# Semi-simple Extension of the (Super) Poincaré Algebra

Authors: Dmitrij V. Soroka; Vyacheslav A. Soroka
Journal: Advances in High Energy Physics
 (2009)

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## Abstract

A semi-simple tensor extension of the Poincaré algebra is proposed for the arbitrary dimensionsD. It is established that this extension is a direct sum of the D-dimensional Lorentz algebra so(D−1, 1) and D-dimensional anti-de Sitter (AdS) algebra so(D−1, 2). A supersymmetric also semi-simple generalization of this extension is constructed in the D=4 dimensions. It is shown that this generalization is a direct sum of the 4-dimensional Lorentz algebra so(3, 1) and orthosymplectic algebra osp(1, 4) (super-AdS algebra).


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## Body

## 1. Introduction

In the papers [1–7] the Poincaré algebra for the generators of the rotations Mab and translations Pa in D dimensions,(1.1)[Mab,Mcd]=(gadMbc+gbcMad)-(c↔d),[Mab,Pc]=gbcPa-gacPb,[Pa,Pb]=0,has been extended by means of the second rank tensor generatorZab in the following way:(1.2)[Mab,Mcd]=(gadMbc+gbcMad)-(c↔d),[Mab,Pc]=gbcPa-gacPb,[Pa,Pb]=cZab,[Mab,Zcd]=(gadZbc+gbcZad)-(c↔d),[Pa,Zbc]=0,[Zab,Zcd]=0,wherec is some constant  (Note that, to avoid the double count under summation over the pair antisymmetric indices, we adopt the rules which are illustrated by the following example:(1.3)[Pa,Pb]=cZab=c2(δacδbd-δadδbc)Zcd=∑c<d‍fabcdZcd=12fabcdZcd,wherefabcd are structure constants, and so on.)Such an extension makes common sense, since it is homomorphic to the usual Poincaré algebra (1.1). Moreover, in the limit c→0 the algebra (1.2) goes to the semidirect sum of the commutative ideal Zab, and Poincaré algebra (1.1).It is  remarkable enough that the momentum square Casimir operator of the Poincaré algebra under this extension ceases to be the Casimir operator, and it is generalized by adding the term linearly dependent on the angular momentum(1.4)PaPa+cZabMba=defXkhklXl,whereXk={Pa,Zab,Mab}. Due to this fact, an irreducible representation of the extended algebra (1.2) has to contain the fields with the different masses [4, 8]. This extension with noncommuting momenta has also something in common with the ideas of the papers [9–11] and with the noncommutative geometry idea [12].It is interesting to note that in spite of the fact that the algebra (1.2) is not semi-simple and therefore has a degenerate Cartan-Killing metric tensor nevertheless there exists another nondegenerate invariant tensor hkl in adjoint representation which corresponds to the quadratic Casimir operator (1.4), where the matrix hkl is inverse to the matrix hkl: hklhlm=δmk.There are other quadratic Casimir operators(1.5)c2ZabZab,(1.6)c2εabcdZabZcd.Note that the Casimir operator (1.6), dependent on the Levi-Civita tensor ϵabcd, is suitable only for the D=4 dimensions.It has also been shown that for the dimensionsD=2,3,4 the extended Poincaré algebra (1.2) allows the following supersymmetric generalization:(1.7){Qκ,Qλ}=-d(σabC)κλZab,[Mab,Qκ]=-(σabQ)κ,[Pa,Qκ]=0,[Zab,Qκ]=0,
with the help of the supertranslation generators Qκ. In (1.7) C is a charge conjugation matrix, d is some constant, and σab=1/4[γa,γb], where γa is the Dirac matrix. Under this supersymmetric generalization the quadratic Casimir operator (1.4) is modified into the following form:(1.8)PaPa+cZabMba-c2dQκ(C-1)κλQλ,while the form of the rest quadratic Casimir operators (1.5), (1.6) remains unchanged.In the present paper we propose another possible semi-simple tensor extension of theD-dimensional Poincaré algebra (1.1) which turns out a direct sum of the D-dimensional Lorentz algebra so(D-1,1) and D-dimensional anti-de Sitter (AdS) algebra so(D-1,2). For the case D=4 dimensions we give for this extension a supersymmetric generalization which is a direct sum of the 4-dimensional Lorentz algebra so(3,1) and orthosymplectic algebra osp(1,4) (super-AdS algebra). In the limit this supersymmetrically generalized extension go to the Lie superalgebra (1.2), (1.7).Let us note that the introduction of the semi-simple extension of the (super) Poincaré algebra is very important for the construction of the models, since it is easier to deal with the nondegenerate space-time symmetry.


## 2. Semi-Simple Tensor Extension

Let us extend the Poincaré algebra (1.1) in the D dimensions by means of the tensor generator Zab in the following way:(2.1)[Mab,Mcd]=(gadMbc+gbcMad)-(c↔d),[Mab,Pc]=gbcPa-gacPb,[Pa,Pb]=cZab,[Mab,Zcd]=(gadZbc+gbcZad)-(c↔d),[Zab,Pc]=4a2c(gbcPa-gacPb),[Zab,Zcd]=4a2c[(gadZbc+gbcZad)-(c↔d)],
where a and c are some constants. This Lie algebra, when the quantities Pa and Zab are taken as the generators of a homomorphism kernel, is homomorphic to the usual Lorentz algebra. It is remarkable that the Lie algebra (2.1) is semi-simple in contrast to the Poincaré algebra (1.1) and extended Poincaré algebra (1.2).The extended Lie algebra (2.1) has the following quadratic Casimir operators:(2.2)C1=PaPa+cZabMba+2a2MabMab=defXkH1klXl,(2.3)C2=c2ZabZab+8a2(cZabMba+2a2MabMab)=defXkH2klXl,(2.4)C3=ϵabcd[c2ZabZcd+8a2(cZbaMcd+2a2MabMcd)].Note that in the limita→0 the algebra (2.1) tends to the algebra (1.2) and the quadratic Casimir operators (2.2), (2.3), and (2.4) are turned into (1.4), (1.5), and (1.6), respectively.The symmetric tensor(2.5)Hkl=sH1kl+tH2kl=Hlkwith arbitrary constantss and t is invariant with respect to the adjoint representation(2.6)Hkl=HmnUmkUnl.Conversely, if we demand the invariance with respect to the adjoint representation of the second rank contravariant symmetric tensor, then we come to the structure (2.5) (see also the relation (32) in [6]).The semi-simple algebra (2.1)(2.7)[Xk,Xl]=fklmXmhas the nondegenerate Cartan-Killing metric tensor(2.8)gkl=fkmnflnm,which is invariant with respect to the coadjoint representation(2.9)gkl=UkmUlngmn.With the help of the inverse metric tensorgkl: gklglm=δmk we can construct the quadratic Casimir operator which, as it turned out, has the following expression in terms of the quadratic Casimir operators (2.2) and (2.3):(2.10)XkgklXl=18a2(D-1)[C1+3-2D8a2(D-2)C2],that corresponds to the particular choice of the constantss and t in (2.5).The extended Poincaré algebra (2.1) can be rewritten in the form(2.11)[Nab,Ncd]=(gadNbc+gbcNad)-(c↔d),(2.12)[LAB,LCD]=(gADLBC+gBCLAD)-(C↔D),(2.13)[Nab,LCD]=0,where the metric tensorgAB has the following nonzero components:(2.14)gAB={gab,gD+1D+1=-1}.The generators(2.15)Nab=Mab-c4a2Zabform the Lorentz algebraso(D-1,1), and the generators(2.16)LAB={Lab=c4a2Zab,LaD+1=-LD+1a=12aPa,LD+1D+1=0}form the algebraso(D-1,2)(Note that in the case D=4 we obtain the anti-de Sitter algebra so(3,2).). The algebra (2.11)–(2.13) is a direct sum so(D-1,1)⊕so(D-1,2) of the D-dimensional Lorentz algebra and D-dimensional anti-de Sitter algebra, correspondingly.The quadratic Casimir operatorsNabNab, LABLAB, and ϵabcdNabNcd of the algebra (2.11)–(2.13) are expressed in terms of the operators C1 (2.2), C2 (2.3), and C3 (2.4) in the following way:(2.17)NabNab-LABLAB=12a2C1,(2.18)NabNab=116a4C2,(2.19)ϵabcdNabNcd=116a4C3.


## 3. Supersymmetric Generalization

In the caseD=4 dimensions the extended Poincaré algebra (2.1) admits the following supersymmetric generalization:(3.1){Qκ,Qλ}=-d[2ac(γaC)κλPa+(σabC)κλZab],[Mab,Qκ]=-(σabQ)κ,[Pa,Qκ]=a(γaQ)κ,[Zab,Qκ]=-4a2c(σabQ)κ,
where Qκ are the supertranslation generators.Under such a generalization the Casimir operator (2.2) is modified by adding a term quadratic in the supertranslation generators(3.2)C̃1=PaPa+cZabMba+2a2MabMab-c2dQκ(C-1)κλQλ=defXKH1KLXL,whereas the form of the rest quadratic Casimir operators (2.3) and (2.4) is not changed. In (3.2) XK={Pa,Zab,Mab,Qκ} is a set of the generators for also the semi-simple extended superalgebra (2.1), (3.1).The tensor(3.3)HKL=vH1KL+wH2KL=(-1)pKpL+pK+pLHLKis invariant with respect to the adjoint representation(3.4)HKL=(-1)(pK+pM)(pL+1)HMNUNLUMK,
where pK=p(K) is a Grassmann parity of the quantity K. In (3.4) v and w are arbitrary constants and nonzero elements of the matrix H2KL equal to the elements of the matrix H2kl followed from (2.3). Again, by demanding the invariance with respect to the adjoint representation of the second rank contravariant tensor HKL=(-1)pKpL+pK+pLHLK, we come to the structure (3.4) (see also the relation (32) in [6]).The semi-simple Lie superalgebra (2.1) (3.1) has the nondegenerate Cartan-Killing metric tensor GKL (see the relation (A.6) in the Appendix A) which is invariant with respect to the coadjoint representation(3.5)GKL=(-1)pK(pL+pN)ULNUKMGMN.With the use of the inverse metric tensorGKL,(3.6)GKLGLM=δMK,we can construct the quadratic Casimir operator (see the relation (A.11) in the Appendix A) which takes the following expression in terms of the Casimir operators (2.3) and (3.2):(3.7)XKGKLXL=120a2(C̃1-932a2C2),that meets the particular choice of the constantsv and w in (3.4).In theD=4 case the extended superalgebra (2.1), (3.1) can be rewritten in the form of the relations (2.11)–(2.13) and the following ones:(3.8){Qκ,Qλ}=-4a2dc(ΣABC)κλLAB,(3.9)[LAB,Qκ]=-(ΣABQ)κ,(3.10)[Nab,Qκ]=0,where(3.11)ΣAB=14[ΓA,ΓB],ΓA={γaγ5,γ5},{γa,γb}=2gab,gab=diag(-1,1,1,1),γ5=γ0γ1γ2γ3.The generatorsNab (2.15) form the Lorentz algebra so(3,1) and the generators LAB (2.16), Qκ form the orthosymplectic algebra osp(1,4). We see that superalgebra (2.11)–(2.13), (3.8)–(3.10) is a direct sum so(3,1)⊕osp(1,4) of the 4-dimensional Lorentz algebra and 4-dimensional super-AdS algebra, respectively.In this case the Casimir operator (2.17) is modified by adding a term quadratic in the supertranslation generators(3.12)NabNab-LABLAB-c4a2dQκ(C-1)κλQλ=12a2C̃1,while the form of the quadratic Casimir operators (2.18) and (2.19) is not changed.


## 4. Conclusion

Thus, we proposed the semi-simple second rank tensor extension of the Poincaré algebra in the arbitrary dimensionsD and super-Poincaré algebra in the D=4 dimensions. It is very important, since under construction of the models, it is more convenient to deal with the nondegenerate space-time symmetry. We also constructed the quadratic Casimir operators for the semi-simple extended Poincaré and super Poincaré algebra.It is interesting to develop the models based on these extended algebra. The work in this direction is in progress.



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*Source: 234147-2009-06-14.xml*