nLab
Hopf algebroid

Idea

A Hopf algebroid is a possibly noncommutative generalization of a structure which is dual to a groupoid in the sense of space-algebra duality. In other words, function algebras on the space of objects and space of morphisms of a groupoid pack together in a structure of Hopf algebroid.

Commutative Hopf algebroids

Given an internal groupoid in the category ${\mathrm{Aff}}_k$ of affine algebraic $k$-schemes, where $k$ is a field, the $k$-algebras of global sections over the scheme of objects and scheme of morphisms have an additional structure of a commutative Hopf algebroid. In fact this is an antiequivalence of categories. Commutative Hopf algebroids are useful also in a version in brave new algebra? (the work of John Rognes).

Noncommutative Hopf algebroids

There are several generalization 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 due Takeuchi. To add an antipode is nontrivial. A definition of Lu from mid 1990s is rather nonselfdual unlike the case of Hopf algebras. So a better solution is to abandon the idea of an antipode and have some replacement for it. There are two approaches, one via monoidal bicategories due to Day and Street, and another due Gabi Böhm, using pairs of a left and right bialgebroid. Gabi has later shown that the two definitions are in fact equivalent.

Literature

The commutative version is classical, and there is an extensive literature. Some of the recent works on commutative case, related to homotopy theory and stacks are

• M. Hovey, Homotopy theory of comodules over a Hopf algebroid, pdf; Morita theory of Hopf algebroids, pdf
• B. Walker, Hopf algebroids and stacks, pdf

The version in brave new algebra?

• A. Baker, A. Jeanneret, Brave new Hopf algebroids and extensions of $MU$-algebras, Homology Homotopy Appl. 4:1 (2002), 163-173, MR1937961, euclid

One can straightforwardly keep the base commutative while having the total algebra noncommutative, and the image of source and target maps are required to commute mutually. This version is due Maltsiniotis; he also generalized this to quasi-Hopf version:

• Georges Maltsiniotis, Groupoïdes quantiques, Comptes R Rendus Acad. Sci. Paris 314, pp. 249-252 (1992) ps
• G. Maltsiniotis, Quasi-groupoïdes quantiques (travail en commun avec A. Bruguières), C.R. Acad. Sci. Paris 319, pp. 933-936 (1994) ps

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
• 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 is discovered in two formally different formalisms, but the two concepts are equivalent:

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

• 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

• G. Böhm, Hopf algebroids, (a chapter of) Handbook of algebra, arxiv:math.RA/0805.3806