A semi-simple tensor extension of the Poincaré algebra is proposed for the arbitrary dimensions D. 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).
1. Introduction
In the papers [1–7] the Poincaré algebra for the generators of the rotations Mab and translations Pa in D dimensions,
where c 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:
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
PaPa+cZabMba=defXkhklXl,
where Xk={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
c2ZabZab,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 dimensions D=2,3,4 the extended Poincaré algebra (1.2) allows the following supersymmetric generalization:
{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:
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 the D-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:
[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:
Note that in the limit a→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
Hkl=sH1kl+tH2kl=Hlk
with arbitrary constants s and t is invariant with respect to the adjoint representation
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)
[Xk,Xl]=fklmXm
has the nondegenerate Cartan-Killing metric tensor
gkl=fkmnflnm,
which is invariant with respect to the coadjoint representation
gkl=UkmUlngmn.
With the help of the inverse metric tensor gkl: 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):
XkgklXl=18a2(D-1)[C1+3-2D8a2(D-2)C2],
that corresponds to the particular choice of the constants s and t in (2.5).
The extended Poincaré algebra (2.1) can be rewritten in the form
where the metric tensor gAB has the following nonzero components:
gAB={gab,gD+1D+1=-1}.
The generators
Nab=Mab-c4a2Zab
form the Lorentz algebra so(D-1,1), and the generators
LAB={Lab=c4a2Zab,LaD+1=-LD+1a=12aPa,LD+1D+1=0}
form the algebra so(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 operators NabNab, 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:
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
HKL=vH1KL+wH2KL=(-1)pKpL+pK+pLHLK
is invariant with respect to the adjoint representation
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
GKL=(-1)pK(pL+pN)ULNUKMGMN.
With the use of the inverse metric tensor GKL,
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):
XKGKLXL=120a2(C̃1-932a2C2),
that meets the particular choice of the constants v and w in (3.4).
In the D=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:
The generators Nab (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
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 dimensions D 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.
AppendixA. Properties of Lie Superalgerbra
Permutation relations for the generators XK of Lie superalgebra are
[XK,XL}=defXKXL-(-1)pKpLXLXK=fKLMXM.
Structure constants fKLM have the Grassmann parity
p(fKLM)=pK+pL+pM=0(mod2),
following symmetry property:
fKLM=-(-1)pKpLfLKM
and obey the Jacobi identities
∑(KLM)(-1)pKpMfKNPfLMN=0,
where the symbol (KLM) means a cyclic permutation of the quantities K, L, and M. In the relations (A.1)–(A.4) every index K takes either a Grassmann-even value k(pk=0) or a Grassmann-odd one κ(pκ=1). The relations (A.1) have the following components:
[Xk,Xl]=fklmXm,{Xκ,Xλ}=fκλmXm,[Xk,Xλ]=fkλμXμ.
The Lie superalgebra possesses the Cartan-Killing metric tensor
As a consequence of the relations (A.3) and (A.4) the tensor with low indices
fKLM=fKLNGNM
has the following symmetry properties:
fKLM=-(-1)pKpLfLKM=-(-1)pKpMfKML.
For a semi-simple Lie superalgebra the Cartan-Killing metric tensor is nondegenerate and therefore there exists an inverse tensor GKL,
GKLGLM=δKM.
In this case, as a result of the symmetry properties (A.9), the quantity
XKGKLXL
is a Casimir operator
[XKGKLXL,XM]=0.
Acknowledgments
The authors are grateful to J.A. de Azcarraga for the valuable remark. They are greatly indebted to the referee for the constructive comments. One of the authors (V.A.S.) thanks the administration of the Office of Associate and Federation Schemes of the Abdus Salam ICTP for the kind hospitality at Trieste where this work has been completed. The research of V.A.S. was partially supported by the Ukrainian National Academy of Science and Russian Fund of Fundamental Research, Grant no. 38/50-2008.
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