Representation theory of Hopf algebras

In abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra.[1][2] 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.

## Contents

* 1 Properties
* 2 Categories of representations as a motivation for Hopf algebras
* 3 Representations on an algebra
* 4 See also
* 5 References

## 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=Δh(v(1)⊗v(2))=h(1)v(1)⊗h(2)v(2),

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

Δ(ab)(v(1)⊗v(2))=(ab)v=a[b[v]]=Δa[Δb(v(1)⊗v(2))]=(Δa)(Δb)(v(1)⊗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 monoidal category with respect to ⊗, V1⊗(V2⊗V3) and (V1⊗V2)⊗V3 must be equivalent and there must be unit object εH, called the trivial module, such that εH ⊗ V, V and V ⊗ εH are equivalent.

This means that for any v in

V1⊗(V2⊗V3)=(V1⊗V2)⊗V3

and for h in H,

((id⁡⊗Δ)Δh)(v(1)⊗v(2)⊗v(3))=h(1)v(1)⊗h(2)(1)v(2)⊗h(2)(2)v(3)=hv=((Δ⊗id⁡)Δh)(v(1)⊗v(2)⊗v(3)).

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

(id⁡⊗Δ)ΔA=(Δ⊗id⁡)ΔA.

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 εH. 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 εH and any h in H,

(ε(h(1))h(2))cv=h(1)c⊗h(2)v=h(c⊗v)=h(cv)=(h(1)ε(h(2)))cv.

The existence of an algebra homomorphism ε satisfying

ε(h(1))h(2)=h=h(1)ε(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 ⟨⋅,⋅⟩ is the usual pairing of dual vector spaces. If the map φ:V⊗V*->εH induced by the pairing is to be an H-homomorphism, then for any h in H, x in V and y in V*,

φ(h(x⊗y))=φ(x⊗S(h(1))h(2)y)=φ(S(h(2))h(1)x⊗y)=hφ(x⊗y)=ε(h)φ(x⊗y),

which is satisfied if

S(h(1))h(2)=ε(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

## References

1. ↑ "Introduction to Hopf algebras and representations". AALTO. https://math.aalto.fi/~kkytola/files_KK/lectures_files_KK/Hopf-lecture_notes.pdf.
2. ↑ Roggenkamp, Klaus W.; Stefanescu, Mirela (2001-08-31). Algebra - Representation Theory. Springer Science & Business Media. ISBN 978-0-7923-7113-7. https://www.google.com/books/edition/Algebra_Representation_Theory/dtsUcQTHPqYC?hl=en&gbpv=1&dq=Representation+theory+of+Hopf+algebras&printsec=frontcover. Retrieved 2025-12-02.

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