In mathematics, in the theory of Hopf algebras, a Hopf algebroid is a generalisation of weak Hopf algebras, certain skew Hopf algebras and commutative Hopf k-algebroids. If k is a field, a commutative k-algebroid is a cogroupoid object in the category of k-algebras; the category of such is hence dual to the category of groupoid k-schemes. This commutative version has been used in 1970-s in algebraic geometry and stable homotopy theory. The generalization of Hopf algebroids and its main part of the structure, associative bialgebroids, to the noncommutative base algebra was introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry (later shown equivalent in nontrivial way to a construction of Takeuchi from the 1970s and another by Xu around the year 2000). They may be loosely thought of as Hopf algebras over a noncommutative base ring, where weak Hopf algebras become Hopf algebras over a separable algebra. It is a theorem that a Hopf algebroid satisfying a finite projectivity condition over a separable algebra is a weak Hopf algebra, and conversely a weak Hopf algebra H is a Hopf algebroid over its separable subalgebra HL. The antipode axioms have been changed by G. Böhm and K. Szlachányi (J. Algebra) in 2004 for tensor categorical reasons and to accommodate examples associated to depth two Frobenius algebra extensions.
Definition
The main motivation behind of the definition of a Hopf algebroidpg301-302 is its a commutative algebraic representation of an algebraic stack which can be presented as affine schemes. More generally, Hopf algebroids encode the data of presheaves of groupoids on the category of affine schemes. That is, if we have a groupoid object of affine schemeswith an identity map giving an embedding of objects into the arrows, we can take as our definition of a Hopf algebroid as the dual objects in commutative rings which encodes this structure. Note that this process is essentially an application of the Yoneda lemma to the definition of the groupoid schemes in the category of affine schemes. Since we may want to fix a base ring, we will instead consider the category of commutative -algebras.
Scheme-theoretic definition
Algebraic objects in the definition
A Hopf algebroid over a commutative ring is a pair of -algebras in such that their functor of points encodes a groupoid in . If we fix as some object in , then is the set of objects in the groupoid and is the set of arrows. This translates to having mapswhere the text on the left hand side of the slash is the traditional word used for the map of algebras giving the Hopf algebroid structure and the text on the right hand side of the slash is what corresponding structure on the groupoid these maps correspond to, meaning their dual maps from the Yoneda embedding gives the structure of a groupoid. For example,corresponds to the source map .
Axioms these maps must satisfy
In addition to these maps, they satisfy a host of axioms dual to the axioms of a groupoid. Note we will fix as some object in giving
, meaning the dual counit map acts as a two-sided identity for the objects in
, meaning composing an arrow with the identity leaves that arrow unchanged
corresponds to the associativity of composition of morphisms
and , translates to inverting a morphism interchanges the source and target
, meaning the inverse of the inverse is the original map
These exists maps encoding the composition of a morphism with its inverse on either side gives the identity morphism. This can be encoded by the commutative diagram below where the dashed arrows represent the existence of these two arrows
where is the map and .
Additional structures
In addition to the standard definition of a Hopf-algebroid, there are also graded commutative Hopf-algebroids which are pairs of graded commutative algebras with graded commutative structure maps given above.
Also, a graded Hopf algebroid is said to be connected if the right and left sub -modules are both isomorphic to