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

International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 632518,9pages

doi:10.1155/2008/632518

Research Article

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

I. Hernandez¹and R. Peniche²

1CIMAT, Apdo. Postal 402, C.P., 36000 Guanajuato, GTO, Mexico

2Facultad de Matem´aticas, Universidad Aut´onoma de Yucat´an, Apdo. Postal 172, C.P., 97110 M´erida, YUC, Mexico

Correspondence should be addressed to R. Peniche,[email protected] Received 18 January 2007; Revised 23 May 2007; Accepted 31 October 2007 Recommended by Nils-Peter Skoruppa

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 superalgebragg0⊕g1which satisfiesg1g0, g0acts ong1via the adjoint representation andg0has a 2-dimensional derived ideal.

Copyrightq2008 I. Hern´andez and R. Peniche. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

A possible notion of superspace associated to a given 3-dimensional Lie algebrag₀ might be a Lie superalgebragg₀⊕g₁. If one is to introduce a minimal set of assumptions, it seems quite natural to consider those superalgebras for whichg₁g₀and for which theg₀action ong₁is the adjoint representation. A complete classification of the real and complex3,3-dimensional Lie superalgebras having precisely this restriction was recently obtained in1. The classification was given in terms of the dimension of the derived idealg₀ 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 in1having a 2-dimensional derived ideal. This is just one step forward in our understanding of the real and complex3,3-dimensional Lie supergroups.

In order to maintain our exposition as self-contained as possible, we will summarize the basic statements from1. Once we fix the adjoint representation, it is well knownsee1,2 that there are as many Lie superalgebras as bilinear symmetric mapsΓ:g₀×g₀ → g₀satisfying

Γ

x, y, z Γ

y,x, z

x,Γy, z

. 1.1

Table 1: Full set of representatives of the isomorphism classes of the 3-dimensional Lie algebrasg0that have 2-dimensional derived ideal. The classes were obtained from the canonical form of the 2×2 invertible matrixA. In the first two cases, the fieldFcan be eitherRorC.

g0 A Constraints

p

F _{1 1}

0 1

— qλ

F _{1 0}

0λ

0<|λ| ≤1

q¹_λR

λ−1 1 λ

λ∈R

Furthermore, the isomorphism class of the Lie superalgebra defined by such a givenΓis completely determined by its orbit under the action of the groupG⊂Autg₀×GLg₀of pairs T, SsatisfyingS◦T⁻¹◦ad· ad·◦S◦T⁻¹given 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 overF. Let{e1, e₂, e₃}be a basis forg₀, and let Sym_adg₀be the real or complex vector space of symmetric, bilinear mapsΓ:g₀×g₀ → g₀satisfying1.1. Thus, we can identify the space Symadg0with the set of triplesΓ¹,Γ²,Γ³of symmetric bilinear forms, for whichΓu, v ₃

i1Γⁱu, veisatisfies1.1.

Letg₀ g₀,g₀be the derived ideal ofg₀. Lie algebras having dimg₀2 can be classified by choosinge1/∈g₀and bringing ade1into a convenient canonical formsee3,4or5for details. Thus,

e1, e₂ ae₂ ce₃,

e1, e3 be2 de3, so that ad e1 _g

0 ←→A

a b c d

, e2, e3 0,

1.3

and, since dimg₀2, thenAmust be an invertible matrix. Now, the classification up to isomor- phism of the Lie algebrasg₀having dimg₀2 can be written as inTable 1, where we followed the notation introduced in1.

It is easy to see that the matrices of adeitake the forms

ad e₁

0 0

0 A

, ad

e₂ −

0 0 Aδ1 0

, ad

e₃ −

0 0 Aδ2 0

, 1.4

where δ1 1

0 and δ2 0

1. It is well known that the associated Lie groups for the Lie algebras we are dealing with here are the so-called unimodular groupsE2andE1,1for q₁Fand q₋₁F, respectively, and for the other Lie algebras, the associated Lie groups are called nonunimodularsee6. We shall denote byG₀Aany of those Lie groups.

It was proved in1that whenAis invertible, any tripleΓ¹,Γ²,Γ³, for whichΓu, v ₃

i1Γⁱu, veisatisfies1.1, is given, up to isomorphism, by

Γ¹p

⎛

⎝1 0 0 0 0 0 0 0 0

⎞

⎠, Γ² p 2

⎛

⎝0 1 0 1 0 0 0 0 0

⎞

⎠, Γ³ p 2

⎛

⎝0 0 1 0 0 0 1 0 0

⎞

⎠, 1.5

I. Hern´andez and R. Peniche

wherepis an arbitrary element of the ground fieldFfor the three classespF,q_λF,andq¹_λR, with no relation between the parameterspandλ.

What we do in this paper is to describe explicitly all the Lie supergroups whose un- derlying 3-dimensional Lie group isG₀Aand having as Lie superalgebras of left-invariant supervector fields the Lie superalgebras given by the tripleΓ¹,Γ²,Γ³above. The method is the one given in7and 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 parameterssee 7.

In more detail, we aim to describe the multiplication law in terms of tetradss, v;σ, θ, wheres, v ∈F×F²are the local coordinates on the 3-dimensional Lie groupG0A, andσ andθare the odd coordinates on the supergroupG0A,ΛE, whereΛEstands for the sheaf of sections of the exterior algebra bundle associated to the rank-3 vector bundleE → G₀A, whose typical fiberg₀can be decomposed asS⊕g₀, withSg₀/g₀. Thus,σis a local section ofG0A×S → G0A, andθis a local section ofG0A×g₀ → G0A. Then, the product s, v;σ, θ∗s, v;σ, θis given byseeTheorem 3.1

s s−p

2σσ, v e^{−s p/2σσA}v p

2σe^−sAθ;σ σ, θ e^{−s p/2σσA}θ σAv

. 1.6

The point of giving such an explicit expression is to actually see the way odd coordi- nates 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 3.2the supermorphisms that define the Lie supergroups within the spirit of7,8.

Finally, inProposition 3.3we 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

A¹ 0 0

0 A

, A² 0 0

δ1 0

, A³ 0 0

δ2 0

, 2.1

whereδ₁andδ₂are defined as above. LetB^randC^rbe 3×3 matricesr1,2,3.

Proposition 2.1. Letg g₀⊕g₁ be a3,3-dimensional Lie superalgebra, whereg₁ g₀,g₀ acts on g₁ via the adjoint representation, and dimg₀ 2. Let{e1, e₂, e₃}and{f1, f₂, f₃}be bases forg₀ and g₁, respectively. LetV V0⊕V1be a3,3-dimensional supervector space and letρ:g → glVbe a linear map such that

ρ ei

Aⁱ 0 0 Aⁱ

, ρ

fi

0 Bⁱ Cⁱ 0

. 2.2

Then, the choices

Bⁱ p

2Aⁱ, CⁱKAⁱ, 2.3

whereK_{0 0}

0A⁻¹

, turnρ:g → glVinto a faithful representation of the Lie superalgebrag.

Proof. Let us write

B^r

β_r u^T_r v_r B_r

, C^r

r x^T_r y_r C_r

, 2.4

whereβ_r, r ∈F,ur, vr, xr, yr ∈F²andBr andCrare 2×2 matrices.

From the definition ofρei, it is straightforward to check that{ρei}defines a Lie alge- bra isomorphic to that described in1.3.

In order to haveg₀acting ong₁via the adjoint representation, we must have

ρ ei

, ρ fj

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩ aρ

f2

cρ f3

, i1, j2, bρ

f₂ dρ

f₃

, i1, j3,

−aρ f₂

−cρ f₃

, i2, j1,

−bρ f2

−dρ f3

, i3, j1,

2.5

butρei, ρfj

_{0 A}iB^j−B^jAⁱ AⁱC^j−C^jAⁱ 0

, so we obtain

B¹ β₁ 0

0 B₁

, B² 0 0

v₂ 0

, B³ 0 0

v₃ 0

,

C¹ ₁ 0

0 C1

, C² 0 0

y2 0

, C³ 0 0

y3 0

,

2.6

whereand1 is the 2×2 identity matrix. Now, we have to satisfy the conditionρfi, ρfj

kΓ^k_ijρek, butρfi, ρfj

_BiC^j B^jCⁱ 0 0 CⁱB^j C^jBⁱ

. Then,

β₁₁0, 2B₁C₁pA2C₁B₁, v₂₁ B₁y₂ p

2δ₁y₂β₁ C₁v₂, v₃₁ B₁y₃ p

2δ₂y₃β₁ C₁v₃.

2.7

I. Hern´andez and R. Peniche

With no loss of generality, we can choose1 0 and y2 A⁻¹δ1, y3 A⁻¹δ2, so that C1 1 andB1 p/2A. Finally, by choosingv2 p/2δ1−β₁y2 andv3 p/2δ2−β₁y3

we find that all equations are satisfied. Actually, we can always chooseβ₁ 0, obtaining the expressions given in the statement.

The choices we have made in the proof ofProposition 2.1produce easier exponential matricesseeTheorem 3.1below. Since the representationρis to be faithful, different choices made in the proof would have produced Lie superalgebras isomorphic toginsideglV. There- fore, the corresponding supergroups obtained via the constructive method used inTheorem 3.1 would have been isomorphic at the end. This is so because Lie’s theory looks for faithfully re- alizinggin terms of vector fields whose integral flows will eventually define the supergroup multiplication law via composition oflocaldiffeomorphisms.

3. Lie supergroups for whichAis invertible

Once we have the different Lie superalgebras represented in glV for some 3,3- dimensional supervector spaceV, 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 forG₀Afrom its Lie al- gebrag₀, where LieG0A g₀. So, let us writes, vas the local coordinates described in the introduction.

Theorem 3.1. Letgg₀⊕g₁be a Lie superalgebra satisfyingg₁ g₀,g₀acting ong₁via the adjoint representation, and having 2-dimensional derived idealg₀ g₀,g₀. The Lie supergroups whose un- derlying Lie superalgebras areghave the following multiplication law for the products ofs, v;σ, θ ands, v;σ, θ:

s s−p

2σσ, v e^{−s p/2σσA}v p

2σe^−sAθ;σ σ, θ e^{−s p/2σσA}θ σAv

. 3.1

Proof. According to7,8we only have to compute the exponential of the matricesρeiand ρfjgiven inProposition 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 beforesee7. If we denote byt1, t2, t3;τ1, τ2, τ3the composition

Exp t₁ρ

e₁

◦Exp t₂ρ

e₂

◦Exp t₃ρ

e₃

◦Exp τ₁ρ

f₁

◦Exp τ₂ρ

f₂

◦Exp τ₃ρ

f₃ , 3.2 we notice thatt1, t₂, t₃;τ₁, τ₂, τ₃

−R PP Q

, where

In order to find the multiplication law, we have to find when the following identity holds:

t₁, t₂, t₃;τ₁, τ₂, τ₃

t₁, t₂, t₃;τ₁, τ₂, τ₃

·

t1, t2, t3;τ1, τ2, τ3

, 3.3

that is, we have to solve P Q

−R P

P Q

−R P

P Q

−R P

PP−QR PQ QP

−RP PR PP−RQ

. 3.4

FromPPP−QR,we obtain e^t1Ae^{t1 t1A}

1−p

2τ₁τ1A

e^{t1 t1−p/2τ1τ1A},

e^t1A t₂ t₃

−p 2τ₁

τ₂ τ₃

e^t1A t₂ t₃

−p 2τ₁

τ₂

τ₃ e^{t1 t1A} t₂ t3

−p 2τ₁

τ₂ τ3

−p 2τ₁

τ₂ τ3 ;

3.5

and fromRRP PR,we obtain τ₁e^t1A τ₁ τ₁e^{t1 t1A},

e^t1AA⁻¹ τ₂

τ₃

e^t1AA⁻¹ τ₂

τ₃

e^{t1 t1A}·A⁻¹ τ2

τ₃

τ₁e^{t1 t1A} t2

t₃

−p 2τ1

τ2

τ₃ .

3.6

Now, it is straightforward to prove that

t₁t1 t₁−p 2τ₁τ1, t₂

t₃

t2

t₃

e^{−t1 p/2τ1τ1A} t₂

t₃ p

2τ1e^−t1A τ₂

τ₃

,

τ₁τ₁ τ₁, t₂

t₃

τ2

τ₃

e^{−t1 p/2τ1τ1A} τ₂

τ₃

τ₁A t2

t₃

,

3.7

and definingst₁,v_t2

t3

,σ τ₁, andθ_t2

t3

,we find the multiplication law given in the statement.

I. Hern´andez and R. Peniche

From3.7, we can write the multiplication law in terms of morphisms as in7,8.

Proposition 3.2. LetGp, Abe the3,3-dimensional supermanifold whose underlying Lie group is G0Aand let{x1, x2, x3;ξ₁, ξ₂, ξ₃}be local coordinates. Fori 1,2,letπi :Gp, A× Gp, A → Gp, Abe the direct product projections. ThenGp, Ais a Lie supergroup endowed with the morphism m:Gp, A× Gp, A → Gp, Adefined by

m^∗x1 π^∗₂x1 π^∗₁x1−p

2π^∗₁ξ₁π^∗₂ξ₁, m^∗x2

m^∗x3

π^∗₂x2 π^∗₂x3

e^{−π∗2x1 p/2π∗1ξ1π∗2ξ1A}

π^∗₁x2 π^∗₁x3

p

2π^∗₂ξ₁e^−π∗2x1A

π^∗₁ξ₂ π^∗₁ξ₃

,

m^∗ξ₁ π^∗₂ξ₁ π^∗₁ξ₁, m^∗ξ₂

m^∗ξ₃

π^∗₂ξ₂ π^∗₂ξ₃

e^{−π∗2x1 p/2π∗1ξ1π∗2ξ1A}

π^∗₁ξ₂ π^∗₁ξ₃

π^∗₁ξ₁A

π^∗₂x2 π^∗₂x3

; 3.8 the morphismε:Gp, A→ Gp, Adefined byε^∗xi ε^∗ξ_i 0, and the morphismi:Gp, A→ Gp, Adefined by

i^∗x1 −x1, i^∗x2

m^∗x3

−e^x1A

⎛

⎜⎝x₂−p 2ξ₁ξ₂ x3−p

2ξ₁ξ₃

⎞

⎟⎠,

i^∗ξ₁ −ξ₁, i^∗ξ₂

m^∗ξ₃

−e^x1A ξ₂

ξ₃

ξ₁Ae^x1A x₂

x₃

.

3.9

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^∗,

3.10

which are the associative law, the identity element, and inverse element properties,hold.

Proposition 3.3. Assuming the hypotheses ofProposition 3.2, the left-invariant supervector fields can be written asXλ1X1 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 p 2

aξ₂ bξ₃ξ₁ ∂

∂ξ₂ cξ₂ dξ₃ξ₁ ∂

∂ξ₃ p

2

ξ₁ ∂

∂x1 ξ₂−ax2 bx3ξ₁ ∂

∂x2 ξ₃−cx2 dx3ξ₁ ∂

∂x3

, Y2 ∂

∂ξ₂, Y3 ∂

∂ξ₃,

3.11

and λi, 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 by1.3and1.5.

Proof. Any supervector field can be written as X

f_i∂/∂xi g_i∂/∂ξ_i andX is a left- invariant supervector field if the supervector field

X

π^∗₂fi ∂

∂π^∗₂xi π^∗₂gi ∂

∂π^∗₂ξ_i 3.12

satisfies

ε²◦X◦p1, m^∗ε²◦p1, m^∗◦X, 3.13 whereε²:Gp, A → Gp, A×Gp, Ais given byε^2∗◦p^∗₁id^∗andε^2∗◦p^∗₂ε^∗, as in7. By Proposition 3.2we have the explicit multiplication morphismsm^∗andε^∗and applying the local coordinates{π^∗₁xi, π^∗₂xi, π^∗₁ξ_i, π^∗₂ξ_i}on both sides of3.13we found the restrictions for fi’s andgi’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◦Xto check that it is precisely the same Lie superalgebra defined by1.3and1.5. By defining the correspondence

e_i→X_i, f_j→Y_j, 3.14

we conclude that they are isomorphic.

I. Hern´andez and R. Peniche 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 Educaci ´on, Grant PROMEP/103.5/2526 PTC-45-D. They would like to thank Pro- fessor Adolfo S´anchez-Valenzuela for enlightening discussions during the genesis of this work, and the kind hospitality received by the authors at CIMAT and Facultad de Matem´aticas, 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.

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