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Description: |
A differential graded algebra on a non-commutative algebra A is interpreted as the space of differential forms on A and is usually referred to as a differential calculus on A. Connections in non-commutative geometry are maps from a module M to M tensored with the space of one-forms that satisfy the Leibinz rule. Motivated by the adjoint relationship between tensor products and homomorphisms, T.Brzezinski introduced the notion of hom-connections as maps with a domain in homomorphisms from one-forms to M and with M as a codomain that are also required to satisfy the Leibniz rule. We will report on the existence of possible differential calculi and non-trivial hom-connections on an algebra A and show that such data can be found if the algebra admits a twisted multi-derivation which consists of an algebra map from A to the space of n by n matrices over A and a derivation from A to the n-tuples of A subject to certain conditions. We will illustrate our construction on several examples and will focus on the case of covariant differential calculi on Hopf algebras and Woronowicz's three-dimensional calculus on the quantum coordinate ring of SL(2). This is joint work with Tomasz Brzezinski from Swansea Univesity and Laichi El Kaoutit from Universidad de Granada.
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