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The cohomology structure of an algebra entwined with a coalgebra

Tomasz Brzezinski
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Abstract

Two cochain complexes are constructed for an algebra A and a coalgebra C entwined with each other via the map $ψ:C\otimes A\to A\otimes C$. One complex is associated to an A-bimodule, the other to a C-bicomodule. In the former case the resulting complex can be considered as a $ψ$-twisted Hochschild complex of A, while for the latter one obtains a $ψ$-twist of the Cartier complex of C. The notion of a {\em weak comp algebra} is introduced by weakening the axioms of the Gerstenhaber comp algebra. It is shown that such a weak comp algebra is a cochain complex with two cup products that descend to the cohomology. It is also shown that the complexes associated to an entwining structure and A or C are examples of a weak comp algebra. Finally both complexes are combined in a double complex whose role in the deformation theory of entwining structures is outlined.

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Tomasz Brzezinski. 1999-09-18. The cohomology structure of an algebra entwined with a coalgebra. https://arxiv.org/abs/math/9909108

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