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

Authors: I. Hernández; R. Peniche
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## 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 superalgebrag=g0⊕g1
which satisfies g1=g0, g0 acts on g1 via the adjoint representation and g0 has
a 2-dimensional derived ideal.


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## Body

## 1. Introduction

A possible notion ofsuperspace 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 𝔤0′=[𝔤0,𝔤0]. The most trivial case was that corresponding to dim𝔤0′=3 whereas the
most difficult case was that of dim𝔤0′=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 satisfying(1.1)Γ([x,y],z)+Γ(y,[x,z])=[x,Γ(y,z)].Furthermore, the isomorphism class of the Lie
superalgebra defined by such a givenΓ 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
(1.2)Γ→(T,S)⋅Γ=T(Γ(S−1(⋅),S−1(⋅))).Now, let𝔽 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 Symad(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 Symad(𝔤0) with the set of
triples (Γ1,Γ2,Γ3) of symmetric
bilinear forms, for which Γ(u,v)=∑i=13Γi(u,v)ei satisfies (1.1).Let𝔤0′=[𝔤0,𝔤0] be the derived
ideal of 𝔤0. Lie algebras having dim𝔤0′=2 can be
classified by choosing e1∉𝔤0′ and bringing ad(e1) into a
convenient canonical form (see [3, 4] or [5] for details). Thus, (1.3)[e1,e2]=ae2+ce3,[e1,e3]=be2+de3,so thatad(e1)|𝔤0′⟷A=(abcd),[e2,e3]=0,
and, since dim𝔤0′=2, then A must be an
invertible matrix. Now, the classification up to isomorphism of the Lie
algebras 𝔤0 having dim𝔤0′=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 the3-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.
𝔤0AConstraints𝔭(𝔽)(1101)—𝔮λ(𝔽)(100λ)0<|λ|≤1𝔮λ1(ℝ)(λ−11λ)λ∈ℝIt is easy to
see that the matrices ofad(ei) take the
forms
(1.4)ad(e1)=(000A),ad(e2)=−(00Aδ10),ad(e3)=−(00Aδ20),
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)=∑i=13Γi(u,v)ei satisfies (1.1),
is given, up to isomorphism, by
(1.5)Γ1=p(100000000),Γ2=p2(010100000),Γ3=p2(001000100),
where p is an arbitrary
element of the ground field 𝔽 for the three
classes 𝔭(𝔽),𝔮λ(𝔽), and 𝔮λ1(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 isG0(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⊕𝔤0′, with S≃𝔤0/𝔤0′. Thus, σ is a local
section of G0(A)×S→G0(A), and θ is a local
section of G0(A)×𝔤0′→G0(A). Then, the product (s′,v′;σ′,θ′)*(s,v;σ,θ) is given by (see Theorem 3.1)
(1.6)(s′+s−p2σ′σ,v+e(−s+(p/2)σ′σ)Av′+p2σe−sAθ′;σ′+σ,θ+e(−s+(p/2)σ′σ)Aθ′+σ′Av).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 Proposition3.2 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 write(2.1)A1=(000A),A2=(00δ10),A3=(00δ20),
where δ1 and δ2 are defined as
above. Let Br and Cr be 3×3 matrices (r=1,2,3).Proposition 2.1.
Let𝔤=𝔤0⊕𝔤1 be a (3,3)-dimensional Lie
superalgebra, where 𝔤1=𝔤0, 𝔤0 acts on 𝔤1 via the adjoint
representation, and dim𝔤0′=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 that(2.2)ρ(ei)=(Ai00Ai),ρ(fi)=(0BiCi0).Then, the choices(2.3)Bi=p2Ai,Ci=KAi,
where K=(000A−1), turn ρ:𝔤→𝔤𝔩(V) into a faithful
representation of the Lie superalgebra 𝔤.Proof.
Let us write(2.4)Br=(βrurTvrBr),Cr=(ϵrxrTyrCr),where βr,ϵr∈𝔽, ur,vr,xr,yr∈𝔽2 and Br and Cr are 2×2 matrices.
From the definition ofρ(ei), it is straightforward to check that {ρ(ei)} defines a Lie
algebra isomorphic to that described in (1.3).
In order to have𝔤0 acting on 𝔤1 via the adjoint
representation, we must have
(2.5)[ρ(ei),ρ(fj)]={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,
but [ρ(ei),ρ(fj)]=(0AiBj−BjAiAiCj−CjAi0), so we obtain
(2.6)B1=(β100B1),B2=(00v20),B3=(00v30),C1=(ϵ100C1),C2=(00y20),C3=(00y30),
where
(2.7)B1=A(v2v3)+β11,C1=A(y2y3)+ϵ11,AB1=B1A,AC1=C1A,Av2=av2+cv3,Av3=bv2+dv3,Ay2=ay2+cy3,Ay3=by2+dy3
and 1 is the 2×2 identity
matrix. Now, we have to satisfy the condition [ρ(fi),ρ(fj)]=∑kΓijkρ(ek), but [ρ(fi),ρ(fj)]=(BiCj+BjCi00CiBj+CjBi). Then,
(2.8)β1ϵ1=0,2B1C1=pA=2C1B1,v2ϵ1+B1y2=p2δ1=y2β1+C1v2,v3ϵ1+B1y3=p2δ2=y3β1+C1v3.
With no loss of generality, we can chooseϵ1=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 Proposition2.1 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 WhichA Is Invertible

Once we have the different Lie superalgebras
represented in𝔤𝔩(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.
Let𝔤=𝔤0⊕𝔤1 be a Lie
superalgebra satisfying 𝔤1=𝔤0, 𝔤0 acting on 𝔤1 via the adjoint
representation, and having 2-dimensional derived ideal 𝔤0′=[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):
(3.1)(s′+s−p2σ′σ,v+e(−s+(p/2)σ′σ)Av′+p2σe−sAθ′;σ′+σ,θ+e(−s+(p/2)σ′σ)Aθ′+σ′Av).Proof.
According to [7, 8] we only have to compute the
exponential of the matrices ρ(ei) and ρ(fj) 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
(3.2)Exp(t1ρ(e1))∘Exp(t2ρ(e2))∘Exp(t3ρ(e3))∘Exp(τ1ρ(f1))∘Exp(τ2ρ(f2))∘Exp(τ3ρ(f3)),
we notice that (t1,t2,t3;τ1,τ2,τ3)=(PQ−RP), where
(3.3)P=(10et1A((t2t3)−p2τ1(τ2τ3))et1A),Q=(00p2et1A(τ2τ3)p2τ1et1AA),R=(00et1AA−1(τ2τ3)τ1et1A).
In order to find the multiplication law, we have to
find when the following identity holds:(3.4)(t1′′,t2′′,t3′′;τ1′′,τ2′′,τ3′′)=(t1′,t2′,t3′;τ1′,τ2′,τ3′)⋅(t1,t2,t3;τ1,τ2,τ3),
that is, we have to solve(3.5)(P′′Q′′−R′′P′′)=(P′Q′−R′P′)(PQ−RP)=(P′P−Q′RP′Q+Q′P−(R′P+P′R)P′P−R′Q).
FromP′′=P′P−Q′R, we obtain(3.6)et1′′A=e(t1′+t1)A(1−p2τ1′τ1A)=e(t1′+t1−(p/2)τ1′τ1)A,et1′′A((t2′′t3′′)−p2τ1′′(τ2′′τ3′′))=et1′A((t2′t3′)−p2τ1′(τ2′τ3′))+e(t1′+t1)A((t2t3)−p2τ1(τ2τ3)−p2τ1′(τ2τ3));
and from R′′=R′P+P′R, we obtain(3.7)τ1′′et1′′A=(τ1′+τ1)e(t1′+t1)A,et1′′AA−1(τ2′′τ3′′)=et1′AA−1(τ2′τ3′)+e(t1′+t1)A⋅A−1(τ2τ3)+τ1′e(t1′+t1)A((t2t3)−p2τ1(τ2τ3)).
Now, it is straightforward to prove that(3.8)t1′′=t1+t1′−p2τ1′τ1,(t2′′t3′′)=(t2t3)++e(−t1+(p/2)τ1′τ1)A(t2′t3′)+p2τ1e−t1A(τ2′τ3′),τ1′′=τ1+τ1′,(t2′′t3′′)=(τ2τ3)+e(−t1+(p/2)τ1′τ1)A(τ2′τ3′)+τ1′A(t2t3),
and defining s=t1, v=(t2t3), σ=τ1, and θ=(t2t3), 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.
Let𝒢(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 πi: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
(3.9)m*(x1)=π2*(x1)+π1*(x1)−p2π1*(ξ1)π2*(ξ1),(m*(x2)m*(x3))=(π2*(x2)π2*(x3))+e(−π2*(x1)+(p/2)π1*(ξ1)π2*(ξ1))A(π1*(x2)π1*(x3))+p2π2*(ξ1)e(−π2*(x1)A(π1*(ξ2)π1*(ξ3)),m*(ξ1)=π2*(ξ1)+π1*(ξ1),(m*(ξ2)m*(ξ3))=(π2*(ξ2)π2*(ξ3))+e(−π2*(x1)+(p/2)π1*(ξ1)π2*(ξ1))A(π1*(ξ2)π1*(ξ3))+π1*(ξ1)A(π2*(x2)π2*(x3));
the morphism ε:𝒢(p,A)→𝒢(p,A) defined by ε*(xi)=ε*(ξi)=0, and the morphism i:𝒢(p,A)→𝒢(p,A) defined by(3.10)i*(x1)=−x1,(i*(x2)m*(x3))=−ex1A(x2−p2ξ1ξ2x3−p2ξ1ξ3),i*(ξ1)=−ξ1,(i*(ξ2)m*(ξ3))=−ex1A(ξ2ξ3)+ξ1Aex1A(x2x3).Proof.
It is straightforward to check that(3.11)(π1,m∘(π2*,π3*))*∘m*=(m∘(π1*,π2*),π3*)*∘m*,(ε,id)*∘m*=id*=(id,ε)*∘m*,(i,id)*∘m*=ε*=(id,i)*∘m*,
which are the associative law, the identity element,
and inverse element properties,
hold.Proposition 3.3.
Assuming the hypotheses of Proposition3.2, the left-invariant
supervector fields can be written as X=λ1X1+λ2X2+λ3X3+μ1Y1+μ2Y2+μ3Y3, where
(3.12)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)∂∂x3],Y2=∂∂ξ2,Y3=∂∂ξ3,
and λi,μj∈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 asX=∑fi(∂/∂xi)+gi(∂/∂ξi) and X is a
left-invariant supervector field if the supervector field
(3.13)X^=∑π2*(fi)∂∂π2*(xi)+π2*(gi)∂∂π2*(ξi)
satisfies
(3.14)ε(2)∘X^∘(p1,m)*=ε2∘(p1,m)*∘X^,
where ε(2):𝒢(p,A)→𝒢(p,A)×𝒢(p,A) is given by ε(2)*∘p1*=id* and ε(2)*∘p2*=ε*, as in [7]. By Proposition 3.2 we have the explicit
multiplication morphisms m* and ε*  and applying the local coordinates {π1*(xi),π2*(xi),π1*(ξi),π2*(ξi)} on both sides
of (3.14) we found the restrictions for fi's and gi'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
(3.15)ei↦Xi,fj↦Yj,we conclude that they are  isomorphic.



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