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Flat connections and (co)modules

Tomasz Brzezinski
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Abstract

The relationship between comodules of a coring and flat connections is reviewed. In particular we specialise to corings which are built on a tensor product of algebra and a coalgebra. Such corings are in one-to-one correspondence with entwining structures, and their comodules are entwined modules. These include Yetter-Drinfeld and anti-Yetter-Drinfeld modules and their generalisations, hence all the modules of interest to Hopf-cyclic cohomology. In this way the interpretation of the latter as modules with flat connections [A Kaygun and M Khalkhali, Hopf modules and noncommutative differential geometry, Lett. Math. Phys. 76 (2006), 77--91] is obtained as a corollary of a more general theory. We also introduce the notions of a connection in a comodule and of a bicomodule connection, and show how comodules with flat connections can be interpreted as modules of a C-ring. In this way all the above mentioned Hopf modules can be interpreted as comodules with flat connections.

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Tomasz Brzezinski. 2007-01-19. Flat connections and (co)modules. https://arxiv.org/abs/math/0608170

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