Z₃-Grading and Ternary Algebraic Structures

Abstract

There is steadily growing interest, during past few years, in the algebraic structures that might replace algebras of functions describing geometries of the continuum. Such non-conventional geometries can be traced back to the introduction of anti-commuting variables (Berezin (1), A and Volkov (2), Wess and Zumino (3)) and the introduction of the supermanifolds (Rogers (4), Gawedzki (5), Sternberg (6)). In these geometries the algebra of functions separates quite naturally into even and odd parts, and can be described locally by ordinary functions on a manifold tensorised with some Grassman algebra spanned by the anti-commuting variables. It turned out that it was possible to generalize almost all the concepts of usual geometries to the non-commutative Grassmannian variables, i.e. the differentiation, the integration, the vector fields, connections and the like.