R-smash products of Hopf quasigroups
The theory of R-smash products for Hopf quasigroups is developed.
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arXiv subjects
Tomasz Brzeziński
Publications and source records attributed to Tomasz Brzeziński.
Actions of Hopf quasigroups
Definitions of actions of Hopf quasigroups are discussed in the context of Long dimodules and smash products. In particular, Long dimodules are defined for Hopf quasigroups and coquasigroups, and solutions to Militaru's D-equation are constructed. A necessary compatibility condition between action and multiplication of a Hopf quasigroup acting on its quasimodule Hopf quasigroup for a smash product construction is derived.
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Integral calculus on $E_q(2)$
The complexes of integral forms on the quantum Euclidean group $E_q(2)$ and the quantum plane are defined and their isomorphisms with the corresponding de Rham complexes are established.
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Hopf modules and the fundamental theorem for Hopf (co)quasigroups
The notion of a Hopf module over a Hopf (co)quasigroup is introduced and a version of the fundamental theorem for Hopf (co)quasigroups is proven.
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On synthetic interpretation of quantum principal bundles
Quantum principal bundles or principal comodule algebras are re-interpreted as principal bundles within a framework of Synthetic Noncommutative Differential Geometry. More specifically, the notion of a noncommutative principal bundle within a braided monoidal category is introduced and it is shown that a noncommutative principal bundle in the category opposite to the category of vector spaces is the same as a faithfully flat Hopf-Galois extension.
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Internal Kleisli categories
A construction of Kleisli objects in 2-categories of noncartesian internal categories or categories internal to monoidal categories is presented.
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The Eilenberg-Moore category and a Beck-type theorem for a Morita context
The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a {\em Morita context} comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a {\em double adjunction} to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the {\em moritability} of a pair of functors with a common domain) are derived.
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