nLab Hopf algebroid

Idea

A Hopf algebroid is an associative bialgebroid with an antipode.

A Hopf algebroid is a (possibly noncommutative) generalization of a structure which is dual to a groupoid (equipped with atlas) in the sense of space-algebra duality. This is the concept that generalizes Hopf algebras with their relation to groups from groups to groupoids.

Specifically commutative Hopf algebroids are the internal groupoids in the opposite category of CRing. These arise notably in stable homotopy theory as generalized dual Steenrod algebras for generalized cohomology.

More generally there are Hopf algebroids over a commutative base, examples of which are convolution algebras of Lie groupoids.

Definition (under construction)

In the general case we should distinguish left and right bialgebroids, see bialgebroid.

In one of the versions, a general Hopf algebroid is defined as a pair of a left algebroid and right algebroid together with a linear map from left to right bialgebroid taking the role of an antipode.

Commutative Hopf algebroids

Given an internal groupoid in the category $Aff_k = Alg_k^{op}$ of affine algebraic $k$-schemes, where $k$ is a field, the $k$-algebras of global sections over the scheme of objects and the scheme of morphisms have an additional structure of a commutative Hopf algebroid. In fact this is an antiequivalence of categories.

These commutative Hopf algebroids play a key role in stable homotopy theory/brave new algebra, as they arise as the dual Steenrod algebras for certain classes of generalized cohomology theories $E$ and as such govern the $E$-Adams spectral sequence.

Noncommutative Hopf algebroids

There are several generalizations to the noncommutative case. A difficult part is to work over the noncommutative base (i.e., the object of objects is noncommutative). The definition of a bialgebroid is not that difficult and there is even a very old definition of an equivalent structure due Takeuchi. To add an antipode is nontrivial. A definition of Lu from mid 1990s is rather nonselfdual, unlike the case of Hopf algebras and introduces rather ad hoc certain section map. So a better solution may be even to abandon the idea of an antipode and have some replacement for it. There are two approaches, one due to Day and Street, and another due Gabi Böhm, using pairs of a left and right bialgebroid. Gabi later showed that the two definitions are in fact equivalent.

Noncommutative Hopf algebroid with invertible antipode

A definition of an antipode avoiding a section map of Lu, but requiring that the antipode is invertible. In this definition, given a left $A$-bialgebroid $(H,\alpha,\beta,\Delta,\epsilon)$, an invertible antipode $S:H\to H$ is an antihomomorphism of algebras with inverse map $S^{-1}:H\to H$ satisfying

$$S\circ\beta = \alpha$$

and for every $h\in H$,

$$(S^{-1} h_{(2)})_{(1)}\otimes_A(S^{-1} h_{(2)})_{(2)}h_{(1)} = S^{-1} h\otimes_A 1_H,$$

$$(S h_{(1)})_{(1)} h_{(2)}\otimes_A(S h_{(1)})_{(2)} = 1_H\otimes_A S h.$$

Some authors call this concept a “full Hopf algebroid”.

Examples

Minimal Hopf algebroid

Let $A$ be a unital associative algebra. Then $A\otimes A^{op}$ has a structure of a Hopf algebroid, a minimal Hopf algebroid over $A$, with source map $\alpha(a) = a\otimes 1$, target map $\beta(b) = 1\otimes b$, comultiplication $\Delta(a\otimes b) = (a\otimes 1)\otimes_A(1\otimes b)$, counit $\epsilon(a\otimes b) = a b$ and antipode $\tau(a\otimes b) = b\otimes a$.

Lu (1996) considers this example an analogue of a Poisson groupoid? structure on $P\times\overline{P}$ where $P$ is a Poisson manifold, which is itself considered an analogue of a set theoretic course groupoid on $X\times X$ where $X$ is a set. Thus she calls this example a coarse Hopf algebroid.

Scalar extension Hopf algebroids

Given a Hopf algebra $B$ and a braided-commutative algebra $A$ in the category of Yetter-Drinfeld modules over $B$, by a result of Brzezinski-Militaru, the smash product algebra $B\sharp A$ is the total algebra of a Hopf algebroid over $A$. This is a noncommutative generalization (of formal dual of) an action groupoid.

This construction is modelled on an earlier variant, first written out by Lu, where $A$ is a braided-commutative algebra in the category of modules over $D(H)$, the Drinfeld double of $H$.

References

The commutative version is classical, and there is an extensive literature, see Hopf algebroids over a commutative base.

Over a noncommutative base ring, there is a nonsymmetric version due J-H. Lu and a similar version is later studied by Ping Xu

• Jiang-Hua Lu, Hopf algebroids and quantum groupoids, Int. J. Math. 7, 1 (1996) pp. 47-70, q-alg/9505024, MR95e:16037, doi; On the Drinfeld double and the Heisenberg double of a Hopf algebra, Duke Math. J. 7:3 (1994) 763-776, MR1277953, doi
• Ping Xu, Quantum groupoids, Commun. Math. Phys., 216:539581, 2001, q-alg/9905192, doi; Quantum groupoids and deformation quantization, q-alg/9708020, Quantum groupoids associated to universal dynamical R-matrice, q-alg/9811172
• blog discussion at Theoretical Atlas

The modern concept over the noncommutative base has been discovered by several different people in several different formalisms. Some of the differences are merely cosmetic, but there are at least two main concepts, depending on the underlying concept of ‘bialgebroid’.

Day and Street have a concept of Hopf algebroid here:

• B. Day, R. Street, Monoidal bicategories and Hopf algebroids, Advances in Mathematics 129, 1 (1997) 99–157

In this they start by taking an algebroid to be an “algebra with several objects” in the sense of a $k$-linear category $A$: that is, a $V$-enriched category with $V = Vect_k$. The 2-category $V Cat$ of $k$-linear categories, functors and natural transformations is monoidal (where the tensor product of $V$-categories is defined by cartesian product on object sets and tensor product on hom-spaces). So, they define a bialgebroid to be a comonoid in $V Cat$. Because the tensor product is cartesian product on object sets, the comultiplication in such a bialgebroid is forced to be the diagonal on objects. Thus, their notion of bialgebroid amounts to a $k$-linear category $A$ equipped with linear maps

$$A(a,b) \to A(a,b) \otimes A(a,b)$$

satisfying coassociativity, a version of the usual bialgebra axiom, and so on. On page 142 of the above reference they define an antipode on a bialgebroid $A$ to be a $k$-linear functor $S: A \to A^{op}$ together with a natural isomorphism

$$A(b,c) \otimes A(a,S b) \cong A(b,c) \otimes A(a,c)$$

A Hopf algebroid is then roughly a bialgebroid with an antipode. With this definition, a Hopf algebra gives a one-object Hopf algebroid.

A different and more widely used concept was developed independently in these two papers, which appeared on the arXiv within a couple of days of each other:

• G. Böhm, An alternative notion of Hopf algebroid; in “Hopf algebras in noncommutative geometry and physics”, 31–53, Lecture Notes in Pure and Appl. Math. 239, Dekker, New York, 2005; math.QA/0301169

• R. Street, B. Day, Quantum categories, star autonomy, and quantum groupoids, in “Galois Theory, Hopf Algebras, and Semiabelian Categories”, Fields Institute Communications 43 (American Math. Soc. 2004) 187-226; arXiv:math/0301209

and also described in:

• G. Böhm, Hopf algebroids, (a chapter of) Handbook of algebra, Vol. 6, ed. by M. Hazewinkel, Elsevier 2009, 173–236 arxiv:math.RA/0805.3806
• G. Böhm, K. Szlachányi, Hopf algebroids with bijective antipodes: axioms, integrals and duals, Comm. Algebra 32 (11) (2004) 4433 - 4464 math.QA/0305136
• T. Brzeziński, G. Militaru, Bialgebroids, $\times_A$-bialgebras and duality, J. Algebra 251: 279-294, 2002 math.QA/0012164
• D. Chikhladze, Category of quantum categories, Theory and Applications of Categories 25 (2011) 1 - 37. (pdf)

This starts with a different concept of bialgebroid, which is discussed here on the nLab. Namely: any $k$-algebra $R$ gives a pseudomonoid $R^e = R^{op} \otimes R$ in the bicategory $Mod_k$ of k-algebras, bimodules, and bimodule homomorphisms, and a bialgebroid is then an opmonoidal monad $A$ on $R^e$. When the fusion (or Galois) operator for this opmonoidal monad is invertible, we say that $A$ is a Hopf algebroid. In G. Böhm’s work this definition is stated in a less compressed, more down-to-earth way.

A class of examples of Hopf algebroids have as underyling algebra the smash product of a Hopf algebra with a Yetter-Drinfeld module algebra over it. These are the scalar extension Hopf algebroids introduced in Brzeziński-Militaru article above modifying slightly a construction of Lu who considered modules over a Drinfeld double instead of Yetter-Drinfeld modules (what is equivalent for finite dimensional Hopf algebras). For a symmetric version see

• M. Stojić, Scalar extension Hopf algebroids, Journal of algebra and applications 2023 doi arXiv:2208.11696

An internal version of associative bialgebroids over a noncommutative base monoid in a symmetric monoidal category admiting coqualizers commuting with the monoidal product is studied in

• G. Böhm, Internal bialgebroids, entwining structures and corings, AMS Contemp. Math. 376 (2005) 207-226 arXiv:math/0311244 doi

The concept of internal bialgebroids over noncommutative base has been refined to an internal version of symmetric Hopf algebroids in

• Martina Stojić, Internal Hopf algebroid, arXiv:2308.14546

This is actually an exposition of a part of the author’s earlier thesis where also a nontrivial class of examples of Hopf algebroids internal to a symmetric monoidal category of filtered cofiltered vector spaces is described:

• M. Stojić, PhD thesis, Completed Hopf algebroids, Ph.D. thesis, University of Zagreb, 2017

A somewhat less canonical version of the same main subexample, written in coordinates, and with somewhat ad hoc treatment of completions (focusing on global cofiltrations only) is in

• S. Meljanac, Z. Škoda, M. Stojić, Lie algebra type noncommutative phase spaces are Hopf algebroids, Lett. Math. Phys. 107:3, 475–503 (2017) enhanced pdf (free for online use) doi arxiv/1409.8188

A definition of a variant of Hopf algebroid which is somewhat similar to Lu’s definition but involves working with a 2-sided ideal, with help of a distinguished „balancing” subalgebra, is in

• Zoran Škoda, Martina Stojić, Hopf algebroids with balancing subalgebra, Journal of Algebra 598 (2022) 445-469 arxiv:1610.03837 doi

A notion of multiplier Hopf algebroid is studied in

• T. Timmermann, A. Van Daele, Multiplier Hopf algebroids. Basic theory and examples, Commun. Alg. 46:5 (2018) arxiv/1307.0769 doi; Multiplier Hopf algebroids arising from weak multiplier Hopf algebras, arxiv/1406.3509
• Frank Taipe, Quantum transformation groupoids: An algebraic and analytical approach, PhD thesis (2018) link

Further examples and counterexamples

• Ana Rovi, Hopf algebroids associated to Jacobi algebras, Int. J. Geom. Methods in Modern Phys. 11: 10 (2014) 1450092 doi
• Ulrich Krähmer, Ana Rovi, A Lie-Rinehart algebra with no antipode, Commun. Alg. 43:10 (2015) doi

The aim of this note is to communicate a simple example of a Lie-Rinehart algebra whose enveloping algebra is not a Hopf algebroid, neither in the sense of Böhm and Szlachányi, nor in the sense of Lu.

On Hopf-Galois extensions of Hopf algebroids

• Xiao Han, Peter Schauenburg. Hopf Galois extensions of Hopf algebroids (2024). (arXiv:2406.11058).

category: algebra