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,

h v = Δ h ( v ( 1 ) ⊗ v ( 2 ) ) = h ( 1 ) v ( 1 ) ⊗ h ( 2 ) v ( 2 ) , {\displaystyle hv=\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,

Δ ( a b ) ( v ( 1 ) ⊗ v ( 2 ) ) = ( a b ) v = a [ b [ v ] ] = Δ a [ Δ b ( v ( 1 ) ⊗ v ( 2 ) ) ] = ( Δ a ) ( Δ b ) ( v ( 1 ) ⊗ v ( 2 ) ) . {\displaystyle \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)}).}

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 ⊗ ( V 2 ⊗ V 3 ) {\displaystyle V_{1}\otimes (V_{2}\otimes V_{3})} and ( V 1 ⊗ V 2 ) ⊗ V 3 {\displaystyle (V_{1}\otimes V_{2})\otimes V_{3}} 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

V 1 ⊗ ( V 2 ⊗ V 3 ) = ( V 1 ⊗ V 2 ) ⊗ V 3 {\displaystyle V_{1}\otimes (V_{2}\otimes V_{3})=(V_{1}\otimes V_{2})\otimes V_{3}}

and for h in H,

( ( id ⊗ Δ ) Δ h ) ( v ( 1 ) ⊗ v ( 2 ) ⊗ v ( 3 ) ) = h ( 1 ) v ( 1 ) ⊗ h ( 2 ) ( 1 ) v ( 2 ) ⊗ h ( 2 ) ( 2 ) v ( 3 ) = h v = ( ( Δ ⊗ id ) Δ h ) ( v ( 1 ) ⊗ v ( 2 ) ⊗ v ( 3 ) ) . {\displaystyle ((\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)}).}

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

( id ⊗ Δ ) Δ A = ( Δ ⊗ id ) Δ A . {\displaystyle (\operatorname {id} \otimes \Delta )\Delta A=(\Delta \otimes \operatorname {id} )\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 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 ) ) c v = h ( 1 ) c ⊗ h ( 2 ) v = h ( c ⊗ v ) = h ( c v ) = ( h ( 1 ) ε ( h ( 2 ) ) ) c v . {\displaystyle (\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.}

The existence of an algebra homomorphism ε satisfying

ε ( h ( 1 ) ) h ( 2 ) = h = h ( 1 ) ε ( h ( 2 ) ) {\displaystyle \varepsilon (h_{(1)})h_{(2)}=h=h_{(1)}\varepsilon (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 ⟩ = ⟨ h y , x ⟩ . {\displaystyle \langle y,S(h)x\rangle =\langle hy,x\rangle .}

where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the usual pairing of dual vector spaces. If the map φ : V ⊗ V ∗ → ε H {\displaystyle \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*,

φ ( h ( x ⊗ y ) ) = φ ( x ⊗ S ( h ( 1 ) ) h ( 2 ) y ) = φ ( S ( h ( 2 ) ) h ( 1 ) x ⊗ y ) = h φ ( x ⊗ y ) = ε ( h ) φ ( x ⊗ y ) , {\displaystyle \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),}

which is satisfied if

S ( h ( 1 ) ) h ( 2 ) = ε ( h ) = h ( 1 ) S ( h ( 2 ) ) {\displaystyle S(h_{(1)})h_{(2)}=\varepsilon (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