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 V₁ ⊗ V₂ 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 Δ : H → H ⊗ H such that for any v in V₁ ⊗ V₂ and any h in H,

$hv=\mathrm{\Delta }h(v_{(1)}\otimes v_{(2)})=h_{(1)}{v}_{(1)}\otimes h_{(2)}{v}_{(2)},$

and for any v in V₁ ⊗ V₂ and a and b in H,

$\mathrm{\Delta }(ab)(v_{(1)}\otimes v_{(2)})=(ab)v=a[b[v]]=\mathrm{\Delta }a[\mathrm{\Delta }b(v_{(1)}\otimes v_{(2)})]=(\mathrm{\Delta }a)(\mathrm{\Delta }b)(v_{(1)}\otimes v_{(2)}).$

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 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,

$((\mathrm{id}\otimes \mathrm{\Delta })\mathrm{\Delta }h)(v_{(1)}\otimes v_{(2)}\otimes v_{(3)})=h_{(1)}{v}_{(1)}\otimes h_{(2)(1)}{v}_{(2)}\otimes h_{(2)(2)}{v}_{(3)}=hv=((\mathrm{\Delta }\otimes \mathrm{id})\mathrm{\Delta }h)(v_{(1)}\otimes v_{(2)}\otimes v_{(3)}).$

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

$(\mathrm{id}\otimes \mathrm{\Delta })\mathrm{\Delta }A=(\mathrm{\Delta }\otimes \mathrm{id})\mathrm{\Delta }A.$

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,

$(\epsilon (h_{(1)})h_{(2)})cv=h_{(1)}c\otimes h_{(2)}v=h(c\otimes v)=h(cv)=(h_{(1)}\epsilon (h_{(2)}))cv.$

The existence of an algebra homomorphism ε satisfying

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

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*,

$⟨y,S(h)x⟩=⟨hy,x⟩.$

where $⟨\cdot ,\cdot ⟩$ is the usual pairing of dual vector spaces. If the map $\phi :V\otimes V^{*}\to {\epsilon }_H$ induced by the pairing is to be an H-homomorphism, then for any h in H, x in V and y in V*,

$\phi (h(x\otimes y))=\phi (x\otimes S(h_{(1)})h_{(2)}y)=\phi (S(h_{(2)})h_{(1)}x\otimes y)=h\phi (x\otimes y)=\epsilon (h)\phi (x\otimes y),$

which is satisfied if

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

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