[Paper Review] Fractional Supersymmetry and F-fold Lie Superalgebras

Michel Rausch de Traubenberg, Marcus J. Slupinski|ArXiv.org|Oct 2, 2001

Algebraic structures and combinatorial models39 references3 citations

TL;DR

This paper introduces F-fold Lie superalgebras as a generalization of Lie superalgebras to describe fractional supersymmetry, using a Z_F-grading structure where the zero-graded part is a Lie algebra and higher-graded parts transform under it. It constructs both infinite-dimensional and finite-dimensional examples via induced representations and symmetric F-ary brackets, with explicit realizations for F=3 and F=4 using sl(2) and osp(m|2n) algebras.

ABSTRACT

We give infinite dimensional and finite dimensional examples of $F-$fold Lie superalgebras. The finite dimensional examples are obtained by an inductive procedure from Lie algebras and Lie superalgebras.

Motivation & Objective

• To generalize Lie superalgebras to F-fold Lie superalgebras for describing fractional supersymmetry beyond standard supersymmetry.
• To provide a systematic construction of F-fold Lie superalgebras using Z_F-graded vector spaces with a Lie algebra in the zero-graded part.
• To demonstrate finite-dimensional examples through inductive procedures from known Lie and Lie superalgebras.
• To establish the algebraic framework for F-fold symmetric brackets that generalize the anticommutator in supersymmetry.
• To explore the structure and representation of F-fold algebras, particularly for F=3 and F=4, using standard Lie algebras and superalgebras as building blocks.

Proposed method

• Define F-fold Lie superalgebras as Z_F-graded complex vector spaces with a Lie algebra in the zero-graded part and representations in non-zero graded parts.
• Introduce an F-ary symmetric, B-equivariant product map {·, ..., ·}: S^F(A_k) → B, generalizing the anticommutator for F=2.
• Ensure closure under Jacobi identities involving the Lie bracket and the F-ary bracket, including a cyclic identity for F+1 generators.
• Construct infinite-dimensional examples using semi-simple Lie algebras and their modules, particularly via root systems and weight lattices.
• Build finite-dimensional examples inductively from Lie algebras (e.g., sl(2)) and Lie superalgebras (e.g., osp(m|2n)) via tensor product structures.
• Explicitly realize the F-ary brackets for F=3 and F=4 using Killing forms and invariant tensors (e.g., δ_ij, Ω_αβ, ε_ij).

Experimental results

Research questions

• RQ1 How can fractional supersymmetry be algebraically described using generalized Lie algebraic structures?
• RQ2 What is the mathematical structure of F-fold Lie superalgebras, and how do they generalize Lie superalgebras?
• RQ3 Can finite-dimensional F-fold Lie superalgebras be systematically constructed from standard Lie and Lie superalgebras?
• RQ4 What are the explicit forms of the F-ary symmetric brackets for F=3 and F=4 in terms of known Lie algebra invariants?
• RQ5 How does the Z_F-grading and the F-ary bracket structure ensure consistency with Jacobi identities and B-equivariance?

Key findings

• The paper constructs infinite-dimensional F-fold Lie superalgebras using semi-simple Lie algebras and their modules via root systems and weight lattices.
• Finite-dimensional F-fold Lie superalgebras are systematically built from Lie algebras (e.g., sl(2)) and Lie superalgebras (e.g., osp(m|2n)) via inductive procedures.
• For F=3, the trilinear bracket of the 3-fold algebra constructed from sl(2) is explicitly given by {A_a, A_b, A_c} = g_ab J_c + g_ac J_b + g_bc J_a, matching a known construction.
• For F=4, the quadrilinear bracket of the 4-fold algebra from osp(m|2n) is derived using invariant tensors δ_ij, Ω_αβ, and ε_ij, with coefficients depending on δ and ε functions.
• The F-fold algebra associated with a Lie algebra is of odd order (F odd), while that from a Lie superalgebra is of even order (F even), as per Remark 4.4.
• The F-ary bracket satisfies a cyclic Jacobi identity for F+1 generators, ensuring consistency with the algebraic structure beyond the standard Lie and super-Lie identities.