Representation of a Hopf algebra

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In abstract algebra, a representation of a Hopf algebra is a representation of the underlying associative algebra of a Hopf 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.

1 Properties
2 Categories of representations as a motivation for Hopf algebras
3 Algebra representations
4 See also

[edit] 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, $hv=\varepsilon(h)v$ , where ε is the counit of H. The subset of all invariant elements of V forms a submodule of V.

[edit] Categories of representations as a motivation for Hopf algebras

For an associative algebra H, the tensor product $V_1\otimes V_2$ 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 $\Delta:H\rightarrow H\otimes H\,$ such that for any v in $V_1\otimes V_2$ and any h in H,

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

and for any v in $V_1\otimes V_2$ and a and b in H,

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

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

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

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

This will hold for any three H-modules if $Δ$ satisfies $(\operatorname{id}\otimes \Delta)\Delta A=(\Delta \otimes \operatorname{id})\Delta A$ .

The trivial module must be one dimensional, and so an algebra homomorphism $\varepsilon:H\rightarrow F$ may be defined such that $hv=\varepsilon(h)v$ for all v in $\varepsilon_H$ . The trivial module may be identified with F, with 1 being the element such that $1\otimes v=v=v\otimes 1$ for all v. It follows that for any v in any H-module V, any c in $\varepsilon_H$ and any h in H,

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

The existence of an algebra homomorphism ε satisfying $\varepsilon(h_{(1)})h_{(2)} = h = h_{(1)}\varepsilon(h_{(2)})$ 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 $\varepsilon$ satisfying these conditions. This is the motivation for the definition of a bialgebra, where $Δ$ is called the comultiplication and $\varepsilon$ 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 $S:H\rightarrow H$ 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^*,

$\varphi\left(h(x\otimes y)\right)=\varphi\left(x\otimes S(h_{(1)})h_{(2)}y\right)=\varphi\left(S(h_{(2)})h_{(1)}x\otimes y\right)=h\varphi(x\otimes y)=\varepsilon(h)\varphi(x\otimes y),$

which is satisfied if $S(h_{(1)})h_{(2)}=\varepsilon(h)=h_{(1)}S(h_{(2)})$ 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.

[edit] Algebra representations

A Hopf algebra also has algebra representations with additional structure.

Let H be a Hopf algebra. If A is an algebra with the product operation $\mu:A\otimes A\rightarrow A$ , then a linear map $\rho:H\otimes A\rightarrow A$ is an algebra representation of H if, in addition to being a (vector space) rep of H, μ is an H-intertwiner. Recall that $A\otimes A$ is also a vector space rep of H. If A happens to be unital, we would require that there is an H-intertwiner from ε_H to A such that the 1 of ε_H maps to the unit of A.