Semi-simple extension of the (super)Poincar\'e algebra

A semi-simple tensor extension of the Poincar\'e algebra is proposed for the arbitrary dimensions $D$. A supersymmetric also semi-simple generalization of this extension is constructed in the D=4 dimensions. This paper is dedicated to the memory of Anna Yakovlevna Gelyukh.


Introduction
In the papers [1,2,3,4,5,6]  where c is some constant 1 . 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 semi-direct sum of the commutative ideal Z ab 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 where X k = {P a , Z ab , M ab }. Due to this fact, an irreducible representation of the extended algebra (1.2) has to contain the fields of the different masses [4,7]. This extension with non-commuting momenta has also something in common with the ideas of the papers [8,9,10] and with the non-commutative geometry idea [11]. 1 Note that, to avoid the double count under summation over the pair antisymmetric indices, we adopt the rules which illustrated by the following example: where f ab cd are structure constants, and so on.
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 non-degenerate invariant tensor h kl in adjoint representation which corresponds to the quadratic Casimir operator (1.3), where the matrix h kl is inverse to the matrix h kl : h kl h lm = δ k m . There are other quadratic Casimir operators Note that the Casimir operator (1.5), dependent on the Levi-Civita tensor ǫ abcd , 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: with the help of the super-translation generators Q κ . In (1.6) C is a charge conjugation matrix, d is some constant and σ ab = 1 4 [γ a , γ b ], where γ a is the Dirac matrix. Under this supersymmetric generalization the quadratic Casimir operator (1.3) is modified into the following form: while the form of the rest quadratic Casimir operators (1.4), (1.5) remains unchanged.
In the present paper we propose another possible semi-simple tensor extension of the Poincaré algebra (1.1) and for the case D = 4 dimensions we give a supersymmetric generalization of this extension. In the limit this supersymmetrically generalized extension go to the Lie superalgebra (1.2), (1.6).

Semi-simple tensor extension
Let us extend the Poincaré algebra (1.1) in the D dimensions by means of the tensor generator Z ab in the following way: where a and c are some constants. This Lie algebra, when the quantities P a and Z ab 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: with arbitrary constants s and t is invariant with respect to the adjoint representation 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) has the non-degenerate Cartan-Killing metric tensor which is invariant with respect to the co-adjoint representation With the help of the inverse metric tensor g kl : g kl g lm = δ k m 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): 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 g AB has the following nonzero components: The generators form the Lorentz algebra so(D − 1, 1) and the generators form the algebra so(D − 1, 2) 2 . The algebra (2.7)-(2.9) is a direct sum so(D − 1, 1) ⊕ so(D − 1, 2). The quadratic Casimir operators N ab N ab , L AB L AB and ǫ abcd N ab N cd of the algebra (2.7)-(2.9) are expressed in terms of the operators C 1 (2.2), C 2 (2.3) and C 3 (2.4) in the following way: (2.14) (2.15)

Supersymmetric generalization
In the case D = 4 dimensions the extended Poincaré algebra (2.1) admits the following supersymmetric generalization: where Q κ are the super-translation generators. Under such a generalization the Casimir operator (2.2) is modified by adding a term quadratic in the super-translation generators whereas the form of the rest quadratic Casimir operators (2.3) and (2.4) is not changed. In (3.2) X K = {P a , Z ab , M ab , Q κ } is a set of the generators for the also semi-simple extended superalgebra (2.1), (3.1).
The tensor is invariant with respect to the adjoint representation where p K = p(K) is a Grassmann parity of the quantity K. In (3.3) v and w are arbitrary constants and nonzero elements of the matrix H KL 2 equal to the elements of the matrix H kl 2 followed from (2.3). Again, by demanding the invariance with respect to the adjoint representation of the second rank contravariant tensor H KL = (−1) p K p L +p K +p L H LK , we come to the structure (3.3) (see also the relation (32) in [6]).
The semi-simple Lie superalgebra (2.1) (3.1) has the non-degenerate Cartan-Killing metric tensor G KL (see the relation (A.5) in the Appendix A) which is invariant with respect to the co-adjoint representation With the use of the inverse metric tensor G KL we can construct the quadratic Casimir operator (see the relation (A.8) in the Appendix A) which takes the following expression in terms of the Casimir operators (2.3) and (3.2): that meets the particular choice of the constants v and w in (3.3).

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 non-degenerate space-time symmetry. We also constructed the quadratic Casimir operators for the semi-simple extended Poincaré and super Poincaré algebras. It is interesting to develop the models based on these extended algebras. The work in this direction is in progress.

As a consequence of the relations (A.3) and (A.4) the tensor with low indices
has the following symmetry properties: For a semi-simple Lie superalgebra the Cartan-Killing metric tensor is non-degenerate and therefore there exists an inverse tensor G KL In this case, as a result of the symmetry properties (A.7), the quantity is a Casimir operator [X K G KL X L , X M ] = 0.