Representation theory of Hopf algebras

In abstract algebra, a representation of a Hopf algebra is a 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 action ''H'' × ''V'' → ''V'' usually denoted by juxtaposition ( that is, the image of (''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 under ''H'' if for all ''h'' in ''H'', ''hv'' = ε(''h'')''v'', where ε is the counit of ''H''. The subset of all invariant elements of ''V'' forms a submodule of ''V''.

Categories of representations as a motivation for Hopf algebras

For an associative algebra ''H'', the tensor product ''V''₁ ⊗ ''V''₂ of two ''H''-modules ''V''₁ and ''V''₂ is a vector space, but not necessarily an ''H''-module. For the tensor product to be a functorial product operation on ''H''-modules, there must be a linear binary operation Δ : ''H'' → ''H'' ⊗ ''H'' such that for any ''v'' in ''V''₁ ⊗ ''V''₂ and any ''h'' in ''H'',

:$hv=\Delta h(v_\otimes v_)=h_v_\otimes h_v_,$

and for any ''v'' in ''V''₁ ⊗ ''V''₂ and ''a'' and ''b'' in ''H'',

:\Delta(ab)(v_\otimes v_)=(ab)v=a [v=\Delta_a[\Delta_b(v_\otimes_v_).html" ;"title=".html" ;"title="[v">[v=\Delta a[\Delta b(v_\otimes v_)">.html" ;"title="[v">[v=\Delta a[\Delta b(v_\otimes v_)(\Delta a )(\Delta b)(v_\otimes v_).

using sumless Sweedler's notation, which is somewhat like an index free form of Einstein's summation convention. This is satisfied if there is a Δ such that Δ(''ab'') = Δ(''a'')Δ(''b'') for all ''a'', ''b'' in ''H''.

For the category of ''H''-modules to be a strict monoidal category with respect to ⊗, $V_1\otimes(V_2\otimes V_3)$ and $(V_1\otimes V_2)\otimes V_3$ must be equivalent and there must be unit object ε_''H'', called the trivial module, such that ε_''H'' ⊗ ''V'', ''V'' and ''V'' ⊗ ε_''H'' are equivalent.

This means that for any ''v'' in

:$V_1\otimes(V_2\otimes V_3)=(V_1\otimes V_2)\otimes V_3$

and for ''h'' in ''H'',

:$((\operatorname\otimes \Delta)\Delta h)(v_\otimes v_\otimes v_)=h_v_\otimes h_v_\otimes h_v_=hv=((\Delta\otimes \operatorname) \Delta h) (v_\otimes v_\otimes v_).$

This will hold for any three ''H''-modules if Δ satisfies

:$(\operatorname\otimes \Delta)\Delta A=(\Delta \otimes \operatorname)\Delta A.$

The trivial module must be one-dimensional, and so an algebra homomorphism ε : ''H'' → ''F'' may be defined such that ''hv'' = ε(''h'')''v'' for all ''v'' in ε_''H''. The trivial module may be identified with ''F'', with 1 being the element such that 1 ⊗ ''v'' = ''v'' = ''v'' ⊗ 1 for all ''v''. It follows that for any ''v'' in any ''H''-module ''V'', any ''c'' in ε_''H'' and any ''h'' in ''H'',

:$(\varepsilon(h_)h_)cv=h_c\otimes h_v=h(c\otimes v)=h(cv)=(h_\varepsilon(h_))cv.$

The existence of an algebra homomorphism ε satisfying

:$\varepsilon(h_)h_ = h = h_\varepsilon(h_)$

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 ''S'' : ''H'' → ''H'' such that for any ''h'' in ''H'', ''x'' in ''V'' and ''y'' in ''V*'',

:$\langle y, S(h)x\rangle = \langle hy, x \rangle.$

where $\langle\cdot,\cdot\rangle$ is the usual pairing of dual vector spaces. If the map $\varphi:V\otimes V^*\rightarrow \varepsilon_H$ induced by the pairing is to be an ''H''-homomorphism, then for any ''h'' in ''H'', ''x'' in ''V'' and ''y'' in ''V*'',

:$\varphi\left(h(x\otimes y)\right)=\varphi\left(x\otimes S(h_)h_y\right)=\varphi\left(S(h_)h_x\otimes y\right)=h\varphi(x\otimes y)=\varepsilon(h)\varphi(x\otimes y),$

which is satisfied if

:$S(h_)h_=\varepsilon(h)=h_S(h_)$

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 with the product operation μ : ''A'' ⊗ ''A'' → ''A'', and 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
Hopf algebras
Representation theory