Noncommutative Riemannian Geometry of Twisted Derivations

Sammanfattning: A twisted derivation is a generalized derivative satisfying a twisted version of the ordinary Leibniz rule for products. In particular, a (σ, τ )-derivation on an algebra A, is a derivation where Leibniz rule is twisted by two endomorphisms σ and τ on A. Such derivations play an important role in the theory of quantum groups, as well as in the context of discretized and deformed derivatives. In this thesis, we develop a (commutative and noncommutative) differential geometry based on (σ, τ )- derivations. To this end, we introduce the notion of (σ, τ )-algebra, consisting of an associative algebra together with a set of (σ, τ )-derivations, to construct connections satisfying a twisted Leibniz rule in analogy with (σ, τ )-derivations. We show that such connections always exist on projective modules and that it is possible to construct connections compatible with a hermitian form. To construct torsion and curvature of (σ, τ )-connections, we introduce the notion of (σ, τ )-Lie algebra and demonstrate that it is possible to construct a Levi-Civita (σ, τ )-connection. Having constructed the framework for studying (σ, τ)-connections, we demonstrate that the framework applied to commutative algebras can help to also give a good understanding of (σ, τ )-derivations on commutative algebras. In particular, we introduce a notion of symmetric (σ, τ )-derivations together with some regularity conditions. For example, we show that strongly regular (σ, τ )-derivations are always inner and there exist a symmetric (σ, τ )-connection on symmetric (σ, τ )- algebras. Finally, we introduce a (σ, τ)-Hochschild cohomology theory which in first degree captures the outer (σ, τ )-derivations of an associative algebra. Along the way, examples including both commutative and noncommutative algebras are presented to illustrate the novel concepts.