Abstract
We provide the classification of real forms of complex D=4 Euclidean algebra ɛ(4;C) = o(4; C)) ⋉ \U0001d4134C as well as (pseudo)real forms of complex D=4 Euclidean superalgebras ɛ(4|N; C) for N=1,2. Further we present our results: N=1 and N=2 supersymmetric D=4 Poincaré and Euclidean r-matrices obtained by using D= 4 Poincaré r-matrices provided by Zakrzewski [1] For N=2 we shall consider the general superalgebras with two central charges.
Highlights
The classification of quantum deformations of Lorentz symmetries described by classical r-matrices was given firstly by Zakrzewski [2], and has been further extended to the classification of classical r-matrices for Poincarealgebra in [1]
The reality conditions (2.8–2.10) imposed on the complex generators M± look as follows
Further the reality constraints on the internal symmetry generators the reality condition (TA) (see (3.5) and (3.8)) and the central charges (Z1,Z2), which are consistent with the relations (3.1–3.2), (3.6) and (3.8) are the following
Summary
The classification of quantum deformations of Lorentz symmetries described by classical r-matrices was given firstly by Zakrzewski [2] (see [3]), and has been further extended to the classification of classical r-matrices for Poincarealgebra in [1]. The list of N = 2 complex supersymmetric r-matrices and their Kleinian o(2,2) real counterparts satisfying suitable (pseudo)reality conditions will be presented in our publications. In the algebraic framework one can extend in odd supercharges sector the conjugation (2.9) as the pseudoconjugation (see Appendix A, (A.3b)) which should be consistent with N = 1 complex Euclidean algebra (2.25–2.27) under the assumption that the generators Pμ, Lμν are real Euclidean, i.e. satisfy the reality conditions (2.9).
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