Editing Representation theory of Hopf algebras

{{Unreferenced|date=December 2009}}
In [[abstract algebra]], a '''representation of a [[Hopf algebra]]''' is a [[algebra representation|representation]] of its underlying [[associative algebra]]. That is, a representation of a Hopf algebra ''H'' over a field ''K'' is a ''K''-[[vector space]] ''V'' with an [[Group action (mathematics)|action]] {{nowrap|''H'' × ''V'' → ''V''}} usually denoted by juxtaposition (that is, the image of {{nowrap|(''h'', ''v'')}} is written ''hv''). The vector space ''V'' is called an ''H''-module.

== Properties ==
The module structure of a representation of a Hopf algebra ''H'' is simply its structure as a module for the underlying associative algebra. The main use of considering the additional structure of a Hopf algebra is when considering all ''H''-modules as a category. The additional structure is also used to define invariant elements of an ''H''-module ''V''. An element ''v'' in ''V'' is [[Invariant (mathematics)|invariant]] under ''H'' if for all ''h'' in ''H'', {{nowrap|1=''hv'' = ''ε''(''h'')''v''}}, where ''ε'' is the [[counit]] of ''H''. The subset of all invariant elements of ''V'' forms a submodule of&nbsp;''V''.

== Categories of representations as a motivation for Hopf algebras ==
For an associative algebra ''H'', the [[tensor product]] {{nowrap|''V''<sub>1</sub> ⊗ ''V''<sub>2</sub>}} of two ''H''-modules ''V''<sub>1</sub> and ''V''<sub>2</sub> is a vector space, but not necessarily an ''H''-module. For the tensor product to be a [[functor]]ial product operation on ''H''-modules, there must be a linear binary operation {{nowrap|Δ : ''H'' → ''H'' ⊗ ''H''}} such that for any ''v'' in {{nowrap|''V''<sub>1</sub> ⊗ ''V''<sub>2</sub>}} and any ''h'' in&nbsp;''H'',
: <math>hv=\Delta h(v_{(1)}\otimes v_{(2)})=h_{(1)}v_{(1)}\otimes h_{(2)}v_{(2)},</math>
and for any ''v'' in ''V''<sub>1</sub> ⊗ ''V''<sub>2</sub> and ''a'' and ''b'' in ''H'',
: <math>\Delta(ab)(v_{(1)}\otimes v_{(2)})=(ab)v=a[b[v]]=\Delta a[\Delta b(v_{(1)}\otimes v_{(2)})]=(\Delta a )(\Delta b)(v_{(1)}\otimes v_{(2)}).</math>
using sumless [[Sweedler's notation]], which is somewhat like an index free form of the [[Einstein summation convention]]. This is satisfied if there is a Δ such that {{nowrap|1=Δ(''ab'') = Δ(''a'')Δ(''b'')}} for all ''a'', ''b'' in&nbsp;''H''.

For the category of ''H''-modules to be a strict [[monoidal category]] with respect to ⊗, <math>V_1\otimes(V_2\otimes V_3)</math> and <math>(V_1\otimes V_2)\otimes V_3</math> must be equivalent and there must be unit object ''ε''<sub>''H''</sub>, called the trivial module, such that {{nowrap|''ε''<sub>''H''</sub> ⊗ ''V''}}, ''V'' and {{nowrap|''V'' ⊗ ε<sub>''H''</sub>}} are equivalent. 

This means that for any ''v'' in 
: <math>V_1\otimes(V_2\otimes V_3)=(V_1\otimes V_2)\otimes V_3</math> 
and for ''h'' in ''H'',
: <math>((\operatorname{id}\otimes \Delta)\Delta h)(v_{(1)}\otimes v_{(2)}\otimes v_{(3)})=h_{(1)}v_{(1)}\otimes h_{(2)(1)}v_{(2)}\otimes h_{(2)(2)}v_{(3)}=hv=((\Delta\otimes \operatorname{id}) \Delta h) (v_{(1)}\otimes v_{(2)}\otimes v_{(3)}).</math>

This will hold for any three ''H''-modules if Δ satisfies 
: <math>(\operatorname{id}\otimes \Delta)\Delta A=(\Delta \otimes \operatorname{id})\Delta A.</math>

The trivial module must be one-dimensional, and so an [[algebra homomorphism]] {{nowrap|''ε'' : ''H'' → ''F''}} may be defined such that {{nowrap|1=''hv'' = ''ε''(''h'')''v''}} for all ''v'' in ''ε''<sub>''H''</sub>. The trivial module may be identified with ''F'', with 1 being the element such that {{nowrap|1=1 ⊗ ''v'' = ''v'' = ''v'' ⊗ 1}} for all ''v''. It follows that for any ''v'' in any ''H''-module ''V'', any ''c'' in ''ε''<sub>''H''</sub> and any ''h'' in&nbsp;''H'',
: <math>(\varepsilon(h_{(1)})h_{(2)})cv=h_{(1)}c\otimes h_{(2)}v=h(c\otimes v)=h(cv)=(h_{(1)}\varepsilon(h_{(2)}))cv.</math>

The existence of an algebra homomorphism ''ε'' satisfying 
: <math>\varepsilon(h_{(1)})h_{(2)} = h = h_{(1)}\varepsilon(h_{(2)})</math> 
is a sufficient condition for the existence of the trivial module. 

It follows that in order for the category of ''H''-modules to be a monoidal category with respect to the tensor product, it is sufficient for ''H'' to have maps Δ and ''ε'' satisfying these conditions. This is the motivation for the definition of a [[bialgebra]], where Δ is called the [[comultiplication]] and ''ε'' is called the [[counit]].

In order for each ''H''-module ''V'' to have a [[dual representation]] ''V'' such that the underlying vector spaces are dual and the operation * is functorial over the monoidal category of ''H''-modules, there must be a linear map {{nowrap|''S'' : ''H'' → ''H''}} such that for any ''h'' in ''H'', ''x'' in ''V'' and ''y'' in ''V''*,
: <math>\langle y, S(h)x\rangle = \langle hy, x \rangle.</math>
where <math>\langle\cdot,\cdot\rangle</math> is the usual [[pairing]] of dual vector spaces. If the map <math>\varphi:V\otimes V^*\rightarrow \varepsilon_H</math> induced by the pairing is to be an ''H''-homomorphism, then for any ''h'' in ''H'', ''x'' in ''V'' and ''y'' in ''V''*,
: <math>\varphi\left(h(x\otimes y)\right)=\varphi\left(x\otimes S(h_{(1)})h_{(2)}y\right)=\varphi\left(S(h_{(2)})h_{(1)}x\otimes y\right)=h\varphi(x\otimes y)=\varepsilon(h)\varphi(x\otimes y),</math>
which is satisfied if 
: <math>S(h_{(1)})h_{(2)}=\varepsilon(h)=h_{(1)}S(h_{(2)})</math> 
for all ''h'' in ''H''.

If there is such a map ''S'', then it is called an ''antipode'', and ''H'' is a Hopf algebra. The desire for a monoidal category of modules with functorial tensor products and dual representations is therefore one motivation for the concept of a Hopf algebra.

== Representations on an algebra ==
A Hopf algebra also has representations which carry additional structure, namely they are algebras.

Let ''H'' be a Hopf algebra.  If ''A'' is an [[algebra over a field|algebra]] with the product operation {{nowrap|''μ'' : ''A'' ⊗ ''A'' → ''A''}}, and {{nowrap|''ρ'' : ''H'' ⊗ ''A'' → ''A''}} is a representation of ''H'' on ''A'', then ''ρ'' is said to be a representation of ''H'' on an algebra if ''μ'' is ''H''-[[equivariant]]. As special cases, Lie algebras, Lie superalgebras and groups can also have representations on an algebra.

== See also ==
* [[Tannaka–Krein reconstruction theorem]]

{{DEFAULTSORT:Representation Theory Of Hopf Algebras}}
[[Category:Hopf algebras]]
[[Category:Representation theory]]