Poisson Structures on the Poincaré Group

Abstract

We present almost complete list of normal forms of classical r-matrices on the Poincaré group.

$ N = \frac{1}{2} $ deformations of chiral superspaces from new quantum Poincaré and Euclidean superalgebras

“…[23]- [26]) supersymmetry. 2 Following the classification of D = 4 Poincaré deformations by the classical rmatrices [33] we shall study their Euclidean counterpart and further the supersymmetric extensions of Poincaré and Euclidean cases. The aim and novelty of our approach is:…”

“…(i) to show that in view of the known list for classical Poincaré r-matrices presented in [33] one can find corresponding list of D = 4 Euclidean r-matrices,…”

“…For the Poincaré algebra the classical r-matrices were almost completely classified already some time ago by S. Zakrzewski in [35] for the Lorentz algebra and in [33] for the Poincaré algebra. We shall briefly remind these results.…”

“…We shall briefly remind these results. It was shown in [33] that each classical r-matrix, r ∈ P(3, 1) ∧ P(3, 1), has a decomposition Table 1. Classification of classical r-matrices for D = 4 Poincaré algebra (see [33]).…”

“…The results of [33] are presented in table 1, where P ± = P 0 ± P 3 and b P + , b P 2 are given by the expressions:…”