nLab
bialgebroid

Idea

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.

Nomenclature

This entry is about “associative” bialgebroid, see also the different concept of a Lie bialgebroid.

Motivation in Tannakian formalism

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 $A$. Sometimes in such cases, the object of endomorphisms of the fiber functor form a bialgebroid over $A$ and the category is equivalent to the category of representations of that bialgebroid.

Definition

Via monoidal categories

Given a unital (possibly noncommutative) ring $R$ an $R$-bialgebroid is an $R$-$R$-bimodule $H$ (object of ${}_R{ℳ}_R$) equipped with a structure of a comonoid in ${}_R{ℳ}_R$ (i.e. an $R$-coring) and of a monoid in ${}_{R^e}{ℳ}_{R^e}$ (i.e. an $R^e$-ring), where $R^e=R^{\mathrm{op}}\otimes R$ is the enveloping ring of $R$; and the structures of a monoid and a comonoid satisfy certain compatibility conditions. These compatibility conditions are equivalent to the fact that the monad ${}_{{\otimes }_{R^e}}H:{ℳ}_{R^e}\to {ℳ}_{R^e}$ is opmonoidal. The category of $R$-comodules is by definition the category of comodules over the underlying $R$-coring.

Via $A\otimes A^{\mathrm{op}}$-rings

All modules and morphisms will be over a fixed ground commutative ring $k$.

A left $A$-bialgebroid is an $A{\otimes }_k{A}^{\mathrm{op}}$-ring $(H,{\mu }_H,\eta )$, together with the $A$-bimodule map “comultiplication” $\Delta :H\to H{\otimes }_{A}H$, which is coassociative and counital with a counit $ϵ$, such that

(i) the $A$-bimodule structure used on $H$ is $a.h.a\prime :=s(a)t(a\prime )h$, where $s:=\eta (-\otimes 1_A):A\to H$ and $t:=\eta (1_A\otimes -):A^{\mathrm{op}}\to H$ are the algebra maps induced by the unit $\eta$ of the $A\otimes A^{\mathrm{op}}$-ring $H$

(ii) the coproduct $\Delta :H\to H{\otimes }_{A}H$ corestricts to the Takeuchi product and the corestriction $\Delta :H\to H{\times }_{A}H$ is a $k$-algebra map, where the Takeuchi product $H{\times }_{A}H$ has a multiplication induced factorwise

(iii) $ϵ$ is a left character on the $A$-ring $(H,{\mu }_H,s)$

Notice that $H{\otimes }_{A}H$ is in general not an algebra, just an $A$-bimodule.

The definition of a right $A$-bialgebroid differs by the $A$-bimodule structure on $H$ given instead by $a.h.a\prime :=hs(a\prime )t(a)$ and the counit $ϵ$ is a right character on the $A$-coring $(H,{\mu }_H,t)$ ($t$ and $s$ can be interchanged in the last requirement).

Literature

Related notions: Hopf algebroid

Commutative case

The commutative case is rather classical. See for example the appendix to

• Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121. Academic Press Inc., Orlando, FL, 1986.

Noncommutative case

The first version of a bialgebroid over a noncommutative base was more narrow:

• M. Sweedler, Groups of simple algebras, Publ. IHES 44:79–189, 1974, numdam

A modern generality, but in different early formalism, is due Takeuchi (who was motivated to generalize the results from the Sweedler’s paper), under the name of ${\times }_A$-bialgebra (as it involves the ${\times }_A$-product, nowdays called Takeuchi product):

• M. Takeuchi, Groups of algebras over $A\times \overline{A}$, J. Math. Soc. Japan 29, 459–492, 1977, MR0506407, euclid

Lu introduces the name bialgebroid for a structure which is equivalent to the Takeuchi’s ${\times }_A$-bialgebra (though differently axiomatized there):

• Jiang-Hua Lu, Hopf algebroids and quantum groupoids, Int. J. Math. 7, 1 (1996) pp. 47-70, q-alg/9505024, MR95e:16037, doi

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, ${\times }_R$-bialgebras and duality, J. Algebra 251: 279-294, 2002, math.QA/0012164
• J. Donin, A. Mudrov, Quantum groupoids and dynamical categories,
  J. Algebra 296 (2006), no. 2, 348–384, math.QA/0311316, MR2007b:17022, doi; MPIM-2004-21, dvi with hyperlinks, ps

There is also a notion of quasibialgebroid, where the coassociativity is weakened by a bialgebroid 3-cocycle. See also Hopf algebroid.