hook length formula

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Combinatorics
+-- {: .hide}
[[!include combinatorics - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[partition]]/[[Young diagram]] $\lambda$ with $n$ boxes/of $n$:

the *hook length formula* expresses both

* the [[number]] of [[standard Young tableaux]] of shape $\lambda$,

* the [[dimension]] of the [[irreducible representation]] of [[symmetric group|Sym(n)]] labelled by $\lambda$ (i.e. of the [[Specht module]] $S^{(\lambda)}$)

in terms of the lengths of all "hooks" inside the Young diagram;

similarly, the *[[hook-content formula]]* expresses both

* the [[number]] of [[semistandard Young tableau]] of shape $\lambda$;

* the [[dimension]] of the [[irreducible representation]] of [[SU(n)]]

in terms of the length of all hooks and the "content" of all of the boxes.

[[!include hook length and content formulas -- table]]

## Preliminaries

Given a [[Young diagram]], the *hook* at any one of its boxes is the collection of boxes to the right and below that box, and including the box itself. We write "$\ell hook_\lambda$" for the *length* of such a hook, i.e. for the number of boxes it contains. Formally:

\begin{defn}\label{HookLength}
**(hook length)** \linebreak
Let $\lambda = (\lambda_1 \geq \cdots \geq \lambda_{rows(\lambda)})$ be a [[partition]]/[[Young diagram]]. Then for

* $i \in \{1, \cdots, rows(\lambda)\}$,

* $j \in \{1, \cdots, \lambda_i\}$

the *hook length* at $(i,j)$ is

$$
  \ell hook_\lambda(i,j)
  \;\coloneqq\;
  1 +
  (\lambda_i - j) + (\lambda'_j - i)
  \,,
$$

where $\lambda'$ denotes the *conjugate partition* (see [there](Young+diagram#Conjugation)).

\end{defn}

\begin{definition}
**(numbers of (semi-)standard Young tableaux)** \linebreak
Given a [[partition]] $\lambda \in Part(n)$, and a [[positive number|positive]] [[natural number]] $N \in \mathbb{N}_+$, consider

* the number of [[standard Young tableaux]]:

  \[
    \label{NumberOfStandardYoungTableauxOfFixedShape}
    \left\vert sYTableaux_\lambda\right\vert
    \;\in\;
    \mathbb{N}_+
  \]

* the number of [[standard Young tableaux]] with bounded entries $T_{i j} \leq N$:

  \[
    \label{NumberOfSemiStandardYoungTableauxOfFixedShape}
    \left\vert ssYTableaux_\lambda(N)\right\vert
    \;\in\;
    \mathbb{N}_+
  \]

of shape $\lambda$.
\end{definition}

## Details

### Counting standard Young tableaux

\begin{prop}
**(hook length formula for standard Young tableaux)** \linebreak
Given a [[partition]] ([[Young diagram]]) $\lambda$ of $n$ (boxes), the number (eq:NumberOfStandardYoungTableauxOfFixedShape) of [[standard Young tableaux]] of shape $\lambda$ equals the [[factorial]] of $n$ over the [[multiplication|product]] of the hook lengths (Def. \ref{HookLength}) at all the boxes of $\lambda$:

\[
  \label{HookLengthFormulaCountingNumberOfStandardYoungTableau}
  \left\vert
    sYTableaux_\lambda
  \right\vert
    \;=\;
  \frac{
    n!
  }{
    \prod_{(i,j)} \ell hook_\lambda(i,j)
  }.
\]

\end{prop}

This is due to [Frame, Robinson & Thrall 54](#FrameRobinsonThrall54). Textbook accounts include [Stanley 99, Cor. 7.21.6](#Stanley99), [Sagan 01 Thm. 3.10.2](#Sagan01).

### Measuring dimension of irreps of $Sym(n)$

The [[dimension]] of the [[irrep]] of the [[symmetric group]] $Sym(n)$ that is labelled by a given [[Young diagram]] $\lambda$ (the [[Specht module]] $S^{(\lambda)}$, see at [[representation theory of the symmetric group]]) equals the number of [[standard Young tableaux]] of shape $\lambda$

\[
  dim(S^{(\lambda)})
  \;=\;
  \left\vert
    sYTableaux_\lambda
  \right\vert
\]

(e.g. [Sagan, Thm. 2.6.5](#Sagan01))

and hence is also given by the hook length formula (eq:HookLengthFormulaCountingNumberOfStandardYoungTableau):

\[
  \label{HookLengthFormulaMeasuresDimensionOfIrrepOfSymmetricGroup}
  dim(S^{(\lambda)})
  \;=\;
  \frac{
    n!
  }{
    \prod_{(i,j)} \ell hook_\lambda(i,j)
  }
  \,.
\]

This is actually the statement of [Frame, Robinson, & Thrall 54, Thm. 1](#FrameRobinsonThrall54). Textbook accounts include [James 78, Thm. 20.1](#James78).

### Counting semi-standard Young tableaux
 {#ForSemiStandardYoungTableaux}

\begin{prop}
**(hook length formula for semi-standard Young tableaux)** \linebreak
Given

  * a [[partition]] ([[Young diagram]]) $\lambda$ of $n$ (boxes),

  * a [[positive number|positive]] [[natural number]] $N \in \mathbb{N}_+$,

the number (eq:NumberOfSemiStandardYoungTableauxOfFixedShape) of semi-standard Young tableaux of shape $\lambda$ and entries $\leq N$ ([hence](semistandard+Young+tableau#eq:RelationToSchurPolynomialsOfFinitelyManyVariables) the value of the [[Schur polynomial]] $s_\lambda(x_1, \cdots, x_N)$ at $x_i = 1$) is:

\[
  \label{TheHookLengthFormulaForSemistandardYoungTableaux}
  \left\vert
    ssYTableaux_\lambda(N)
  \right\vert
  \;=\;
  s_\lambda
  \big(
    \underset{
      \mathclap{
        N\; arguments
      }
    }{
    \underbrace{
      1, \cdots, 1
    }
    }
  \big)
    \;=\;
  \underset{(i,j)}{\prod}
  \frac{
    N - i + j
  }{
    \ell hook_\lambda(i,j)
  }
  \,.
\]

\end{prop}

Here

$$
  content(i,j) \;\coloneqq\; j - i
$$

is also called the *content* of the box $(i,j)$, whence (eq:TheHookLengthFormulaForSemistandardYoungTableaux) is also called a *[[hook-content formula]]* ("hook length and box content"):

$$
  \left\vert
    ssYTableaux_\lambda(N)
  \right\vert
  \;=\;
  s_\lambda
  \big(
    \underset{
      \mathclap{
        N\; arguments
      }
    }{
    \underbrace{
      1, \cdots, 1
    }
    }
  \big)
    \;=\;
  \underset{(i,j)}{\prod}
  \frac{
    N + content(i,j)
  }{
    \ell hook_\lambda(i,j)
  }
  \,.
$$

### Measuring of dimension of irreps of $GL(n, \mathbb{C})$

The [[dimension]] of the [[irrep]] $V^{(\lambda)}$ of the [[general linear group]] $GL(n, \mathbb{C})$ that is labelled by a given [[Young diagram]] $\lambda$ (see at [[representation theory of the general linear group]]), is also given by the hook-content formula (eq:TheHookLengthFormulaForSemistandardYoungTableaux):

$$
  dim\big(V^{(\lambda)}\big)
  \;=\;
  \underset{(i,j)}{\prod}
  \frac{
    N + content(i,j)
  }{
    \ell hook_\lambda(i,j)
  }
$$

This appears as [Sternberg 94 (C.27)](#Sternberg94)

## Related concepts

* [[hook-content formula]]

* [[Plancherel measure]]

* [[Schur-Weyl measure]]

## References

### For standard Young tableaux

The original proof is due to:

* {#FrameRobinsonThrall54} [[James Sutherland Frame]], [[Gilbert de Beauregard Robinson]], [[Robert McDowell Thrall]], *The Hook Graphs of the Symmetric Group*, Canadian Journal of Mathematics, Volume 6, 1954, pp. 316 - 324 ([doi:10.4153/CJM-1954-030-1](https://doi.org/10.4153/CJM-1954-030-1))

Textbook accounts:

* {#James78} [[G. D. James]], Thm. 20.1 in: *The Representation Theory of the Symmetric Groups*,  Lecture Notes in Mathematics, volume 682, Springer 1978 ([doi:10.1007/BFb0067708](https://link.springer.com/book/10.1007/BFb0067708), [pdf](http://www-users.math.umn.edu/~webb/oldteaching/Year2010-11/the-representation-theory-of-the-symmetric-groups-SLN.pdf))

* {#Stanley99} [[Richard Stanley]], Cor. 7.21.6 in: *Enumerative combinatorics 2*, Cambridge University Press (1999, 2010) ([doi:10.1017/CBO9780511609589](https://doi.org/10.1017/CBO9780511609589), [webpage](http://www-math.mit.edu/~rstan/ec/))

* {#Sagan01} [[Bruce Sagan]], Thm. 3.10.2 & Thm. 2.6.5 in: _The symmetric group_, Springer 2001 ([doi:10.1007/978-1-4757-6804-6](https://link.springer.com/book/10.1007/978-1-4757-6804-6), [pdf](http://math.sfsu.edu/federico/Clase/RepTh/sagan.pdf))

Further review:

* Alex Ghorbani, Section 4.4 of: *Applications of representation theory to combinatorics* ([pdf](http://math.uchicago.edu/~may/REU2020/REUPapers/Ghorbani.pdf))

* [[Yufei Zhao]], Section 4.4. of _Young Tableaux and the Representations of the Symmetric Group_ ([pdf](https://yufeizhao.com/research/youngtab-hcmr.pdf), [[ZhaoYoungTableaux.pdf:file]])

* Shiyue and Andrew, *Young Tableaux and Probability*, 2019 ([pdf](http://www.mit.edu/~lindrew/hooklength.pdf))

See also:

* Wikipedia, *[Hook length formula](https://en.wikipedia.org/wiki/Hook_length_formula)*

Alternative proofs:

* Jean-Christophe Novelli, Igor Pak, Alexander V. Stoyanovskii, *A direct bijective proof of the hook-lengthformula*, Discrete Mathematics and Theoretical Computer Science1, 1997, 53–67 ([pdf](https://www.math.ucla.edu/~pak/papers/bij.pdf))

* Kenneth Glass, Chi-Keung Ng, *A Simple Proof of the Hook Length Formula*, The American Mathematical Monthly Vol. 111, No. 8 (Oct., 2004), pp. 700-704 ([jstor:4145043](https://www.jstor.org/stable/4145043))

* Jason Bandlow, *An elementary proof of the hook formula*, The Electronic Journal of Combinatorics **15** (2008) ([pdf](http://www.kurims.kyoto-u.ac.jp/EMIS/journals/EJC/Volume_15/PDF/v15i1r45.pdf))

* [[Bruce Sagan]], *Probabilistic proofs of the hook length formulas involving trees*, Séminaire Lotharingien de Combinatoire 61A (2009) ([pdf](https://users.math.msu.edu/users/bsagan/Papers/Old/pph.pdf))

Generalizations:

* Ionuţ Ciocan, Fontanine Matjaž, Konvalinka, Igor Pak, *The weighted hook length formula*, Journal of Combinatorial Theory, Series A Volume 118, Issue 6, August 2011, Pages 1703-1717 ([doi:10.1016/j.jcta.2011.02.004](https://doi.org/10.1016/j.jcta.2011.02.004))

* Alejandro Morales, Igor Pak, Greta Panova, *Hook formulas for skew shapes I. q-analogues and bijections*, Journal of Combinatorial Theory Series A 154 (2018), pp 350--405 ([arXiv:1512.08348](https://arxiv.org/abs/1512.08348))

### For semistandard Young tableaux

The original proof:

* [[Richard Stanley]], Theorem 15.3 in: *Theory and application of plane partitions 2*, Studies in Applied Math. **50** 3 (1971), 259-279 ([pdf](http://www-math.mit.edu/~rstan/pubs/pubfiles/12-2.pdf), [[StanleyPlanePartitions2.pdf:file]])

Review:

* {#Stanley99} [[Richard Stanley]], Thm. 7.21.2 in: *Enumerative combinatorics 2*, Cambridge University Press (1999, 2010) ([doi:10.1017/CBO9780511609589](https://doi.org/10.1017/CBO9780511609589), [webpage](http://www-math.mit.edu/~rstan/ec/))

* *A proof of the HCF* ([pdf](https://someproofsandstuff.files.wordpress.com/2015/03/hookcon.pdf), [[AProofOfTheHCF.pdf:file]])

### For dimensions of irreps

Textbook accounts:

* {#Sternberg94} [[Shlomo Sternberg]], Section 5.4 and Appendix C.7 of: *Group Theory and Physics*, Cambridge University Press 1994

Review for [[Sym(n)]]:

* James Stevens, Section 2.2 of: *Schur-Weyl duality* ([pdf](http://math.uchicago.edu/~may/REU2016/REUPapers/Stevens.pdf))

Review for [[SU(n)]]:

* {#Peluse14} [[Sarah Peluse]], Section 1 of: *Irreducible representations of $SU(n)$ with prime power degree*, Séminaire Lotharingien de Combinatoire 71 (2014), Article B71d ([pdf](https://www.emis.de/journals/SLC/wpapers/s71peluse.pdf))

* *Some Notes on Young Tableaux as useful for irreps of $\mathrm{su}(n)$* ([pdf](http://www.physics.mcgill.ca/~keshav/673IV/youngtableaux.pdf))

* Section 2 of: *Group Theory primer $SU(n)$* ([pdf](http://web.physics.ucsb.edu/~phys229B/s2013/Lectures_files/Chap13grouptheory.pdf))

[[!redirects hook length formulas]]

[[!redirects hook-length formula]]
[[!redirects hook-length formulas]]


hook-content formula

> For disambiguation of *[[content]]* see there.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Combinatorics
+-- {: .hide}
[[!include combinatorics - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[combinatorics]] and [[representation theory]], the *hook-content formula* expresses the [[number]] of [[semistandard Young tableaux]] with fixed underlying [[Young diagram]] and upper bound on its labels in terms of the "hook lengths" and the "content of boxes" of the underlying diagram.

If the dependency in the "contents" is removed, the formula reduces to the *[[hook length formula]]* that counts [[standard Young tableaux]] of the given shape.

Both of these formulas equivalently give [[dimensions]] of [[irreps]] in [[representation theory]]:

[[!include hook length and content formulas -- table]]

See at *[[hook length formula]]* for more on this.

## Preliminaries

Given a [[Young diagram]], the *hook* at any one of its boxes is the collection of boxes to the right and below that box, and including the box itself. We write "$\ell hook$" for the *length* of such a hook, i.e. for the number of boxes it contains. Formally:

\begin{defn}
**(hook length)** \linebreak
Let $\lambda = (\lambda_1 \geq \cdots \geq \lambda_{rows(\lambda)})$ be a [[partition]]/[[Young diagram]]. Then for

* $i \in \{1, \cdots, rows(\lambda)\}$,

* $j \in \{1, \cdots, \lambda_i\}$

the *hook length* at $(i,j)$ is

$$
  \ell hook(i,j)
  \;\coloneqq\;
  1 +
  (\lambda_i - j) + (\lambda'_j - 1)
  \,,
$$

where $\lambda'$ denotes the *conjugate partition* (see [there](Young+diagram#Conjugation)).

\end{defn}

Alongside this terminology one says that

$$
  content(i,j)
  \;\coloneqq\;
  j - i
  \;\;
  \in
  \mathbb{Z}
$$

is the *content* of the box $(i,j)$.

\linebreak

Consider in the following:

* $n \in \mathbb{N}$ a [[natural number]];

* $\lambda = (\lambda_1 \geq \cdots \geq \lambda_{rows(\lambda)})$ a [[partition]] of $n$, $\underset{i}{\sum} \lambda_i = n$,

  equivalently a [[Young diagram]] with $n$ boxes;

* $N \in \mathbb{N}_+$ a [[positive number|positive]] [[natural number]];

Write:

* $ssYT_\lambda(N)$ for the [[set]] of [[semistandard Young tableaux]] $T$

  * of shape (i.e. with underlying [[Young diagram]]) $\lambda$,

  * with labels $T_{i,j} \leq N$ (i.e. with $T_{i,j} \in \{1, \cdots, N\}$).

* $n(T) = \sum_{i,j} T_{i,j}$ for the [[sum]] of all the labels.

## Statement

### Standard hook-content formula

\begin{prop}
  For $\lambda$ a [[partition]]/[[Young diagram]] and $N \in \mathbb{N}_+$, the [[number]] $\left\vert ssYTableaux_\lambda(N)\right\vert$ of [[semistandard Young tableaux]] $T$ of shape $\lambda$
([hence](Schur+function#eq:SchurPolynomialCountsNumberOfSemiStandardYoungTableaux) the value of the [[Schur polynomial]] $s_\lambda$ on $N$ unit argument)
and entries bounded by $T_{i,j} \leq N$ is:

\[
  \left\vert ssYTableaux_\lambda(N)\right\vert
  \;=\;
  s_{\lambda}
  \big(
    x_1 \!=\! 1,
    \cdots,
    x_N \!=\! 1
  \big)
  \;\;
    =
  \;\;
  \underset{
    (i,j)
  }{\prod}
  \frac{
     N + content(i,j)
  }{
    \ell hook_\lambda(i,j)
  }
  \,.
\]
\end{prop}

([Stanley 71, Thm. 15.3](#Stanley71), [Stanley 99, Thm. 7.21.2](#Stanley99))

### $q$-Deformed hook-content formula

With the above ingredients, we have the following [[equalities]] of [[polynomials]] in a [[variable]] $q$:

$$
  \underset{
    T \in ssYT_\lambda(N)
  }{\sum}
  q^{ n(T) }
  \;\;=\;\;
  q^{ \sum_i i \cdot \lambda_i }
    \cdot
  \!\!\!\!\!\!\!
  \underset{
    { i \in \{1, \cdots, rows(\lambda)\} }
    \atop
    {j \in \{1, \cdots, \lambda_j\}}
  }{\prod}
  \frac{
    1 - q^{N + content(i,j)}
  }{
    1 - q^{\ell hook(i,j)}
  }
$$

([Krattenthaler 98, Thm. 1](#Krattenthaler98))

## Related concepts

* [[hook length formula]]

* [[Plancherel measure]], [[Schur-Weyl measure]]

## References

### Standard

The original statement and proof:

* {#Stanley71} [[Richard Stanley]], Theorem 15.3 in: *Theory and application of plane partitions 2*, Studies in Applied Math. **50** 3 (1971), 259-279 ([pdf](http://www-math.mit.edu/~rstan/pubs/pubfiles/12-2.pdf), [[StanleyPlanePartitions2.pdf:file]])

Review:

* {#Stanley99} [[Richard Stanley]], Thm. 7.21.2 in: *Enumerative combinatorics 2*, Cambridge University Press (1999, 2010) ([doi:10.1017/CBO9780511609589](https://doi.org/10.1017/CBO9780511609589), [webpage](http://www-math.mit.edu/~rstan/ec/))

See also:

* Graham Hawkes, *An Elementary Proof of the Hook Content Formula* ([arXiv:1310.5919](https://arxiv.org/abs/1310.5919))

### Generalized
 {#Generalized}

* [[Christian Krattenthaler]], *An Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula*, Discrete Mathematics and Theoretical Computer Science **3**, 1998, 11–32 ([hal:hal-00958904](https://hal.inria.fr/hal-00958904), [pdf](https://hal.inria.fr/hal-00958904/document))

* {#Krattenthaler98} [[Christian Krattenthaler]], *Another involution principle-free bijective proof of Stanley's hook-content formula*, J. Combin. Theory Ser. A 88 (1999), 66-92 ([arXiv:math/9807068](https://arxiv.org/abs/math/9807068))

See also:

* Mark Wildon, *A corollary of Stanley's Hook Content Formula* ([arXiv:1904.08904](https://arxiv.org/abs/1904.08904))

[[!redirects hook-content formulas]]

[[!redirects hook/content formula]]
[[!redirects hook/content formulas]]

[[!redirects hook content formula]]
[[!redirects hook content formulas]]


Hopf action

In a strict symmetric monoidal category $C$ with symmetry $\tau$, a map $\triangleright : B\otimes A\to A$, where $(B,\Delta_B)$ is a comonoid and $(A,\mu_A)$ a monoid, is a [[measuring|measuring]] if
$$
B\triangleright \mu_A = \mu_A\circ(\triangleright \otimes \triangleright)\circ (B\otimes \tau\otimes A)\circ (\Delta_B\otimes A\otimes A) : B\otimes A\otimes A\to A
$$
where we wrote $B=\mathrm{id}_B$ etc. If $B$ is in fact a bimonoid and if the measuring $\triangleright:B\otimes A\to A$ is an [[action]], then $\triangleright$ is said to be a __Hopf action__. In the $k$-linear case, a $k$-algebra $(A,\mu_A)$ equipped with a Hopf action is called also a __left $B$-module algebra__; it is the same as a monoid (=algebra) in the monoidal category of left $B$-modules, where the monoidal structure is induced by the coaction $\Delta_B$. It is straightforward to modify the condition above to the case of non-strict symmetric monoidal categories. A dual concept is a *Hopf coaction* or equivalently, a notion of a $B$-[[comodule algebra]].


Hopf adjunction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $C$ and $D$ be [[monoidal categories]], and $F \,\colon\, C \rightleftarrows D \,\colon\, G$ a _comonoidal adjunction_, i.e. an [[adjunction]] $F\dashv G$ in the [[2-category]] of [[colax 2-functor|colax]] [[monoidal functors]].  (By [[doctrinal adjunction]], this is actually equivalent to requiring that $G$ is a [[strong monoidal functor]].)  This adjunction is a **Hopf adjunction** if the canonical morphisms

$$
  F(x \otimes G y)
   \longrightarrow
  F x \otimes y
$$

$$
  F(G y \otimes x)
    \longrightarrow
  y \otimes F x
$$

are [[isomorphisms]] for any $x\in C$ and $y\in D$.

Of course, if $C$, $D$, $F$, and $G$ are [[symmetric monoidal category|symmetric]], then it suffices to ask for one of these.  If $C$ and $D$ are moreover [[cartesian monoidal category|cartesian monoidal]], then any adjunction is comonoidal, and the condition is also (mis?)named *[[Frobenius reciprocity]]*.

## Properties

* If $C$ and $D$ are closed, then by the calculus of [[mates]], saying that $F\dashv G$ is Hopf is equivalent to asking that $G$ be a [[closed monoidal functor]], i.e. preserve [[internal-homs]] up to isomorphism.

* If $F\dashv G$ is a Hopf adjunction, then its induced monad $G F$ is a [[Hopf monad]].  Conversely, the [[Eilenberg-Moore category|Eilenberg-Moore]] adjunction of a Hopf monad is a Hopf adjunction.

## Related concepts

* [[adjunction]]

* [[Frobenius reciprocity]]

* [[Hopf monad]]

* [[Hopf algebra]]

## References

* [[Alain Bruguières]], [[Steve Lack]], [[Alexis Virelizier]], *Hopf monads on monoidal categories*,  Adv. Math. __227__ 2 (2011) 745-800 [[arXiv:1003.1920](https://arxiv.org/abs/1003.1920), [doi:10.1016/j.aim.2011.02.008](https://doi.org/10.1016/j.aim.2011.02.008)]

* Adriana Balan, *On Hopf adjunctions, Hopf monads and Frobenius-type properties*, Appl. Categ. Structures **25** 5 (2017) 747-774 [[arxiv:1411.2236](https://arxiv.org/abs/1411.2236), [doi:10.1007/s10485-016-9428-0](https://doi.org/10.1007/s10485-016-9428-0)]

* Harshit Yadav, *Frobenius monoidal functors from (co)Hopf adjunctions* [[arXiv:2209.15606v2](https://arxiv.org/abs/2209.15606)]

[[!redirects Hopf adjunctions]]


Hopf algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _Hopf algebra_ is an abstraction of the properties of

* the [[group algebra]] of a [[group]];

* the [[algebra of functions]] on a finite group, and more generally, the algebra of regular functions on an affine algebraic $k$-group;

* the [[universal enveloping algebra]] of a [[Lie algebra]],

where not only the [[associative algebra]] structure is remembered, but also the natural [[coalgebra]] structure, making it a [[bialgebra]], as well as the algebraic structure induced by the [[inverse]]-operation in the group, called the _[[antipode]]_.

More intrinsically, a Hopf algebra structure on an [[associative algebra]] is precisely the structure such as to make its [[category of modules]] into a [[rigid monoidal category]] equipped with a [[fiber functor]] -- this is the statement of _[[Tannaka duality]]_ for Hopf algebras.

Hopf algebras and their generalization to [[Hopf algebroids]] arise notably as [[groupoid convolution algebras]]. Another important source of Hopf algebras is [[combinatorics]], see at _[[combinatorial Hopf algebras]]_.

There is a wide variety of variations of the notion of Hopf algebra, relaxing [[properties]] or adding [[structure]]. Examples are _[[weak Hopf algebras]]_, _[[quasi-Hopf algebras]]_, _([[quasi-triangular Hopf algebra|quasi]]-)[[triangular Hopf algebras]]_, _[[quantum groups]]_, _[[hopfish algebras]]_ etc. Most of these notions are systematized via [[Tannaka duality]] by the properties and structures on the coresponding [[categories of modules]], see at _[Tannaka duality](#TannakaDuality)_ below.

## Definition

+-- {: .num_defn #Antipode}
###### Definition
**([[antipode]])**

A $k$-[[bialgebra]] $(A,m,\eta,\Delta,\epsilon)$ with multiplication $m$, comultiplication $\Delta$, unit $\eta \colon k\to A$ and counit $\epsilon \colon A\to k$ is called a **Hopf algebra** if there exists a $k$-[[linear function]]

$$
  S \colon A \longrightarrow A
  \,,
$$

then called the **[[antipode]]** or **coinverse**, such that

$$
  m\circ(\mathrm{id}\otimes S)\circ \Delta
  \;=\;
  m\circ(S\otimes\mathrm{id})\circ\Delta = \eta\circ\epsilon
$$

(as a map $A\to A$).

=--

+-- {: .num_defn #InvolutiveHopfAlgebra}
###### Definition

If the antipode of a Hopf algebra (Def. \ref{Antipode}) is an [[anti-involution]], in that $S \circ S = id$, then one speaks of an *[[involutive Hopf algebra]]*.

=--

+-- {: .num_remark}
###### Remark

If an antipode exists then it is unique, just the way that if inverses exist in a [[monoid]] then they are unique. One sometimes prefers to have a __skew-antipode__ $\tilde{S}$ such that $h_{(2)}\tilde{S}(h_{(1)}) = \tilde{S}(h_{(2)}) h_{(1)} =
(\eta\circ\epsilon) (h)$. If $S$ is an invertible antipode then $\tilde{S} - S^{-1}$ is a skew-antipode. $H$ is a bialgebra with a skew-antipode iff $H^{op}$ (the same vector space, opposite product, the same coproduct) is a Hopf algebra.

The unit of a Hopf algebra is a [[grouplike element]],  hence $S(1)1=1$, therefore $S(1)=1$. By linearity of $S$ this implies that $S\circ\eta\circ\epsilon = \eta\circ\epsilon$.

=--

+-- {: .num_prop #AntipodeIsAnAntihomomorphism}
###### Proposition

The antipode is an [[anti-homomorphism]] both of algebras and coalgebras (i.e. a homomorphism $S \colon A\to A^{cop}_{op}$).

In particular, an [[involutive Hopf algebra]] (Def. \ref{InvolutiveHopfAlgebra}) is a [[star-algebra]].

=--

+-- {: .proof}
###### Proof (algebra part)
In Sweedler notation, for any $g,h\in A$,

$$S((h g)_{(1)}) (h g)_{(2)} = \epsilon(h g)$$

$$S((h g)_{(1)}) h_{(2)}g_{(2)} = \epsilon(h)\epsilon(g)$$

$$S((h g)_{(1)}) h_{(2)}g_{(2)} S g_{(3)} S h_{(3)} = \epsilon(h_{(1)})\epsilon(g_{(1)}) S g_{(2)} S h_{(2)}$$

$$S(h_{(1)}g_{(1)}) \epsilon(h_{(2)})\epsilon(g_{(2)}) = (S g) (S h)$$

$$S(h_{(1)}\epsilon(h_{(2)})g_{(1)}\epsilon(g_{(2)})) = (S g)(S h)$$

Therefore $S(h g) = (S g) (S h)$.

For the coalgebra part, notice first that $\epsilon(h)1\otimes 1 = \tau\circ\Delta(\epsilon(h)1)=\tau\circ\Delta(S h_{(1)}h_{(2)})$. Expand this as

$$ (S h_{(1)}\otimes S h_{(2)})(h_{(4)}\otimes h_{(3)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(h_{(4)}\otimes h_{(3)})$$

$$ (S h_{(1)}\otimes S h_{(2)})(h_{(4)}\otimes h_{(3)}) (S h_{(5)}\otimes S h_{(6)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(h_{(3)}\otimes h_{(2)})(S h_{(4)}\otimes S h_{(5)})$$

$$ ((S h_{(1)}\otimes S h_{(2)})\epsilon(h_{(3)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(1\otimes\epsilon(h_{(2)}))$$

$$ ((S h_{(1)}\otimes S h_{(2)})\epsilon(h_{(3)}) = (\tau\circ\Delta)(S h_{(1)})(1\otimes\epsilon(h_{(2)})1) = (\tau\circ\Delta)(S h_{(1)}\epsilon(h_{(2)}))$$

$$ (S h_{(1)}\otimes S h_{(2)})=(\tau\circ\Delta)(S h) = (S h)_{(2)}\otimes (S h)_{(1)}.$$
=--

The axiom that must be satisfied by the antipode looks like a $k$-linear version of the identity satisfied by the inverse map in a group bimonoid: taking a group element $g$, duplicating by the diagonal map $\Delta$ to obtain $(g,g)$, taking the inverse of either component of this ordered pair, and then multiplying the two components, we obtain the identity element of our group.

Just as an [[associative unital algebra|algebra]] is a [[monoid]] in [[Vect]] and a [[bialgebra]] is a [[bimonoid]] in $Vect$, a Hopf algebra is a [[Hopf monoid]] in $Vect$.

+-- {: .num_remark}
###### Remark
**Caution: convention in topology**

In [[algebraic topology]], it is common to define Hopf algebras without mentioning the antipode, since in many topological cases of interest it exists automatically.  For example, this is the case if it is [[graded object|graded]] and "connected" in the sense that its degree-0 part is just the [[ground field]] (a property possessed by the homology or cohomology of any connected space).  In algebraic topology also the strict coassociativity is not always taken for granted.

=--

## Properties

### Relation to Hopf groups

Note that the definition of Hopf algebra (or, really, of [[Hopf monoid]]) is [[duality|self-dual]]: a Hopf monoid in a [[symmetric monoidal category]] $V$ is the same as a Hopf monoid in the [[opposite category]] $V^{op}$ (i.e. a "Hopf comonoid").  Thus we can view a Hopf algebra as "like a group" in two different ways, depending on whether the group multiplication corresponds to the multiplication or the comultiplication of the Hopf algebra.  The formal connections between Hopf monoids and group objects are:

1. A Hopf monoid in a [[cartesian monoidal category]] $V$ is the same as a group object in $V$.  Such Hopf monoids are always _cocommutative_ (that is, their underlying comonoid is cocommutative).  This is because every object of a cartesian monoidal category is a cocommutative comonoid object in a unique way, and every morphism is a comonoid homomorphism.

1. A _commutative_ Hopf monoid in a [[symmetric monoidal category]] $V$ is the same as a [[group object]] in the [[opposite category]] $CMon(V)^{op}$, where $CMon(V)$ is the category of commutative monoids in $V$, hence a [[cogroup]] object in $CMon(V)$ (a point highlighted by [[Haynes Miller]], see ([Ravenel 86, appendix A1](#Ravenel86))).  This works because the tensor product of commutative algebras is the categorical [[coproduct]], and hence the [[product]] in its [[opposite category]]. In particular, a [[commutative Hopf algebra]] is the same as a group object in the category $Alg^{op}$ of affine schemes.

Corresponding to these two, an ordinary group $G$ gives us two different Hopf algebras (here $k$ is the [[ground ring]]):

1. The [[group algebra]] $k[G]$ (the free vector space on the set $G$), with multiplication given by the group operation of $G$ and comultiplication given by the diagonal $g\mapsto g\otimes g$.  This Hopf algebra is always cocommutative, and is commutative iff $G$ is abelian.  It can be viewed as the result of applying the [[strong monoidal functor]] $k[-]:Set \to k Mod$ to the Hopf monoid $G$ in $Set$.

1. The function algebra $k(G)$ (the set of functions $G\to k$), with comultiplication given by precomposition with the group operation
$$k(G) \to k(G\times G) \cong k(G)\otimes k(G),$$
and multiplication given by pointwise multiplication in $k$.  In this case we need some finiteness or algebraicity of $G$ in order to guarantee $k(G\times G) \cong k(G)\otimes k(G)$.  This Hopf algebra is always commutative, and is cocommutative iff $G$ is abelian.

Note that if $G$ is finite, then $k[G]\cong k(G)$ as $k$-modules, but the Hopf algebra structure is quite different.

###  The theorem of Hopf modules

Hopf algebras can be characterized among bialgebras by the [fundamental theorem of Hopf modules](https://ncatlab.org/nlab/show/Hopf+module#fundamental_theorem_on_hopf_modules): the category of Hopf modules over a bialgebra is canonically equivalent to the category of vector spaces over the ground ring iff the bialgebra is a Hopf algebra.  This categorical fact enables a definition of Hopf monoids in some setups that do not allow a sensible definition of antipode.

### Relation to Lie algebras

* [[Milnor-Moore theorem]]

### Tannaka duality
 {#TannakaDuality}

The [[category of modules]] (finite dimensional) over the underlying [[associative algebra]] of a Hopf algebra canonically inherits the structure of an [[rigid monoidal category|rigid]] [[monoidal category]] such that the forgetful [[fiber functor]] to [[vector spaces]] over the ground field is a [[strict monoidal functor]].

The statement of [[Tannaka duality]] for Hopf algebras is that this property characterizes Hopf algebras. (See for instance ([Bakke](#Bakke)))

For generalization of this characterization to [[quasi-Hopf algebras]] and [[hopfish algebras]] see ([Vercruysse](#Vercruysse)).

[[!include structure on algebras and their module categories - table]]

### As 3-vector spaces

A Hopf algebra structure on an [[associative algebra]] $A$ canonically defines on $A$ the structure of an algebra object [[internalization|internal]] to the [[2-category]] of algebras, [[bimodule]]s and bimodule homomorphisms.

By the discussion at [[n-vector space]] this allows to identify Hopf algebras with certain _3-vector spaces_ .

(For instance ([FHLT, p. 27](http://ncatlab.org/nlab/show/Topological+Quantum+Field+Theories+from+Compact+Lie+Groups))).

More general 3-vector spaces are given by _[[hopfish algebras]]_ and generally by [[sesquiunital sesquialgebras]].

### Relation to Frobenius algebras

Both Hopf algebras and [[Frobenius algebras]] are at the same time an [[algebra]] and a [[coalgebra]], albeit with different additional structures and properties. Nevertheless, one may ask whether there is any relation between these two. This leads to the following result due to [Larson & Sweedler (1969)](#LS69).

+-- {: .num_prop #HopfAlgebraIsFrobeniusAlgebra}
###### Proposition

Any [[finite dimensional vector space|finite-dimensional]] Hopf algebra can be endowed with the structure of a Frobenius algebra.

=--

The finite-dimensional condition is expected since all Frobenius algebras are finite-dimensional. A nontrivial part of this result is that, while a finite-dimensional Hopf algebra $A$ already has a counit, this is not be the same counit that realizes $A$ as a Frobenius algebra. Rather, it is an *integral* (see [Larson & Sweedler (1969)](#LS69) for details on this), which for finite-dimensional Hopf algebras always exist and are unique up to scaling.

In fact, more is true. In [Fuchs, Schweigert & Stigner (2011)](#FSS11) a [symmetric special Frobenius algebra](https://ncatlab.org/nlab/show/Frobenius+algebra#special_frobenius_algebras) is constructed from a particular kind of Hopf algebra.

+-- {: .num_prop #DualHopfAlgebraIsSSFrobeniusAlgebra}
###### Proposition

Let $H$ be a finite-dimensional factorizable ribbon Hopf algebra, with multiplication $\mu$, counit $\epsilon$, comultiplication $\Delta$, and antipode $S$. Let $\Lambda$ be a left-integral element of $H$ and $\lambda$ a right-cointegral of $H$ (since $H$ is finite-dimensional, this is equivalent to an integral of the dual Hopf algebra $H^*$) such that $\lambda\circ\Lambda=1$. Then the dual vector space $H^*$ endowed with: a unit $\epsilon^*$, a multiplication $\Delta^*$, a counit $\Lambda^*$, and comultiplication

$$\Delta_F=((\text{id}_H\otimes (\lambda\circ \mu))\circ (\text{id}_H\otimes S\otimes\text{id}_H)\circ (\Delta\otimes\text{id}_H))^*$$

is a symmetric Frobenius object in the category $\text{Bimod}(H,H)$ of bimodules of $H$, and it is furthermore special iff $H$ is is semisimple.

=--

## Examples

* [[group algebra]]

* [[Steenrod algebra]]

* the [[ordinary homology]] of a [[finite type]] [[H-space]] (for instance a [[based loop space]]) is a [[Hopf algebra]] via its [[Pontrjagin ring]]-[[mathematical structure|structure]] -- see at _[[homology of loop spaces]]_ for more

### The Kac-Paljutkin $H_8$ Hopf algebra

The Kac-Paljutkin $H_8$ Hopf algebra was first described in [Kac & Paljutkin (1966)](#KP66) and it is the Hopf algebra with the smallest dimension that is [[semisimple object|semisimple]], [[commutative algebra|noncommutative]], and [[cocommutative coalgebra|noncocommutative]]. It is also [[self-dual object|self-dual]] (see e.g. [Burciu (2017)](#Burciu17)). One presentation  is in terms of generators $\{x,y,z\}$ satisfying the relations $x^2=y^2=z^2=1$, $xz=zx$, $zy=yz$, $xyz=yx$, along with comultiplication

$$\Delta(x)=xe_0\otimes x+xe_1\otimes y$$

 $$\Delta(y)=ye_1\otimes x+ye_0\otimes y$$

$$\Delta(z)=z\otimes z$$

 (where $e_0=\frac{1}{2}(1+z)$, $e_1=\frac{1}{2}(1-z)$) whose counit is $\epsilon(x)=\epsilon(y)=\epsilon(z)=1$. The antipode map is

$$S(x)=xe_0+ye_1$$

 $$S(y)=xe_1+ye_0$$

 $$S(z)=z$$

. Its [[category of representations]] is the unique $\mathbb{Z}_2\times\mathbb{Z}_2$ [[Tambara-Yamagami category]] that admits a [[fiber functor]] and is not the representation category of some group.

## Related concepts

* [[quasi-Hopf algebra]]

* [[triangular Hopf algebra]]

* [[quasitriangular Hopf algebra]]/[[quantum group]]

* [[hopfish algebra]]

* [[Hopf algebroid]]

  * [[commutative Hopf algebroid]]

* [[Hopf C-star algebra]]

* [[Hopf monoid]]

* [[Hopf monoidal category]]

* [[Hopf monad]]

* [[change of rings theorem]]

* [[Hopf ring spectrum]]

* [[Pontrjagin product]]

[[!include structure on algebras and their module categories - table]]

## References

* [[Kenneth Brown]], _Hopf algebras_, lectures, [pdf](http://www.maths.gla.ac.uk/~kab/Hopf%20lects%201-8.pdf)

* Nicolas Andruskiewitsch, Walter Ferrer Santos, *The beginnings of the theory of Hopf algebras*, Acta Appl Math **108** (2009) 3-17 [[arXiv:0901.2460](https://arxiv.org/abs/0901.2460)]

The diagrammatic definition of a Hopf algebra, is also in the [Wikipedia entry](http://en.wikipedia.org/wiki/Hopf_algebra#Formal_definition).

* Eiichi Abe, _Hopf algebras_, Cambridge UP 1980.

* [[Pierre Cartier]], _A primer on Hopf algebras_, Frontiers in number theory, physics, and geometry II, 537--615, preprint IHÉS 2006-40, 81p ([doi:10.1007/978-3-540-30308-4_12](https://doi.org/10.1007/978-3-540-30308-4_12),  [pdf](http://preprints.ihes.fr/2006/M/M-06-40.pdf),  [pdf](http://www.math.osu.edu/~kerler.2/VIGRE/InvResPres-Sp07/Cartier-IHES.pdf), [[CartierHopfAlgebras.pdf:file]])

* V. G. [[Drinfel'd]], _Quantum groups_, Proceedings of the International Congress of Mathematicians 1986, Vol. 1, 2 798--820, AMS 1987, [djvu:1.3 M](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.djvu), [pdf:2.5 M](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.pdf)

* G. Hochschild, _Introduction to algebraic group schemes_, 1971

* [[Shahn Majid]], _Foundations of quantum group theory_, Cambridge University Press 1995, 2000.

* {#MilnorMoore65} [[John Milnor]], [[John Moore]], _On the structure of Hopf algebras_, Annals of Math. __81__ (1965), 211-264 ([doi:10.2307/1970615](https://doi.org/10.2307/1970615), [pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf))

* Susan Montgomery, _Hopf algebras and their actions on rings_, AMS 1994, 240p.

* B. Parshall, J.Wang, _Quantum linear groups_, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.

* [[Moss Sweedler]], _Hopf algebras_, Benjamin 1969.

* William C. Waterhouse, _Introduction to affine group schemes_, Graduate Texts in Mathematics __66__, Springer 1979. xi+164 pp.

[[Tannaka duality]] for Hopf algebras and their generalization is alluded to in

* {#Vercruysse} Joost Vercruysse, _Hopf algebras---Variant notions and reconstruction theorems_ ([arXiv:1202.3613](http://arxiv.org/abs/1202.3613))

and discussed in detail in

* {#Bakke} Tørris Koløen Bakke, _Hopf algebras and monoidal categories_ (2007) ([pdf](https://munin.uit.no/bitstream/handle/10037/1084/finalthesis.pdf))

Discussion in [[algebraic topology]] with an eye towards [[stable homotopy theory]], [[Steenrod algebra]] and the [[Adams spectral sequence]]:

* [[Douglas Ravenel]], [[W. Stephen Wilson]], _The Hopf ring for complex cobordism_,  Bull. Amer. Math. Soc. 80 (6) 1185 - 1189, November 1974 (doi:10.1016/0022-4049(77)90070-6, [euclid](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-80/issue-6/The-Hopf-ring-for-complex-cobordism/bams/1183536024.full?tab=ArticleLink), [pdf](https://people.math.rochester.edu/faculty/doug/mypapers/hopfring.pdf))

  > (for [[MU]])

* {#Ravenel86} [[Douglas Ravenel]], appendix A1 of: _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986 ([pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenelA1.pdf))

* [[Neil Strickland]], _Bott periodicity and Hopf rings_, 1992 ([pdf](https://neil-strickland.staff.shef.ac.uk/research/thesis.pdf), [[StricklandHopfRings.pdf:file]])

  > (for [[topological K-theory]])

* [[W. Stephen Wilson]], _Hopf rings in algebraic topology_, Expositiones Mathematicae, 18:369–388, 2000 ([pdf](https://math.jhu.edu/~wsw/papers2/math/39-hopf-rings-expo-2000.pdf))

* [[Christoph Schweigert]], *Hopf algebras, quantum groups and topological field theory* (2022) [[pdf](https://www.math.uni-hamburg.de/home/schweigert/skripten/hskript.pdf)]

For Hopf algebras in generative linguistics, see:

* [[Matilde Marcolli]], [[Noam Chomsky]], Robert Berwick, *Mathematical Structure of Syntactic Merge* ([arXiv:2305.18278](https://arxiv.org/abs/2305.18278))

* [[Matilde Marcolli]], Robert Berwick, [[Noam Chomsky]], *Old and New Minimalism: a Hopf algebra comparison* ([arXiv:2306.10270](https://arxiv.org/abs/2306.10270))

The construction of a Frobenius algebra structure on finite-dimensional Hopf algebras due to

* {#LS69} Richard Larson, [[Moss Sweedler]]. *An Associative Orthogonal Bilinear Form for Hopf Algebras*. American Journal of Mathematics, Vol. 91, No. 1 (Jan., 1969), pp. 75-94 (20 pages). ([doi](https://doi.org/10.2307/2373270))

The proof that $H^*$ can be endowed with the structure of a symmetric special Frobenius object in $\text{Bimod}(H,H)$ is in

* {#FSS11} [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Carl Stigner]]. *Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms*. Journal of Algebra **363**, 1 August 2012, Pages 29-72. ([doi](https://doi.org/10.1016/j.jalgebra.2012.04.008))

On the $H_8$ Hopf algebra

* {#KP66} [[G. I. Kac]], V. G. Paljutkin, *Finite ring groups_, Trans. Moscow Math. Soc. 15 (1966), 251–294.

* {#Burciu17} Sebastian Burciu. *Representations and conjugacy classes of semisimple quasitriangular Hopf algebras*. (2017) (([arXiv:1709.02176
](https://arxiv.org/abs/1709.02176)).

[[!redirects Hopf algebras]]

[[!redirects Hopf ring]]
[[!redirects Hopf rings]]

[[!redirects skew-antipode]]
[[!redirects skew-antipodes]]
[[!redirects skew antipode]]
[[!redirects skew antipodes]]


Hopf algebroid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A Hopf algebroid is an associative [[bialgebroid]] with an [[antipode]].

A _Hopf algebroid_ is a (possibly [[noncommutative geometry|noncommutative]]) generalization of a structure which is dual to a [[groupoid]] (equipped with [[atlas]]) in the sense of [[Isbell duality|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 [[dual equivalence|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 $h\in H$,
$$
(S^{-1} h_{(2)})_{(1)}\otimes_A(S^{-1} h_{(2)})_{(2)}h_{(1)} = S^{-1} h\otimes_A 1_H,
$$
$$
(S h_{(1)})_{(1)} h_{(2)}\otimes_A(S h_{(1)})_{(2)} = 1_H\otimes_A S h.
$$

## 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](http://arXiv.org/abs/q-alg/9505024), [MR95e:16037](http://www.ams.org/mathscinet-getitem?mr=95e:16037), [doi](https://doi.org/10.1142/S0129167X96000050); _On the Drinfeld double and the Heisenberg double of a Hopf algebra_, Duke Math. J. __7__:3 (1994) 763-776, [MR1277953](https://mathscinet.ams.org/mathscinet-getitem?mr=1277953), [doi](https://doi.org/10.1215/S0012-7094-94-07428-0)
* [[Ping Xu]], _Quantum groupoids_, Commun. Math. Phys., 216:539581, 2001, [q-alg/9905192](https://arxiv.org/abs/math/9905192), [doi](https://doi.org/10.1007/s002200000334); _Quantum groupoids and deformation quantization_, [q-alg/9708020](https://arxiv.org/abs/q-alg/9708020), _Quantum groupoids associated to universal dynamical R-matrice_, [q-alg/9811172](https://arxiv.org/abs/math/9811172)
* blog discussion at [Theoretical Atlas](http://theoreticalatlas.wordpress.com/2009/03/18/a-note-on-quantum-groupoids)

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; <a href="http://arxiv.org/abs/math.QA/0301169"> math.QA/0301169 </a>

* [[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](https://arxiv.org/abs/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](https://arxiv.org/abs/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](https://arxiv.org/abs/math.QA/0305136)
* [[T. Brzeziński]], G. Militaru, _Bialgebroids, $\times_A$-bialgebras and duality_,  J. Algebra __251__: 279-294, 2002 [math.QA/0012164](https://arxiv.org/abs/math.QA/0012164)
* D. Chikhladze, Category of quantum categories, Theory and Applications of Categories __25__ (2011) 1 - 37.  ([pdf](http://www.tac.mta.ca/tac/volumes/25/1/25-01.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 algebroid]]s 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](https://doi.org/10.1142/S0219498824501147) [arXiv:2208.11696](https://arxiv.org/abs/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](https://arxiv.org/abs/math/0311244) [doi](https://doi.org/10.48550)

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](https://arxiv.org/abs/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](http://em.rdcu.be/wf/click?upn=KP7O1RED-2BlD0F9LDqGVeSHnmX9Ae5OTohRXxZ3Kuwuk-3D_fxWtIqIinsN9WJMaEHmGcrXGHheD-2FQtZ0bATf5S8uPqTE1vT5wX4Z3MHOr3LESbPy9zd7ZFfFy-2BgrWYwBzwnkToXTLfpXlfVLbWgoyxh3yG3X87hmFOy6WBB8-2BfFT6d5xCnkUAl78V3yBYjEJE6K0TYy2-2FYZewOjCVwZXxVneWJX8yW4YWbnghVE5d2d5zhz6lkzaj9LLNSVj-2B3QvKR6Dg-3D-3D)
 (free for online use) [doi](https://doi.org/10.1007/s11005-016-0908-9) [arxiv/1409.8188](https://arxiv.org/abs/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](https://arxiv.org/pdf/1610.03837) [doi](https://doi.org/10.1016/j.jalgebra.2022.01.027)

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](https://arxiv.org/abs/1307.0769) [doi](https://doi.org/10.1080/00927872.2017.1363220); _Multiplier Hopf algebroids arising from weak multiplier Hopf algebras_, [arxiv/1406.3509](https://arxiv.org/abs/1406.3509)
* Frank Taipe, _Quantum transformation groupoids: An algebraic and analytical approach_, PhD thesis (2018) [link](https://hal.archives-ouvertes.fr/tel-02288186)

Further examples and counterexamples

* Ana Rovi, _Hopf algebroids associated to Jacobi algebras_, Int. J. Geom. Methods in Modern Phys. __11__: 10 (2014) 1450092 [doi](https://doi.org/10.1142/S0219887814500923)
* [[Ulrich Krähmer]], Ana Rovi, _A Lie-Rinehart algebra with no antipode_, Commun. Alg. __43__:10 (2015) [doi](https://doi.org/10.1080/00927872.2014.896375)

> 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.

category: algebra
[[!redirects Hopf algebroids]]


Hopf algebroid over a commutative base

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

Hopf algebroids have a base and a total algebra (and some other data).
There are 3 levels of generality: commutative Hopf algebras (classical case, both base and the total space are commutative), Hopf algebras with commutative base (studied from late 1980s: Maltsionitis, later Connes' school etc) and genuinely noncommutative case for which see [[Hopf algebroid]] and [[bialgebroid]].

## Idea

Just as for [[groups]] and their [[Hopf algebras]], there are _two_ ways to assign a __Hopf algebroid over a commutative base algebra__ to a [[groupoid]] $\mathcal{G}_\bullet$:

1. ([[commutative Hopf algebroids]]) with **commutative but non-co-commutative total algebra**: Form the [[commutative algebra|commutative]] [[algebras of functions]] $C(\mathcal{G}_1)$ and $C(\mathcal{G}_0)$ and regard the operation induced by the partially defined [[composition]] in $\mathcal{G}_\bullet$ as an in general non-co-commutative [[coalgebra]] structure on $C(\mathcal{G}_1)$ over $C(\mathcal{G}_0)$; the graded
commutative case appears in algebraic topology and is classical
(Steenrod algebra and other examples);

1. with **non-commutative but co-commutative total algebra**: Form the in general non-commutative [[groupoid convolution algebra]] $C_{conv}(\mathcal{G})$ and regard it as a co-commutative [[coalgebra]] over $C(\mathcal{G}_0)$.

Given an internal [[groupoid]] in the category $Aff_k$ 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 [[dual equivalence|antiequivalence of categories]]. Commutative Hopf algebroids are useful also in a version in [[brave new algebra]] (the work of John Rognes).

## Higher groupoid convolution algebras and n-vector spaces/n-modules

> under construction

We discuss here a natural generalization of the notion of [[groupoid convolution algebras]] to [[higher algebra|higher algebras]] for [[n-groupoid|higher groupoids]].

There may be several sensible such generalizations. The one discussed now follows the principle of iterated [[internalization]] and naturally matches to the concept of [[n-modules]] ([[n-vector spaces]]) as they appear in [[extended prequantum field theory]].

In order to disentangle conceptual from technical aspects, we first discuss [[discrete ∞-groupoid|geometrically discrete higher groupoids]]. The results of this discussion then in particular help to suggest what the right definition of "higher Lie groupoid" in the ontext of higher convolution algebras should be in the first place.

The consideration are based on the following

+-- {: .num_remark}
###### Remark

By the discussion at _[[2-module]]_ we may think of the [[2-category]] $k Alg_b$ of $k$-[[associative algebras]] and [[bimodules]] between them as a model for the 2-category [[2Mod]] of $k$-[[2-modules]] that admit a 2-[[basis]] ([[2-vector spaces]]). Hence the groupoid convolution algebra constructiuon is a 2-functor

$$
  C \;\colon\; Grpd \to 2 Mod
  \,.
$$

There is then the following systematic refinement of this to higher groupoids and higher algebra: by the discussion at _[[n-module]]_, 3-modules are algebra objects in [[2Mod]] and maps between them are [[bimodule]] objects in there. An algebra object in  $k Alg_b$ is equivalently a [[sesquialgebra]], an algebra equipped with a second algebra structure up to coherent homotopy, that is exhibited by structure bimodules.

Special cases of this are [[bialgebras]], for which these structure bimodules come from actual algebra homomorphisms. Examples of these in turn are [[Hopf algebras]]. These we naturally re-discover as special higher groupoid convolution higher algebras in example \ref{DoubleGroupoid2AlgebraOfDeloopingOfFiniteGroup} below.

=--

This iterated internalization on the codomain of the groupoid convolution algebra functor has a natural analog on its domain: a 2-groupoid we may present by a [[double groupoid]], namely a [[groupoid object in an (∞,1)-category]] in [[Grpd]] which is 3-[[coskeleton|coskeletal]] as a [[simplicial object]] in [[Grpd]].

+-- {: .num_remark}
###### Remark

Given a [[groupoid object in an (∞,1)-category|groupoid object]] $\mathcal{G}_\bullet$ in the [[(2,1)-topos]] [[Grpd]] hence a [[double groupoid]], applying the groupoid convolution algebra $(2,1)$-functor $C$ to the corresponding [[simplicial object]] $\mathcal{G}_\bullet \in Grpd^{\Delta^{op}}$ yields:

* groupoid convolution algebras $C(\mathcal{G}_0)$ and $C(\mathcal{G}_1)$,

* a $C(\mathcal{G}_1) \otimes_{C(\mathcal{G}_{0,1})} C(\mathcal{G}_1)-C(\mathcal{G}_{0})$-bimodule, assigned to the [[composition]] functor $\partial_1 \colon \mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 \to \mathcal{G}_1$.

Under the 2-functoriality of $C$, the [[Segal conditions]] satisfied by $\mathcal{G}_\bullet$ make this bimodule exhibi a [[sesquialgebra]] structure over $C(\mathcal{G}_{0,1})$.

This sesquialgebra we call the the **double groupoid convolution 2-algebra** of $\mathcal{G}_\bullet$.

(Here we make invariant sense of the [[tensor product]] by evaluating on a [[Reedy model structure|Reedy fibrant]] representative.)

=--

+-- {: .num_example #DoubleGroupoid2AlgebraOfDeloopingOfFiniteGroup}
###### Example

Let $G$ be a [[finite group]]. Write $\mathbf{B}G$ for its [[delooping]] [[groupoid]] (the connected groupoid with $\pi_1 = G$). There are two natural ways to present $\mathbf{B}G$ as a [[double groupoid]]:

1. $\underset{\longrightarrow}{\lim}(\cdots \mathbf{B}G \stackrel{\overset{id}{\to}}{\underset{id}{\to}} \mathbf{B}G) \simeq \mathbf{B}G$;

1. $\underset{\longrightarrow}{\lim}(\cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *) \simeq \mathbf{B}G$.

Applying the [[groupoid convolution algebra]] functor to the first presentation yields the groupoid convolution algebra $C(\mathbf{B}G)$ equipped with a trivial multiplication bimodule, hence just the group convolution algebra $C(\mathbf{B}G) \simeq C_{conv}(G)$.

Applying however the [[groupoid convolution algebra]] functor to the second presentation yields the _commutative_ algebra of functions $C(G)$ equipped with the multiplication bimodule which is $C(G \times G)$ regarded as a $(C(G\times G), C(G))$-bimdodule, where the right action is induced by pullback along the group product map $G \times G \to G$.

This bimodule is in the image of the functor $Alg \to Alg_b$ that sends algebra homomorphisms to their induced bimodules, by sending $f \colon A \to B$ to $A$ regarded as an $(A,B)$-bimodule with the canonical left action on itself and the right action induced by $f$. Namely this bimdoule correspondonds to the map

$$
  \Delta \colon C(G) \to C(G \times G) \simeq C(G) \otimes C(G)
$$

given on $\phi \in C(G)$ and $g_1, g_2 \in G$ by

$$
  \Delta \phi \colon (g_1, g_2) \mapsto \phi
  \,.
$$

In summary this means that (for $G$ a finite group)

1. if we regard $\mathbf{B}G$ as presented as a  double groupoid constant on $\mathbf{B}G$, then the corresponding groupoid convolution [[sesquialgebra]] (basis for a [[n-module|3-module]]) is the convolution algebra of $G$;

1. if instead we regard $\mathbf{B}G$ as presented as the double groupoid which is degreewise constant as a groupoid, then the corresponding groupoid convolution sesquialgebra is the standard ("dual") [[Hopf algebra]] structure on the commutative pointwise product algebra of functions on $G$.

=--
## Properties

### Steenrod operations on coTor groups

For $\Gamma$ a suitable commutative Hopf algebroid and $N_1, N_2$
two $\Gamma$-[[comodules]], then the co[[Tor]] groups $CoTor_\Gamma(N_1, N_2)$
form a _[[Steenrod algebra]]_. See there for details and citations.

For [[MU]] this is the content of the [[Landweber-Novikov theorem]].

### Generalized dual Steenrod algebra
 {#GeneralizedDualSteenrodAlgebra}

For $E$ a suitable [[E-infinity ring]] [[spectrum]], its [[homotopy groups]]
$\pi_\bullet(E)$ and [[generalized homology]] $E_\bullet(E)$
form a Hopf algebroid of [[spectra]], the dual $E$-[[Steenrod algebra]].
(These examples have also been called [[brave new Hopf algebroids]].)
See at _[Steenrod algebra -- Hopf algebroid structure](Steenrod+algebra#HopfAlgebraStructure)_.

The general statement is this:

+-- {: .num_lemma #SelfHomologyIsModuleOverCohomologyRing}
###### Lemma

Let $R$ be an [[E-∞ ring]] and let $A$ an [[E-∞ algebra]] over $R$.
The self-[[generalized homology]] $A^R_\bullet(A)$
is naturally a [[module]] over the [[cohomology ring]] $A_\bullet$
via applying the [[homotopy groups]] $\infty$-functor $\pi_\bullet$
to the canonical inclusion

$$
  A \stackrel{\simeq}{\rightarrow}
  A \underset{R}{\wedge} R
  \stackrel{}{\rightarrow}
  A \underset{R}{\wedge} A
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

Let $R$ be an [[E-∞ ring]] and let $A$ an [[E-∞ algebra]] over $R$.
If the the $A_\bullet$-[[module]] $A^R_\bullet(A)$ of
lemma \ref{SelfHomologyIsModuleOverCohomologyRing} is a [[flat module]],
then

1. $(A_\bullet, A_\bullet(A))$ is a [[Hopf algebroid]] over $R_\bulllet$;

1. $A^R_\bullet(X)$ is a left $A^R_\bullet(A)$-module for every $R$-[[∞-module]] $X$.

=--

This is due to ([Baker-Lazarev 01](#BakerLazarev01)), further discussed  in ([Baker-Jeanneret 02](#BakerJeanneret02)) (there expressed in terms of the presentation by [[commutative monoids]] in [[symmetric spectra]]).
A review is also in ([Ravenel, chapter 2, prop. 2.2.8](#Ravenel)).

## References

The generalization of [[commutative Hopf algebroids]] where the base is kept tcommutative while having the total algebra noncommutative, and the image of source and target maps are required to commute mutually is due Maltsiniotis; he also generalized this to [[quasi-Hopf algebra|quasi-Hopf]] version:

* [[Georges Maltsiniotis]], _Groupoïdes quantiques_, Comptes R
Rendus Acad. Sci. Paris __314__, pp. 249-252 (1992) [ps](http://people.math.jussieu.fr/~maltsin/ps/GRN-1.PS)
* G. Maltsiniotis, _Quasi-groupoïdes quantiques_ (travail en commun avec A. Bruguières), C.R. Acad. Sci. Paris __319__, pp. 933-936 (1994) [ps](http://people.math.jussieu.fr/~maltsin/ps/N-QSBIG.PS)

A [[Tannaka duality]]-type theorem relating certain subcategory of
commutative Hopf algebroids to discrete groupoids is in

* [[Laiachi EL Kaoutit]], _Representative functions on discrete groupoids and duality with Hopf algebroids_, [arxiv/1311.3109](http://arxiv.org/abs/1311.3109)

For the relation to [[groupoid convolution algebras]] see also at _[groupoid convolution algebra -- References -- Convolution Hopf algebroids](category%20algebra#ReferencesConvolutionHopfAlgebroids)_.

* [[Mark Hovey]], _Morita theory for Hopf algebroids and presheaves of groupoids_ ([arXiv:math/0105137](http://arxiv.org/abs/math/0105137))

category: algebra

[[!redirects Hopf algebroids over a commutative base]]


Hopf C-star-algebra

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Hopf $C^\ast$-algebra_ according to ([Vaes-VanDale](#VaesDaele)) is a [[C-star algebra]] equipped with [[stuff, structure, property|structure and property]] analogous to that of a [[Hopf algebra]] structure on the underlying [[associative algebra]].

A _weak $C^\ast$-Hopf algebra_ according to ([Böhm-Szlachanyi](#BohmSzlachanyi)) is a [[star-algebra|star]]-[[weak Hopf algebra]] such that has a faithful [[star-representation]] on a [[Hilbert space]].

With suitable definitions the central [[Tannaka duality]]-property of [[Hopf algebras]] (that their [[representation category]] is a [[rigid monoidal category]] with [[fiber functor]]) is lifted to the [[operator algebra]] context: the $C^\ast$-[[representation category]] of a (weak) $C^\ast$-Hopf algebra is a rigid monoidal _[[C-star-category]]_ with [[fiber functor]]. ([Böhm-Szlachanyi](#BohmSzlachanyi)).

## References

$C^\ast$ algebras equipped with a suitable [[coproduct]], but without an [[antipode]] -- hence just $C^\ast$-[[bialgebras]] -- , are considered in

* S. Baaj, [[Georges Skandalis]], _Unitaires multiplicatifs et dualité pour les produits croisés de $C^\ast$-algèbres. Ann. scient.  Ec. Norm. Sup., 4e série, 26 (1993), 425–488.

* J.-M. Vallin, _$C^\ast$-algèbres de Hopf et $C^\ast$-algèbres de Kac._ Proc. London Math. Soc. (3)50 (1985), 131–174.

The issue of how to add the definition of the [[antipode]] in the $C^\ast$-context is discussed in

* Stefaan Vaes, Alfons Van Daele, _Hopf $C^\ast$-algebras_, Proc. London Math. Soc. (3) 82, 2001, 337-384. ([arXiv:math/9907030](http://arxiv.org/abs/math/9907030))
 {#VaesDaele}

Weak $C^\ast$-Hopf algebras and their [[C-star categories]] of [[representations]] are discussed in

* [[Gabi Böhm]], [[Kornel Szlachanyi]], _A Coassociative C*-Quantum Group with Non-Integral Dimensions_ ([arXiv:q-alg/9509008](http://arxiv.org/abs/q-alg/9509008))

  _Weak $C^\ast$-Hopf  algebras_ ([pdf](http://matwbn.icm.edu.pl/ksiazki/bcp/bcp40/bcp4011.pdf))
 {#BohmSzlachanyi}

[[!redirects Hopf C-star-algebras]]

[[!redirects Hopf C*-algebra]]
[[!redirects Hopf C*-algebras]]

[[!redirects Hopf C-star algebra]]
[[!redirects Hopf C-star algebras]]

[[!redirects Hopf-C*-algebra]]
[[!redirects Hopf-C*-algebras]]

[[!redirects Hopf-C-star algebra]]
[[!redirects Hopf-C-star algebras]]

[[!redirects C-star Hopf algebra]]
[[!redirects C-star Hopf algebras]]
[[!redirects C* Hopf algebra]]
[[!redirects C* Hopf algebras]]
[[!redirects C-* Hopf algebra]]
[[!redirects C-* Hopf algebras]]

[[!redirects weak C* Hopf algebra]]
[[!redirects weak C* Hopf algebras]]
[[!redirects weak C-* Hopf algebra]]
[[!redirects weak C-* Hopf algebras]]


Hopf construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Let $X$ be an [[H-space]].  The _Hopf construction_ ([Hopf 35](#Hopf35)) on $X$ is a [[fibration]]

$$
  X \hookrightarrow X\ast X \to \Sigma X
$$

whose [[fiber]] is $X$, whose base space is the [[suspension]] of $X$, and whose total space is the [[join of topological spaces|join]] of $X$ with itself. ([Stasheff 70, chapter 1](#Stasheff70)).

Specialized to $X$ the [[sphere]] of [[dimension]] [[0-sphere|0]], [[1-sphere|1]], [[3-sphere|3]], or [[7-sphere|7]], the Hopf construction yields the _[Hopf fibrations](#HopfFibrations)_. (And by the [[Hopf invariant one theorem]] these are the only dimensions for in which spheres are H-spaces.)

## Definition
 {#Definition}

+-- {: .num_defn #SuspensionJoin}
###### Definition

Write $I \coloneqq [0,1]$ for the unit interval, regarded as a [[topological space]].

Let $X,Y $ be [[topological spaces]].

1. The [[suspension]] $\Sigma X$ is the [[quotient space]]

   $$
     \Sigma X \coloneqq (X \times I)_{/\sim}
   $$

   by the [[equivalence relation]]  given by

   $$
     (x_1,0) \sim (x_2,0) \;\,,\;\;  (x_1, 1) \sim (x_2, 1) \;\;\; \forall x_1,x_2 \in X
   $$

1. The [[join of topological spaces|join]] $X \ast Y$ is the [[quotient space]]

   $$
      X \ast Y \coloneqq (X \times I \times Y)_{/\sim}
   $$

   by the [[equivalence relation]]

   $$
     (x, 0, y_1) \simeq (x,0,y_2) \;\;,\;\; (x_1,1,y) \sim (x_2, 1, y)
     \,.
   $$

=--

+-- {: .num_defn #HopfConstruction}
###### Definition

Given a [[continuous function]] of the form

$$
  f \colon X \times Y \longrightarrow Z
$$

its _Hopf construction_ is the continuous function

$$
  H_f \colon X \ast Y \longrightarrow \Sigma Z
$$

out of the [[join of topological spaces|join]] into the [[suspension]], given in the coordinates of def. \ref{SuspensionJoin} by

$$
  H_f \colon (x,t,y) \mapsto (f(x,y), t)
  \,.
$$

=--

## Properties
 {#Properties}

### Homotopy fiber
  {#HomotopyFiber}

If $X$ is an "grouplike H-space", in that it is a topological [[magma]] such that left multiplication acts by [[weak homotopy equivalences]], then the [[homotopy fiber]] of the Hopf construction $X \ast X \to \Sigma X$ over any point is
[[weak homotopy equivalence|weakly homotopy equivalent]] to $X$ ([here](https://nforum.ncatlab.org/discussion/3391/hopf-fibration/?Focus=76088#Comment_76088)).

Beware that it may not generally be true that the ordinary [[fibers]] of the Hopf construction are [[weak homotopy equivalence|weakly homotopy equivalent]] to the [[homotopy fibers]], see also the discussion of [[quasifibrations]] [below](#RealizationAsAQuasiFibration). But in the classical examples it Happens to be the case, see at [[Hopf fibration]].

### Relation to Hopf invariant

Consider $X = S^{n-1}$ a [[sphere]]

+-- {: .num_prop #HopfInvariantFromProductOfDegrees}
###### Proposition

Given a [[continuous function]]

$$
  f \colon S^{n-1}\times S^{n-1} \longrightarrow S^{n-1}
$$

the [[degree of a continuous function|degrees]]

$$
  \alpha \coloneqq deg(f(x,-)) \;\;\; \beta \coloneqq deg(f(-,x))
$$

are independent of the choice of $x \in S^{n-1}$. The [[Hopf invariant]]  $h$ of the Hopf construction $H_f$ of $f$, def. \ref{HopfConstruction}, is the product of these two:

$$
  h(H_f) = \alpha \beta
  \,.
$$

=--

([Mosher-Tangora, exercises to section 4, page 38](#MosherTangora))

### Realization as a quasi-fibration
 {#RealizationAsAQuasiFibration}

Beware that [Stasheff 70, theorem 1.2](#Stasheff70) claims that Sugawara claimed that the Hopf construction for any [[CW-complex|CW]] H-space is necessarily a [[quasifibration]]. But it seems ([here](https://nforum.ncatlab.org/discussion/3391/hopf-fibration/?Focus=75986#Comment_75986)) that Sugawara never actually claimed this and also ([here](https://nforum.ncatlab.org/discussion/3391/hopf-fibration/?Focus=75985#Comment_75985)) that it is not actually the case.

A different but homotopy-equivalent realization of the Hopf construction, which over grouplike H-spaces is guaranteed to be a [[quasifibration]], is maybe given in [Dold-Lashof 59](#DoldLashof59), see also [Stasheff 70, theorem 1.4](#Stasheff70).

## Examples

### Hopf fibrations
 {#HopfFibrations}

When $X$ is a [[sphere]] that is an $H$-space, namely, one of the [[groups]] $S^0 = \mathbb{Z}/2$ the [[group of order 2]], $S^1 = U(1)$ the [[circle group]], the [[3-sphere]] [[special unitary group]] $S^3 = SU(2)$;  or the [[7-sphere]] $S^7$ with its [[Moufang loop]] structure, then the Hopf construction produces the four _[[Hopf fibrations]]_:

1. $S^0 \hookrightarrow S^1 \to S^1 $ -- [[real Hopf fibration]]
1. $ S^1 \hookrightarrow S^3 \to S^2 $ -- [[complex Hopf fibration]]
1. $ S^3 \hookrightarrow S^7 \to S^4 $ -- [[quaternionic Hopf fibration]]
1. $ S^7 \hookrightarrow S^{15} \to S^8 $ -- [[octonionic Hopf fibration]]

In detail, let $A \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\}$
be one of the real [[normed division algebras]] and write

$$
  n \coloneqq dim_{\mathbb{R}}(A) \in \{1,2,4,8\}
$$

for its [[dimension]] as a real vector space. Then the $S^{n-1}$-sphere may be identified with the subspace of unit norm elements in $A$:

$$
  S^{n-1} \simeq
  \left\{
    x \in A
    \,;
    {\vert x\vert}^2 = 1
  \right\}
  \,.
$$

Consider then the pairing map

$$
  f \colon S^{n-1}\times S^{n-1} \longrightarrow S^{n-1}
$$

which is the restriction to these unit norm elements of the product in $A$:

$$
  f \colon (x,y) \mapsto x \cdot y
$$

This is well defined by the very property that for [[normed division algebras]] the norm is multiplicative.

Accordingly, the [[join of topological spaces|join]] of two such spheres is naturally parameterized as follows

$$
  S^{n-1}\ast S^{n-1}
  =
  (S^{n-1}\times I \times S^{n-1})_{/\sim}
  \simeq
  \left\{
    (x,t,y)
    \,,
    {\vert x \vert}^2 = 2t
    \,,\;
    {\vert y\vert}^2 = 2 - 2t
  \right\}
$$

which makes manifest that

$$
  S^{n-1} \ast S^{n-1} \simeq S^{2n-1}
$$

Similarly, the [[suspension]] is parameterized by

$$
  \Sigma S^{n-1}
  =
  (S^{n-1}\times I)_{/\sim}
  \simeq
  \left\{
    (z,t)
    \,,\;
    {\vert z \vert}^2 + (1 - 2t)^2   = 1
  \right\}
$$

where we take $I = [0,1]$ and $t \in I$. This makes manifest that

$$
  \Sigma S^{n-1} \simeq S^n
  \,.
$$

Moreover, in this parameterization the Hopf construction, def. \ref{HopfConstruction}, which is given by

$$
  (x,y) \mapsto x \cdot \overline{y}
$$

manifestly gives the Hopf fibration map.

Notice that it is again the multiplicativity of the norm in division algebras which makes this work: if  ${\vert x \vert}^2 = 2t$ and ${\vert y\vert}^2 = 2 - 2t$ then it follows that

$$
  \begin{aligned}
     {\vert x \cdot \overline{y}\vert}^2 + (1- 2t)^2
     & =
     {\vert x \vert}^2 {\vert y \vert}^2
     + (1-2t)^2
     \\
     & = 2t (2-2t) + (1 - 2t)^2
     \\
     & = 1
  \end{aligned}
  \,,
$$

hence that indeed we have a well-defined map like so:

$$
  \array{
      S^7 & \longrightarrow & S^4
      \\
     \left\{
       (x,t,y)
        \,,
       {\vert x \vert}^2 = 2t
        \,,\;
        {\vert y\vert}^2 = 2 - 2t
     \right\}
     &\stackrel{{(x,y) \mapsto z \coloneqq x \cdot \overline{y}}\atop{t \mapsto t}}{\longrightarrow}&
    \left\{
       (z,t)
       \,,\;
       {\vert z \vert}^2 + (1 - 2t)^2   = 1
    \right\}
  }
  \,.
$$

## See also

* [[Hopf construction in homotopy type theory]]

## References

The original sources are

* {#Hopf35} [[Heinz Hopf]], _Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension_, Fund. Math. 25: 427–440 (1935) ([Euclid](https://eudml.org/doc/212801))

* [[George Whitehead]], _On the homotopy groups of spheres and rotation groups_, Annals of Mathematics. Second Series 43 (4): 634–640, (1942) ([JSTOR](http://www.jstor.org/stable/1968956))

* {#DoldLashof59} [[Albrecht Dold]], [[Richard Lashof]], _Principal quasifibrations and fibre homotopy equivalence of bundles_, Illinois J. Math. Volume 3, Issue 2 (1959), 285-305 ([euclid:1255455128](https://projecteuclid.org/euclid.ijm/1255455128))

Review inclides

* {#Stasheff70} [[Jim Stasheff]], chapter 1 in  _H-Spaces from a Homotopy point of view_, Lecture Notes in Mathematics Volume 161 1970

Textbook accounts include

* {#MosherTangora} [[Robert Mosher]], [[Martin Tangora]], p. 38 of _Cohomology Operations and Application in Homotopy Theory_, Harper and Row (1968) ([pdf](http://math.jhu.edu/~anakade1/notes/Dec%202014/Mosher%20and%20Tangora%20Notes/mosher-tangora.pdf))

* {#AguilarGitlerPrieto02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 10.6 of _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

See also

* {#Moreno04} Guillermo Moreno, _Hopf construction map in higher dimensions_ ([arXiv:math/0404172](https://arxiv.org/abs/math/0404172))

*  Feza Guersey, Chia-Hsiung Tze, (4b.2) in _On the role of Division, Jordan and Related algebras in Particle Physics_

Discussion of the situation in [[parameterized homotopy theory]] includes

* {#CookCrabb93} A. L. Cook, M.C. Crabb, _Fiberwise Hopf structures on sphere bundles_, J. London Math. Soc. (2) 48 (1993) 365-384 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/crabbcook.pdf))

* {#Iriye95} Kouyemon Iriye, _Equivariant Hopf structures on a sphere_, J. Math. Kyoto Univ. Volume 35, Number 3 (1995), 403-412 ([Euclid](http://projecteuclid.org/euclid.kjm/1250518704))

See also

* Wikipedia, _[Hopf construction](https://en.wikipedia.org/wiki/Hopf_construction)_

[[!redirects Hopf constructions]]

[[!redirects Hopf map]]
[[!redirects Hopf maps]]


Hopf construction in homotopy type theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In classical [[algebraic topology]] we have four [[Hopf fibrations]] (of spheres):

1. $S^0 \hookrightarrow S^1 \to S^1$ The [[real Hopf fibration]]
2. $S^1 \hookrightarrow S^3 \to S^2$ The usual [[complex Hopf fibration]]
3. $S^3 \hookrightarrow S^7 \to S^4$ The [[quaternionic Hopf fibration]]
4. $S^7 \hookrightarrow S^15 \to S^8$ The [[octonionic Hopf fibration]]

These can be constructed in [[homotopy type theory]] as part of a more general construction:

An [[H-space]] [[structure]] on a [[pointed type]] $A$ gives a [[fibration]] over $\Sigma A$ via the [[Hopf construction]]. This fibration can be written classically as: $A \to A\ast A \to \Sigma A$ where $A\ast A$ is the join of $A$ and $A$. This is all done in the [[HoTT book]]. Note that $\Sigma A$ can be written as a [[homotopy pushout]] $\Sigma A \coloneqq \mathbf 1 \sqcup^A \mathbf 1 $, and there is a lemma in the [[HoTT book]] allowing you to construct a fibration on a pushout (the equivalence $A \to A$ needed is simply the multiplication from the H-space $\mu(a,-)$).

Thus the problem of constructing a [[Hopf fibration]] reduces to finding a [[H-space]] structure on the [[spheres]]: the $S^1$, $S^3$ and $S^7$.

* The space $S^0=\mathbf 2(\equiv Bool)$ is not [[connected]] so we cannot perform the construction from the book on it. However it is very easy to construct a [[family]] $S^1 \to \mathcal{U}$ with fiber $Bool$ by induction on $S^1$.  (Note: loop maps to $ua(neg)$ where $neg$ is the equivalence of negation and $ua$ is the [[univalence axiom]].

* For $S^1$ [[Peter Lumsdaine]] gave the construction in 2012 and [[Guillaume Brunerie]] proved it was correct in 2013. By [[induction]] on the [[circle]] we can define the multiplication: $\mu(base)\equiv id_{S^1}$, and $ap_\mu(loop)\equiv funext(h)$ where $h : (x : S^1) \to (x = x)$ is also defined by [[circle]] [[induction]]: $h(base) = loop$ and $ap_h(loop) = refl$. $funext$ denotes [[functional extensionality]].

* For $S^3$ [Buchholtz-Rijke 16](#BuchholtzRijke16) solved this through a [[homotopy theory|homotopy  theoretic]] version of the [[Cayley-Dickson construction]]. This has been formalised in Lean.

* For [[7-sphere|$S^7$]] this is still an open problem.

It is still an open problem to show that these are the only spheres to have a H-space structure. This would be done by showing these are the only spheres with [[Hopf invariant]] $1$ which has been defined in [Brunerie 2016](#Brunerie16)

## Related concepts

* [[homotopy groups of spheres in homotopy type theory]]

* [[cohomology in homotopy type theory]]

* [[internal category in homotopy type theory]]

## References

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

* {#Brunerie16} [[Guillaume Brunerie]], *[[On the homotopy groups of spheres in homotopy type theory]]* $[$[arxiv:1606.05916](https://arxiv.org/abs/1606.05916)$]$

* {#BuchholtzRijke16} [[nLab:Ulrik Buchholtz]], [[nLab:Egbert Rijke]], _The Cayley-Dickson Construction in Homotopy Type Theory_ ([arXiv:1610.01134](https://arxiv.org/abs/1610.01134))

category: homotopy theory