Poincaré and sl(2) algebras of order 3

Abstract

In this paper we initiate a general classification for Lie algebras of order 3 and we give all Lie algebras of order 3 based on sl(2, C) and iso(1, 3) the Poincaré algebra in four-dimensions. We then set the basis of the theory of the deformations (in the Gerstenhaber sense) and contractions for Lie algebras of order 3.

Extension of the Poincaré Symmetry and Its Field Theoretical Implementation

“…We present here such results for 4-dimensional representations, other results also to be found in [24]. Notice here that, when considering both our extension and the SUSY extension 2 , one uses representations of the Poincaré algebra of dimension 4.…”

“…In this section we give definitions for the algebraic setting used; general aspects are briefly discussed (for more details one may refer to [6,7] and [24]). In [6,7], a complex Lie algebra of order F (F ∈ N * ) is defined as a Z F -graded C-vector space g = g 0 ⊕ g 1 ⊕ g 2 ⊕ · · · ⊕ g F −1 satisfying the following conditions:…”

Cubic extentions of the Poincaré algebra

“…[P µ , g 1 ] = 0 [12]. The case where g 1 is reducible is more involved as can be seen of the following two examples:…”

“…The general case can be more complicated and his synthesize in Lemma 5.1 of [12]. Here, we just recall the main properties of this technical Lemma.…”

“…A general study of the possible non-trivial extensions of the (1+3)D Poincaré algebra has been undertaken. In this paper we summarize some of the results established in [12] and we give a systematic way to construct all possible extensions of the Poincaré algebra when F = 3.…”