ar X iv : q - a lg / 9 60 30 19 v 1 2 4 M ar 1 99 5 DUAL STRUCTURES IN NON - COMMUTATIVE DIFFERENTIAL ALGEBRAS

The non-commutative algebraic analog of the moduli of vector and covector fields is built. The structure of moduli of derivations of non-commutative algebras are studied. The canonical coupling is introduced and the conditions for appropriate moduli to be reflexive are obtained. FOREWORD The duality problem we are going to tackle stems from the non-commutative generalization of differential geometry of manifolds. The variety of structures it deals with is defined in terms of two basic objects: the algebra A of smooth functions on a manifold and the Lie algebra V of smooth vector fields. In standard differential geometry V is always a reflexive A-module which enables the tensor analysis to be successively built. Whereas it is worthy to note that the fundamental geometrical notions are formulated in pure algebraic terms of commutative algebra A [2]. The attempts of direct generalization of these notions starting from a non-commutative algebra A produce non-trivial algebraic problems: we focus on two of them.