Construction of Lie algebras and Lie superalgebras from ternary algebras

Abstract

Ternary algebras are algebras which close under suitable triple products. They have been shown to be building blocks of ordinary Lie algebras. They may acquire a deep physical meaning in fundamental theories given the important role played by Lie (super) algebras in theoretical physics. In this paper we introduce the concept of superternary algebras involving Bosers and Fermi variables. Using them as building blocks, we give a unified construction of Lie algebras and superalgebras in terms of (super) ternary algebras. We prove theorems that must be satisfied for the validity of this construction, which is a generalization of Kantor’s results. A large number of examples and explicit constructions of the Lie algebras A_n, B_n, F_n, F_n, F₄, F₆, F₇, F₈, and Lie superalgebras A (m,n), B (m,n), D (m,n), P (n), Q (n) are given. We speculate on possible physical applications of (super) ternary algebras.