nLab bialgebroid


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.

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


Via monoidal categories

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

An explicit definition

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

Via AA opA\otimes A^{op}-rings

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

A left AA-bialgebroid is an A kA opA\otimes_k A^{op}-ring (H,μ H,η)(H,\mu_H,\eta), together with the AA-bimodule map “comultiplication” Δ:HH AH\Delta : H\to H\otimes_A H, which is coassociative and counital with a counit ϵ\epsilon, such that

(i) the AA-bimodule structure used on HH is a.h.a:=s(a)t(a)ha.h.a':= s(a)t(a')h, where s:=η(1 A):AHs := \eta(-\otimes 1_A):A\to H and t:=η(1 A):A opHt:=\eta(1_A\otimes -):A^{op}\to H are the algebra maps induced by the unit η\eta of the AA opA\otimes A^{op}-ring HH

(ii) the coproduct Δ:HH AH\Delta : H\to H\otimes_A H corestricts to the Takeuchi product and the corestriction Δ:HH× AH\Delta : H\to H\times_A H is a kk-algebra map, where the Takeuchi product H× AHH\times_A H has a multiplication induced factorwise

(iii) ϵ\epsilon is a left character on the AA-ring (H,μ H,s)(H,\mu_H,s).

Notice that H AHH\otimes_A H is in general not an algebra, just an AA-bimodule. That is why (ii) is needed. An equivalent condition to (ii) is the following: the formula h.( ik il i)= ih (1)k ih (2)l ih.(\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 HH on H AHH\otimes_A H.

The definition of a right AA-bialgebroid differs by the AA-bimodule structure on HH given instead by a.h.a:=hs(a)t(a)a.h.a':= h s(a')t(a) and the counit ϵ\epsilon is a right character on the AA-coring (H,μ H,t)(H,\mu_H,t) (tt and ss can be interchanged in the last requirement).


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 × A\times_A-bialgebra (as it involves the × A\times_A-product, nowdays called Takeuchi product):

  • M. Takeuchi, Groups of algebras over A×A¯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 × A\times_A-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, × R\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.

Last revised on October 31, 2021 at 14:50:01. See the history of this page for a list of all contributions to it.