Finite-dimensional Lie algebras of order F

Abstract

F −Lie algebras are natural generalisations of Lie algebras (F = 1) and Lie superalgebras (F = 2). When F > 2 not many finite-dimensional examples are known. In this paper we construct finitedimensional F −Lie algebras F > 2 by an inductive process starting from Lie algebras and Lie superalgebras. Matrix realisations of F −Lie algebras constructed in this way from su(n), sp(2n) so(n) and sl(n|m), osp(2|m) are given. We obtain non-trivial extensions of the Poincaré algebra by Inönü-Wigner contraction of certain F −Lie algebras with F > 2.

Poincaré and sl(2) algebras of order 3

“…and we refer to it as a 3−bracket. Non-trivial examples of Lie algebras of order F (finite and infinite-dimensional) are given in [26] and [27]. We now give some examples of finite-dimensional Lie algebras of order 3, which will be relevant in the sequel.…”

“…In this section we recall the definition and some basic properties of Lie algebras of order F introduced in [26] and [27] and we define the algebraic variety of these algebraic structures.…”

“…In [26] an inductive process for the construction of Lie algebras of order F starting from a Lie algebra of order F 1 with 1 ≤ F 1 < F is given. In this paper we are especially concerned by deformation and classification problems.…”

“…In the same way it is known that the N −extended supersymmetric extension of the Poincaré algebra can be obtained through a contraction of the superalgebra osp(4|N ). Similarly the extension of the Poincaré algebra studied in [20,21,22,29] was obtained through a contraction of a certain Lie algebra of order 3 [26].…”

“…Several attempts to construct models based on different algebras were proposed. Here we focus on one of the possible extensions called fractional supersymmetry [3,4,5,6,18,19,20,21,22,24,25,28,29] together with its associated underlying algebraic structure named F −Lie algebra or Lie algebra of order F [26,27] (note that a different approach has been proposed in [17]). Lie algebras of order F lead to new models of symmetry of space-time i.e.…”