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 ${}_R \mathcal{M}_R$) equipped with a structure of a comonoid in ${}_R \mathcal{M}_R$ (i.e. an $R$-coring) and of a monoid in ${}_{R^e}\mathcal{M}_{R^e}$ (i.e. an $R^e$-ring), where $R^e = R^{op}\otimes R$ 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 ${}_{\otimes_{R^e}} H : \mathcal{M}_{R^e}\to \mathcal{M}_{R^e}$ is opmonoidal. The category of $R$-comodules is by definition the category of comodules over the underlying $R$-coring.

An explicit definition

If $A$ is an associative algebra over some ground field $k$, then a left associative $A$-bialgebroid is another associative $k$-algebra $H$ together with the following additional maps: an algebra map $\alpha:A\to H$ called the source map, an algebra map $\beta:A^{op}\to H$ called the target map, so that the elements of the images of $\alpha$ and $\beta$ commute in $H$, therefore inducing an $A$-bimodule structure on $H$ via the rule $a.h.b = \alpha(a)\beta(b) h$ for $a,b\in A, h\in H$; an $A$-bimodule morphism $\Delta:H\to H\otimes_A H$ which is required to be a counital coassociative comultiplication on $H$ in the monoidal category of $A$-bimodules with monoidal product $\otimes_A$. The map $H\otimes A\ni h\otimes a\mapsto \epsilon(h\alpha(a))\in$ must be a left action extending the multiplication $A\otimes A\to A$ along $\alpha\otimes id_A$. Furthermore, a compatibility between the comultiplication $\Delta$ and multiplications on $H$ and on $H\otimes H$ is required. For a noncommutative $L$ the tensor square $H\otimes_A$ is not an algebra, hence asking for a bialgebra-like compatibility that $\Delta:H\to H\otimes_A H$ is a morphism of $k$-algebras does not make sense. Instead, one requires that $H\otimes_A H$ has a $k$-subspace $T$ which contains the image of $\Delta$ and has a well-defined multiplication induced from its preimage under the projection from the usual tensor square algebra $H\otimes H$. Then one requires that the corestriction $\Delta|^T :H\to T$ is a homomorphism of unital algebras. Under these conditions, one can make a canonical choice for $T$, namely the so called Takeuchi’s product $H\times_A H\subset H\otimes_A H$, which always inherits an associative multiplication along the projection from $H\otimes H$.

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

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

A left $A$-bialgebroid is an $A\otimes_k A^{op}$-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 $\epsilon$, such that

(i) the $A$-bimodule structure used on $H$ is $a.h.a':= s(a)t(a')h$, where $s := \eta(-\otimes 1_A):A\to H$ and $t:=\eta(1_A\otimes -):A^{op}\to H$ are the algebra maps induced by the unit $\eta$ of the $A\otimes A^{op}$-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) $\epsilon$ 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. That is why (ii) is needed. An equivalent condition to (ii) is the following: the formula $h.(\sum_i k_i \otimes l_i) = \sum_i h_{(1)}\cdot k_i \otimes h_{(2)} \cdot l_i$ defines a well-defined action of $H$ on $H\otimes_A H$.

The definition of a right $A$-bialgebroid differs by the $A$-bimodule structure on $H$ given instead by $a.h.a':= h s(a')t(a)$ and the counit $\epsilon$ 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 to 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 $A \times \bar{A}$, 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.