Editing Hopf algebra (section)

== Hopf algebras in braided monoidal categories ==
The definition of Hopf algebra is naturally extended to arbitrary [[braided monoidal category|braided monoidal categories]].{{sfn|Turaev|Virelizier|2017|loc=6.2}}{{sfn|Akbarov|2009|p=482}} A Hopf algebra in such a category <math>(C,\otimes,I,\alpha,\lambda,\rho,\gamma)</math> is a sextuple <math>(H,\nabla,\eta,\Delta,\varepsilon,S)</math> where <math>H</math> is an object in <math>C</math>, and

: <math>\nabla:H\otimes H\to H</math> (multiplication),

: <math>\eta:I\to H</math> (unit),

: <math>\Delta:H\to H\otimes H</math> (comultiplication),

: <math>\varepsilon:H\to I</math> (counit),

: <math>S:H\to H</math> (antipode)

— are morphisms in <math>C</math> such that

:1) the triple <math>(H,\nabla,\eta)</math> is a [[Monoid (category theory)|monoid]] in the monoidal category <math>(C,\otimes,I,\alpha,\lambda,\rho,\gamma)</math>, i.e. the following diagrams are commutative:{{efn|name=left-and-right-units|Here <math>\alpha_{H,H,H}:(H\otimes H)\otimes H\to H\otimes (H\otimes H)</math>, <math>\lambda_H:I\otimes H\to H</math>, <math>\rho_H:H\otimes I\to H</math> are the natural transformations of associativity, and of the left and the right units in the monoidal category <math>(C,\otimes,I,\alpha,\lambda,\rho,\gamma)</math>.}}
<div style="text-align: center;">
[[File:Minoid.png|600px|monoid in a monoidal category]]
</div>

:2) the triple <math>(H,\Delta,\varepsilon)</math> is a [[Monoid (category theory)|comonoid]] in the monoidal category <math>(C,\otimes,I,\alpha,\lambda,\rho,\gamma)</math>, i.e. the following diagrams are commutative:{{efn|name=left-and-right-units}}
<div style="text-align: center;">
[[File:Cominoid.png|600px|comonoid in a monoidal category]]
</div>

:3) the structures of monoid and comonoid on <math>H</math> are compatible: the multiplication <math>\nabla</math> and the unit <math>\eta</math> are morphisms of comonoids, and (this is equivalent in this situation) at the same time the comultiplication <math>\Delta</math> and the counit <math>\varepsilon</math> are morphisms of monoids; this means that the following diagrams must be commutative:
<div style="text-align: center;">
[[File:Multiplication-comultiplication.png|600px|coherence between multiplication and comultiplication]]
</div> 
<div style="text-align: center;">
[[File:Unit-counit.png|350px|unit and counit in bialgebras]]
</div> 
<div style="text-align: center;">
[[File:Uinit-counit-1.png|150px|unit and counit in bialgebras]]
</div>
: where <math>\lambda_I:I\otimes I\to I</math> is the left unit morphism in <math>C</math>, and <math>\theta</math> the natural transformation of functors <math>(A\otimes B)\otimes (C\otimes D)\stackrel{\theta}{\rightarrowtail} (A\otimes C)\otimes (B\otimes D)</math> which is unique in the class of natural transformations of functors composed from the structural transformations (associativity, left and right units, transposition, and their inverses) in the category <math>C</math>.

The quintuple <math>(H,\nabla,\eta,\Delta,\varepsilon)</math> with the properties 1),2),3) is called a '''bialgebra''' in the category <math>(C,\otimes,I,\alpha,\lambda,\rho,\gamma)</math>;



:4) the diagram of antipode is commutative:
<div style="text-align: center;">
[[File:Antipode-1.png|300px|unit and counit in bialgebras]]
</div>

The typical examples are the following.

* '''Groups'''. In the monoidal category <math>(\text{Set},\times,1)</math> of [[set (mathematics)|sets]] (with the [[cartesian product]] <math>\times</math> as the tensor product, and an arbitrary singletone, say, <math>1=\{\varnothing\}</math>, as the unit object) a triple <math>(H,\nabla,\eta)</math> is a [[Monoid (category theory)|monoid in the categorical sense]] if and only if it is a [[monoid|monoid in the usual algebraic sense]], i.e. if the operations <math>\nabla(x,y)=x\cdot y</math> and <math>\eta(1)</math> behave like usual multiplication and unit in <math>H</math> (but possibly without the invertibility of elements <math>x\in H</math>). At the same time, a triple <math>(H,\Delta,\varepsilon)</math> is a comonoid in the categorical sense iff <math>\Delta</math> is the diagonal operation <math>\Delta(x)=(x,x)</math> (and the operation <math>\varepsilon</math> is defined uniquely as well: <math>\varepsilon(x)=\varnothing</math>). And any such a structure of comonoid <math>(H,\Delta,\varepsilon)</math> is compatible with any structure of monoid <math>(H,\nabla,\eta)</math> in the sense that the diagrams in the section 3 of the definition always commute. As a corollary, each monoid <math>(H,\nabla,\eta)</math> in <math>(\text{Set},\times,1)</math> can naturally be considered as a bialgebra <math>(H,\nabla,\eta,\Delta,\varepsilon)</math> in <math>(\text{Set},\times,1)</math>, and vice versa. The existence of the antipode <math>S:H\to H</math> for such a bialgebra <math>(H,\nabla,\eta,\Delta,\varepsilon)</math> means exactly that every element <math>x\in H</math> has an inverse element <math>x^{-1}\in H</math> with respect to the multiplication <math>\nabla(x,y)=x\cdot y</math>. Thus, in the category of sets <math>(\text{Set},\times,1)</math> Hopf algebras are exactly [[group (mathematics)|groups]] in the usual algebraic sense.
* '''Classical Hopf algebras'''. In the special case when <math>(C,\otimes,s,I)</math> is the category of vector spaces over a given field <math>K</math>, the Hopf algebras in <math>(C,\otimes,s,I)</math> are exactly the classical Hopf algebras [[#Formal definition|described above]].
* '''Functional algebras on groups'''. The standard [[Stereotype algebra#Examples|functional algebras]] <math>{\mathcal C}(G)</math>, <math>{\mathcal E}(G)</math>, <math>{\mathcal O}(G)</math>, <math>{\mathcal P}(G)</math> (of continuous, smooth, holomorphic, regular functions) on groups are Hopf algebras in the category ('''Ste''',<math>\odot</math>) of [[stereotype space]]s,{{sfn|Akbarov|2003|loc=10.3}}
* '''Group algebras'''. The [[stereotype group algebra]]s <math>{\mathcal C}^\star(G)</math>, <math>{\mathcal E}^\star(G)</math>, <math>{\mathcal O}^\star(G)</math>, <math>{\mathcal P}^\star(G)</math> (of measures, distributions, analytic functionals and currents) on groups are Hopf algebras in the category ('''Ste''',<math>\circledast</math>) of [[stereotype space]]s.{{sfn|Akbarov|2003|loc=10.3}} These Hopf algebras are used in the [[Pontryagin duality#Dualities for non-commutative topological groups|duality theories for non-commutative groups]].{{sfn|Akbarov|2009}}