Hopf algebroid

A Hopf algebroid over a commutative ring ${\displaystyle k}$ is a pair of ${\displaystyle k}$-algebras ${\displaystyle (A,\Gamma )}$ in ${\displaystyle {\text{CRing}}_{k}}$ such that their functor of points

${\displaystyle {\text{Hom}}_{k}(\Gamma ,-)\rightrightarrows {\text{Hom}}_{k}(A,-)}$

encodes a groupoid in ${\displaystyle {\text{Aff}}}$. If we fix ${\displaystyle B}$ as some object in ${\displaystyle {\text{CRing}}_{k}}$, then ${\displaystyle {\text{Hom}}_{k}(A,B)}$ is the set of objects in the groupoid and ${\displaystyle {\text{Hom}}_{k}(\Gamma ,B)}$ is the set of arrows. This translates to having maps

${\displaystyle {\begin{aligned}\eta _{L}&:A\to \Gamma &{\text{ left unit/ source map}}\\\eta _{R}&:A\to \Gamma &{\text{ right unit/ target map}}\\\Delta &:\Gamma \to \Gamma \otimes _{A}\Gamma &{\text{ coproduct/ composition map}}\\\varepsilon &:\Gamma \to A&{\text{ counit/ identity map }}\\c&:\Gamma \to \Gamma &{\text{ conjugation/ inverse map}}\end{aligned}}}$

where 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

${\displaystyle {\text{Hom}}_{k}(\Gamma ,-)\rightrightarrows {\text{Hom}}_{k}(A,-)}$

these maps correspond to, meaning their dual maps from the Yoneda embedding gives the structure of a groupoid. For example,

${\displaystyle \eta _{L}^{*}:{\text{Hom}}_{k}(\Gamma ,-)\rightarrow {\text{Hom}}_{k}(A,-)}$

corresponds to the source map ${\displaystyle s}$.

In addition to these maps, they satisfy a host of axioms dual to the axioms of a groupoid. Note we will fix ${\displaystyle B}$ as some object in ${\displaystyle {\text{CRing}}_{k}}$ giving

1. ${\displaystyle \varepsilon \eta _{L}=\varepsilon \eta _{R}={\text{Id}}_{A}}$, meaning the dual counit map ${\displaystyle \varepsilon ^{*}}$ acts as a two-sided identity for the objects in ${\displaystyle {\text{Hom}}_{k}(A,B)}$
2. ${\displaystyle ({\text{Id}}_{\Gamma }\otimes \varepsilon )\Delta =(\varepsilon \otimes {\text{Id}}_{\Gamma })\Delta ={\text{Id}}_{\Gamma }}$, meaning composing an arrow with the identity leaves that arrow unchanged
3. ${\displaystyle ({\text{Id}}_{\Gamma }\otimes \Delta )\Delta =(\Delta \otimes {\text{Id}}_{\Gamma })\Delta ={\text{Id}}_{\Gamma }}$ corresponds to the associativity of composition of morphisms
4. ${\displaystyle c\eta _{R}=\eta _{L}}$ and ${\displaystyle c\eta _{L}=\eta _{R}}$, translates to inverting a morphism interchanges the source and target
5. ${\displaystyle cc={\text{Id}}_{\Gamma }}$, meaning the inverse of the inverse is the original map
6. These exists maps ${\displaystyle \Gamma \otimes _{A}\Gamma \rightrightarrows \Gamma }$ 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 ${\displaystyle c\cdot \Gamma }$ is the map ${\displaystyle c\cdot \Gamma (\gamma _{1}\otimes \gamma _{2})=c(\gamma _{1})\gamma _{2}}$ and ${\displaystyle \Gamma \cdot c(\gamma _{1}\otimes \gamma _{2})=\gamma _{1}c(\gamma _{2})}$.

In addition to the standard definition of a Hopf-algebroid, there are also graded commutative Hopf-algebroids which are pairs of graded commutative algebras ${\displaystyle (A,\Gamma )}$ with graded commutative structure maps given above.

Also, a graded Hopf algebroid ${\displaystyle (A,\Gamma )}$ is said to be connected if the right and left sub ${\displaystyle A}$-modules ${\displaystyle \Gamma _{0}\hookrightarrow \Gamma }$ are both isomorphic to ${\displaystyle A}$

A left Hopf algebroid (H, R) is a left bialgebroid together with an antipode: the bialgebroid (H, R) consists of a total algebra H and a base algebra R and two mappings, an algebra homomorphism s: R → H called a source map, an algebra anti-homomorphism t: R → H called a target map, such that the commutativity condition s(r₁) t(r₂) = t(r₂) s(r₁) is satisfied for all r₁, r₂ ∈ R. The axioms resemble those of a Hopf algebra but are complicated by the possibility that R is a non-commutative algebra or its images under s and t are not in the center of H. In particular a left bialgebroid (H, R) has an R-R-bimodule structure on H which prefers the left side as follows: r₁ ⋅ h ⋅ r₂ = s(r₁) t(r₂) h for all h in H, r₁, r₂ ∈ R. There is a coproduct Δ: H → H ⊗_R H and counit ε: H → R that make (H, R, Δ, ε) an R-coring (with axioms like that of a coalgebra such that all mappings are R-R-bimodule homomorphisms and all tensors over R). Additionally the bialgebroid (H, R) must satisfy Δ(ab) = Δ(a)Δ(b) for all a, b in H, and a condition to make sure this last condition makes sense: every image point Δ(a) satisfies a₍₁₎ t(r) ⊗ a₍₂₎ = a₍₁₎ ⊗ a₍₂₎ s(r) for all r in R. Also Δ(1) = 1 ⊗ 1. The counit is required to satisfy ε(1_H) = 1_R and the condition ε(ab) = ε(as(ε(b))) = ε(at(ε(b))).

The antipode S: H → H is usually taken to be an algebra anti-automorphism satisfying conditions of exchanging the source and target maps and satisfying two axioms like Hopf algebra antipode axioms; see the references in Lu or in Böhm-Szlachányi for a more example-category friendly, though somewhat more complicated, set of axioms for the antipode S. The latter set of axioms depend on the axioms of a right bialgebroid as well, which are a straightforward switching of left to right, s with t, of the axioms for a left bialgebroid given above.

One of the main motivating examples of a Hopf algebroid is the pair ${\displaystyle (\pi _{*}(E),E_{*}(E))}$ for a spectrum ${\displaystyle E}$.[3] For example, the Hopf algebroids ${\displaystyle (\pi _{*}({\text{MU}}),{\text{MU}}_{*}({\text{MU}}))}$, ${\displaystyle (\pi _{*}({\text{BP}}),{\text{BP}}_{*}({\text{BP}}))}$, for the spectra representing complex cobordism and Brown-Peterson homology, and truncations of them are widely studied in algebraic topology. This is because of their use in the Adams-Novikov spectral sequence for computing the stable homotopy groups of spheres.

There is a Hopf-algebroid which corepresents the stack of formal group laws ${\displaystyle {\mathcal {M}}_{FG}}$ which is constructed using algebraic topology.[4] If we let ${\displaystyle {\text{MP}}}$ denote the spectrum

${\displaystyle {\text{MP}}=\bigvee _{m\in \mathbb {Z} }\Sigma ^{2m}{\text{MU}}}$

there is a Hopf algebroid

${\displaystyle {\text{MP}}_{0}\rightrightarrows {\text{MP}}_{0}({\text{MP}})}$

corepresenting the stack ${\displaystyle {\mathcal {M}}_{FG}}$. This means, there is an isomorphism of functors

${\displaystyle {\mathcal {M}}_{FG}(-)\cong [{\text{MP}}_{0}\rightrightarrows {\text{MP}}_{0}({\text{MP}})](-)}$

where the functor on the right sends a commutative ring ${\displaystyle B}$ to the groupoid

${\displaystyle {\text{MP}}_{0}(B)\rightrightarrows {\text{MP}}_{0}({\text{MP}})(B)}$

As an example of left bialgebroid, take R to be any algebra over a field k. Let H be its algebra of linear self-mappings. Let s(r) be left multiplication by r on R; let t(r) be right multiplication by r on R. H is a left bialgebroid over R, which may be seen as follows. From the fact that H ⊗_R H ≅ Hom_k(R ⊗ R, R) one may define a coproduct by Δ(f)(r ⊗ u) = f(ru) for each linear transformation f from R to itself and all r, u in R. Coassociativity of the coproduct follows from associativity of the product on R. A counit is given by ε(f) = f(1). The counit axioms of a coring follow from the identity element condition on multiplication in R. The reader will be amused, or at least edified, to check that (H, R) is a left bialgebroid. In case R is an Azumaya algebra, in which case H is isomorphic to R ⊗ R, an antipode comes from transposing tensors, which makes H a Hopf algebroid over R. Another class of examples comes from letting R be the ground field; in this case, the Hopf algebroid (H, R) is a Hopf algebra.