Representation theory of Hopf algebras

{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= Template:Ambox }} 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 Template:Nowrap usually denoted by juxtaposition (that is, the image of Template:Nowrap 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, Template:Nowrap, 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 Template:Nowrap of two H-modules V₁ and V₂ 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 Template:Nowrap such that for any v in Template:Nowrap and any h in 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 V₁ ⊗ V₂ 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 Template:Nowrap for all a, b in 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_H, called the trivial module, such that Template:Nowrap, V and Template:Nowrap 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 Template:Nowrap may be defined such that Template:Nowrap for all v in ε_H. The trivial module may be identified with F, with 1 being the element such that Template:Nowrap for all v. It follows that for any v in any H-module V, any c in ε_H and any h in 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 Template:Nowrap 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 with the product operation Template:Nowrap, and Template:Nowrap 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