We propose a generalization of non-commutative geometry and gauge theories based on ternary Z 3 -graded structures. In the new algebraic structures we define, we leave all products of two entities free, imposing relations on ternary products only. These relations reflect the action of the Z 3 -group, which may be either trivial, i.e. abc = bca = cab, generalizing the usual commutativity, or non-trivial, i.e. abc = jbca, with j = e (2πi)/3 . The usual Z 2 -graded structures such as Grassmann, Lie and Clifford algebras are generalized to the Z 3 -graded case. Certain suggestions concerning the eventual use of these new structures in physics of elementary particles are exposed.

We investigate a particular realization of generalized q-differential calculus of exterior forms on a smooth manifold based on the assumption that d N ϭ0 while d k 0 for kϽN. It implies the existence of cyclic commutation relations for the differentials of first order and their generalization for the differentials of higher order. Special attention is paid to the cases Nϭ3 and Nϭ4. A covariant basis of the algebra of such q-grade forms is introduced, and the analogs of torsion and curvature of higher order are considered. We also study a Z N -graded exterior calculus on a generalized Clifford algebra.

Abstract. We propose a notion of a super n-Lie algebra and construct a super n-Lie algebra with the help of a given binary super Lie algebra which is equipped with an analog of a supertrace. We apply this approach to the super Lie algebra of a Clifford algebra with even number of generators and making use of a matrix representation of this super Lie algebra given by a supermodule of spinors we construct a series of super 3-Lie algebras labeled by positive even integers. Mathematics Subject Classification (2010). Primary 17B56; Secondary 15A66.

Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper we show that this approach can be extended to the infinitedimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras gl (m, n) if instead of the trace of a matrix we make use of the super trace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the super trace satisfies a graded ternary Filippov-Jacobi identity. In two particular cases of gl (1, 2) and gl(2, 2) we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

We construct the graded q-differential algebra on a Z N -graded algebra by means of a graded q-commutator. We apply this construction to a reduced quantum plane and study the first order differential calculus on a reduced quantum plane induced by the N -differential of the graded q-differential algebra.

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