nLab
Hopf algebroid

Contents

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.

More details on the case of the Hopf algebroids associated to groupoids, i.e. the convolution Hopf algebroid and groupoid Hopf algebroid, as well as stable homotopy examples like Steenrod algebra, see entry Hopf algebroid over a commutative base.

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 kAff_k of affine algebraic kk-schemes, where kk is a field, the kk-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. Commutative Hopf algebroids are useful also in a version in brave new algebra (the work of John Rognes).

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 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 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.

Examples

Scalar extension Hopf algebroids

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

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

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 kk-linear category AA: that is, a VV-enriched category with V=Vect kV = Vect_k. The 2-category VCatV Cat of kk-linear categories, functors and natural transformations is monoidal (where the tensor product of VV-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 VCatV 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 kk-linear category AA equipped with linear maps

A(a,b)A(a,b)A(a,b) 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 AA to be a kk-linear functor S:AA opS: A \to A^{op} together with a natural isomorphism

A(b,c)A(a,Sb)A(b,c)A(a,c) 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 and 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:0301209

and also described in:

  • G. Böhm, Hopf algebroids, (a chapter of) Handbook of algebra, 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, × A\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)

A class of examples of such Hopf algebroids internally in a monoidal category of cocomplete cofiltered vector spaces is in

  • S. Meljanac, Z. Škoda, Lie algebra type noncommutative phase spaces are Hopf algebroids, arxiv/1409.8188

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

category: algebra

Revised on September 30, 2014 13:37:13 by Zoran Škoda (161.53.130.104)