Summary: The thesis is a study in differential geometry. One of the central notions in differential geometry are cochain complex and its cohomology. For instance, the cohomologies of the complex of differential forms defined on a smooth manifold are analytical characteristic of manifold’s topology. The idea of cohomology is based on the following important property of the differential: it gives zero when applied twice. The algebraic model of this geometrical structure is differential module with differential satisfying the property that differential squared is equal to zero. In the present thesis we study the generalization of differential module where differential satisfies the property that differential to the \(q\)-th power is equal to zero, where \(q\) is an integer greater than or equal to two. This generalization was proposed and studied within the framework of noncommutative geometry by M. Kapranov and M. Dubois-Violette with R. Kerner in 1990th.
The theory is based on the following \(q\)-deformed structures used in noncommutative geometry: graded \(q\)-commutator, graded \(q\)-derivation, graded \(q\)-Leibniz rule, where \(q\) is a primitive \(N\)th root of unity. In elaboration of this theory following structures are developed: \(N\)-differential complex, \(N\)-simplicial complex, cochain \(N\)-complex and its generalized cohomologies, graded \(N\)-differential algebra. A connection and its curvature are basic elements of the theory of fibre bundles, and they play an important role not only in modern differential geometry but also in theoretical physics namely in gauge field theory. Within the framework of noncommutative geometry was constructed the generalization of the theory of connection, called connection on a module.
In the present thesis we propose the generalization of a concept of connection on a module, called \(N\)-connection, our approach is based on a graded \(q\)-differential algebra. We define the notion of curvature of \(N\)-connection and proove that it satisfies the analog of Bianchi identity. We prove that every projective module admits an \(N\)-connection. We study the local structure of \(N\)-connection and its curvature. We express the components of the curvature of a \(N\)-connection in the terms of the matrix of \(N\)-connection. We propose a possible approach to how one can construct a Chern character in the theory of \(N\)-connection.