symmetric monoidal (∞,1)-category of spectra
A bialgebroid may be viewed as a multiobject generalization of a concept of a bialgebra, or a possibly noncommutative generalization of a space-algebra dual version of the concept of an internal category in spaces.
This entry is about “associative” bialgebroid, see also the different concept of a Lie bialgebroid.
When a monoidal category has a fiber functor to a category of vector spaces over a field, one tries to “reconstruct” the category as the category of representations of the endomorphism object of a fiber functor. One often does not have a fiber functor to vector spaces but only to bimodules over some base algebra . Sometimes in such cases, the object of endomorphisms of the fiber functor form a bialgebroid over and the category is equivalent to the category of representations of that bialgebroid.
Given a unital (possibly noncommutative) ring an -bialgebroid is an --bimodule (object of ) equipped with a structure of a comonoid in (i.e. an -coring) and of a monoid in (i.e. an -ring), where is the enveloping ring of ; and the structures of a monoid and a comonoid satisfy certain compatibility conditions. These compatibility conditions are equivalent to the fact that the monad is opmonoidal. The category of -comodules is by definition the category of comodules over the underlying -coring.
If is an associative algebra over some ground field , then a left associative -bialgebroid is another associative -algebra together with the following additional maps: an algebra map called the source map, an algebra map called the target map, so that the elements of the images of and commute in , therefore inducing an -bimodule structure on via the rule for ; an -bimodule morphism which is required to be a counital coassociative comultiplication on in the monoidal category of -bimodules with monoidal product . The map must be a left action extending the multiplication along . Furthermore, a compatibility between the comultiplication and multiplications on and on is required. For a noncommutative the tensor square is not an algebra, hence asking for a bialgebra-like compatibility that is a morphism of -algebras does not make sense. Instead, one requires that has a -subspace which contains the image of and has a well-defined multiplication induced from its preimage under the projection from the usual tensor square algebra . Then one requires that the corestriction is a homomorphism of unital algebras. Under these conditions, one can make a canonical choice for , namely the so called Takeuchi’s product , which always inherits an associative multiplication along the projection from .
All modules and morphisms will be over a fixed ground commutative ring .
A left -bialgebroid is an -ring , together with the -bimodule map “comultiplication” , which is coassociative and counital with a counit , such that
(i) the -bimodule structure used on is , where and are the algebra maps induced by the unit of the -ring
(ii) the coproduct corestricts to the Takeuchi product and the corestriction is a -algebra map, where the Takeuchi product has a multiplication induced factorwise
(iii) is a left character on the -ring .
Notice that is in general not an algebra, just an -bimodule. That is why (ii) is needed. An equivalent condition to (ii) is the following: the formula defines a well-defined action of on .
The definition of a right -bialgebroid differs by the -bimodule structure on given instead by and the counit is a right character on the -coring ( and can be interchanged in the last requirement).
Related notions: Hopf algebroid
The commutative case is rather classical. See for example the appendix to
The first version of a bialgebroid over a noncommutative base was more narrow:
A modern generality, but in different early formalism, is due to Takeuchi (who was motivated to generalize the results from the Sweedler’s paper), under the name of -bialgebra (as it involves the -product, nowdays called Takeuchi product):
Lu introduces the name bialgebroid for a structure which is equivalent to the Takeuchi’s -bialgebra (though differently axiomatized there):
Modern treatments are in
Gabriella Böhm, Internal bialgebroids, entwining structures and corings, math.QA/0311244, in: Algebraic structures and their representations, 207–226, Contemp. Math. 376, Amer. Math. Soc. 2005.
G. Böhm, Hopf algebroids, (a chapter of) Handbook of algebra, arxiv:math.RA/0805.3806
Kornél Szlachányi, The monoidal Eilenberg–Moore construction and bialgebroids, J. Pure Appl. Algebra 182, no. 2–3 (2003) 287–315; Fiber functors, monoidal sites and Tannaka duality for bialgebroids, arxiv/0907.1578
T. Brzeziński, G. Militaru, Bialgebroids, -bialgebras and duality, J. Algebra 251: 279-294, 2002, math.QA/0012164
J. Donin, Andrey Mudrov, Quantum groupoids and dynamical categories, J. Algebra 296 (2006), no. 2, 348–384, math.QA/0311316, MR2007b:17022, doi
There is also a notion of quasibialgebroid, where the coassociativity is weakened by a bialgebroid 3-cocycle. See also Hopf algebroid.
Last revised on September 28, 2024 at 13:58:07. See the history of this page for a list of all contributions to it.