₂𝜏₃+𝜏^′₁𝑒^{(𝑡′1+𝑡1)𝐴}(𝑡₂𝑡₃−𝑝2𝜏₁𝜏₂𝜏₃). (3.7)
Now, it is straightforward to prove that𝑡₁^′′=𝑡₁+𝑡^′₁−𝑝2𝜏^′₁𝜏₁,𝑡₂^′′𝑡₃^′′=𝑡₂𝑡₃++𝑒^{(−𝑡1+(𝑝/2)𝜏′1𝜏1)𝐴}𝑡^′₂𝑡^′₃+𝑝2𝜏₁𝑒^−𝑡1𝐴𝜏^′₂𝜏^′₃,𝜏₁^′′=𝜏₁+𝜏^′₁,𝑡₂^′′𝑡₃^′′=𝜏₂𝜏₃+𝑒^{(−𝑡1+(𝑝/2)𝜏′1𝜏1)𝐴}𝜏^′₂𝜏^′₃+𝜏^′₁𝐴𝑡₂𝑡₃, (3.8) and defining 𝑠=𝑡₁, 𝑣=(^𝑡2_𝑡3), 𝜎=𝜏₁, and 𝜃=(^𝑡2_𝑡3), we find the multiplication law given in the statement.

From (3.8), we can write the multiplication law in terms of morphisms as in [7, 8].

Proposition 3.2. Let 𝒢(𝑝,𝐴) be the (3,3)-dimensional supermanifold whose underlying Lie group is 𝐺₀(𝐴) and let {𝑥₁,𝑥₂,𝑥₃;𝜉₁,𝜉₂,𝜉₃} be local coordinates. For 𝑖=1,2, let 𝜋_𝑖∶𝒢(𝑝,𝐴)×𝒢(𝑝,𝐴)→𝒢(𝑝,𝐴) be the direct product projections. Then 𝒢(𝑝,𝐴) is a Lie supergroup endowed with the morphism 𝑚∶𝒢(𝑝,𝐴)×𝒢(𝑝,𝐴)→𝒢(𝑝,𝐴) defined by 𝑚^∗(𝑥₁)=𝜋^∗₂(𝑥₁)+𝜋^∗₁(𝑥₁𝑝)−2𝜋^∗₁(𝜉₁)𝜋^∗₂(𝜉₁𝑚),_∗(𝑥₂)𝑚^∗(𝑥₃)=𝜋^∗₂(𝑥₂)𝜋^∗₂(𝑥₃)+𝑒^{(−𝜋∗2(𝑥1)+(𝑝/2)𝜋∗1(𝜉1)𝜋∗2(𝜉1))𝐴}𝜋^∗₁(𝑥₂)𝜋^∗₁(𝑥₃)+𝑝2𝜋^∗₂(𝜉₁)𝑒^{(−𝜋∗2(𝑥1)𝐴}𝜋^∗₁(𝜉₂)𝜋^∗₁(𝜉₃),𝑚^∗(𝜉₁)=𝜋^∗₂(𝜉₁)+𝜋^∗₁(𝜉₁𝑚),_∗(𝜉₂)𝑚^∗(𝜉₃)=𝜋^∗₂(𝜉₂)𝜋^∗₂(𝜉₃)+𝑒^{(−𝜋∗2(𝑥1)+(𝑝/2)𝜋∗1(𝜉1)𝜋∗2(𝜉1))𝐴}𝜋^∗₁(𝜉₂)𝜋^∗₁(𝜉₃)+𝜋^∗₁(𝜉₁𝜋)𝐴^∗2(𝑥₂)𝜋^∗₂(𝑥₃); (3.9) the morphism 𝜀∶𝒢(𝑝,𝐴)→𝒢(𝑝,𝐴) defined by 𝜀^∗(𝑥_𝑖)=𝜀^∗(𝜉_𝑖)=0, and the morphism 𝑖∶𝒢(𝑝,𝐴)→𝒢(𝑝,𝐴) defined by𝑖^∗(𝑥₁)=−𝑥₁,𝑖^∗(𝑥₂)𝑚^∗(𝑥₃)=−𝑒^𝑥1𝐴𝑥₂−𝑝2𝜉₁𝜉₂𝑥₃−𝑝2𝜉₁𝜉₃,𝑖^∗(𝜉₁)=−𝜉₁,𝑖^∗(𝜉₂)𝑚^∗(𝜉₃)=−𝑒^𝑥1𝐴𝜉₂𝜉₃+𝜉₁𝐴𝑒^𝑥1𝐴𝑥₂𝑥₃. (3.10)

Proof. It is straightforward to check that (𝜋₁,𝑚∘(𝜋^∗₂,𝜋^∗₃))^∗∘𝑚^∗=(𝑚∘(𝜋^∗₁,𝜋^∗₂),𝜋^∗₃)^∗∘𝑚^∗,(𝜀,𝑖𝑑)^∗∘𝑚^∗=𝑖𝑑^∗=(𝑖𝑑,𝜀)^∗∘𝑚^∗,(𝑖,𝑖𝑑)^∗∘𝑚^∗=𝜀^∗=(𝑖𝑑,𝑖)^∗∘𝑚^∗, (3.11) which are the associative law, the identity element, and inverse element properties, hold.

Proposition 3.3. Assuming the hypotheses of Proposition 3.2, the left-invariant supervector fields can be written as 𝑋=𝜆₁𝑋₁+𝜆₂𝑋₂+𝜆₃𝑋₃+𝜇₁𝑌₁+𝜇₂𝑌₂+𝜇₃𝑌₃, where 𝑋₁=𝜕𝜕𝑥₁−(𝑎𝑥₂+𝑏𝑥₃)𝜕𝜕𝑥₂−(𝑐𝑥₂+𝑑𝑥₃)𝜕𝜕𝑥₃−(𝑎𝜉₂+𝑏𝜉₃)𝜕𝜕𝜉₂−(𝑐𝜉₂+𝑑𝜉₃)𝜕𝜕𝜉₃,𝑋₂=𝜕𝜕𝑥₂+𝑎𝜉₁𝜕𝜕𝜉₂+𝑐𝜉₁𝜕𝜕𝜉₃,𝑋₃=𝜕𝜕𝑥₃+𝑏𝜉₁𝜕𝜕𝜉₂+𝑑𝜉₁𝜕𝜕𝜉₃,𝑌₁=𝜕𝜕𝜉₁+𝑝2[(𝑎𝜉₂+𝑏𝜉₃)𝜉₁𝜕𝜕𝜉₂+(𝑐𝜉₂+𝑑𝜉₃)𝜉₁𝜕𝜕𝜉₃]+𝑝2[𝜉₁𝜕𝜕𝑥₁+(𝜉₂−(𝑎𝑥₂+𝑏𝑥₃)𝜉₁)𝜕𝜕𝑥₂+(𝜉₃−(𝑐𝑥₂+𝑑𝑥₃)𝜉₁)𝜕𝜕𝑥₃𝑌],₂=𝜕𝜕𝜉₂,𝑌₃=𝜕𝜕𝜉₃, (3.12) and 𝜆_𝑖,𝜇_𝑗∈𝔽. Furthermore, the Lie superalgebra defined by the left-invariant supervector fields {𝑋₁,𝑋₂,𝑋₃,𝑌₁,𝑌₂,𝑌₃} is isomorphic to the Lie superalgebra given by (1.3) and (1.5).

Proof. Any supervector field can be written as 𝑋=∑𝑓_𝑖(𝜕/𝜕𝑥_𝑖)+𝑔_𝑖(𝜕/𝜕𝜉_𝑖) and 𝑋 is a left-invariant supervector field if the supervector field 𝑋=𝜋^∗₂(𝑓_𝑖)𝜕𝜕𝜋^∗₂(𝑥_𝑖)+𝜋^∗₂(𝑔_𝑖)𝜕𝜕𝜋^∗₂(𝜉_𝑖) (3.13) satisfies 𝜀⁽²⁾∘𝑋∘(𝑝₁,𝑚)^∗=𝜀²∘(𝑝₁,𝑚)^∗∘𝑋, (3.14) where 𝜀⁽²⁾∶𝒢(𝑝,𝐴)→𝒢(𝑝,𝐴)×𝒢(𝑝,𝐴) is given by 𝜀^(2)∗∘𝑝^∗₁=𝑖𝑑^∗ and 𝜀^(2)∗∘𝑝^∗₂=𝜀^∗, as in [7]. By Proposition 3.2 we have the explicit multiplication morphisms 𝑚^∗ and 𝜀^∗ and applying the local coordinates {𝜋^∗₁(𝑥_𝑖),𝜋^∗₂(𝑥_𝑖),𝜋^∗₁(𝜉_𝑖),𝜋^∗₂(𝜉_𝑖)} on both sides of (3.14) we found the restrictions for 𝑓_𝑖's and 𝑔_𝑖's and they are written as in the statement.
Finally, in order to prove that {𝑋₁,𝑋₂,𝑋₃,𝑌₁,𝑌₂,𝑌₃} defines the Lie superalgebra given in the beginning, we just have to compute the Lie superbrackets given for the supervector fields, namely, [𝑋,𝑍]=𝑋∘𝑍−(−1)^|𝑋||𝑍|𝑍∘𝑋 to check that it is precisely the same Lie superalgebra defined by (1.3) and (1.5). By defining the correspondence 𝑒_𝑖↦𝑋_𝑖,𝑓_𝑗↦𝑌_𝑗, (3.15)we conclude that they are isomorphic.

Acknowledgments

The authors would like to acknowledge with thanks the partial support received from the following grants: CONACYT Grant 46274, and Programa del Mejoramiento del Profesorado, Secretara de Educación, Grant PROMEP/103.5/2526 PTC-45-D. They would like to thank Professor Adolfo Sánchez-Valenzuela for enlightening discussions during the genesis of this work, and the kind hospitality received by the authors at CIMAT and Facultad de Matemáticas, at UADY. Last but not least, they would like to thank the referees for their comments, criticism, and recommendations contributed to clear various passages of the original manuscript.