# 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", $hv=varepsilon\left(h\right)v$, 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 $V_1otimes V_2$ of two "H"-modules "V"1 and "V"2 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 $Delta:H ightarrow Hotimes H,$ such that for any "v" in $V_1otimes V_2$ and any "h" in "H",

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

and for any "v" in $V_1otimes V_2$ and "a" and "b" in "H",

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

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 &Delta; such that $Delta\left(ab\right)=Delta\left(a\right)Delta\left(b\right)$ for all "a" and "b" in "H".

For the category of "H"-modules to be a strict monoidal category with respect to $otimes$, $V_1otimes\left(V_2otimes V_3\right)$ and $\left(V_1otimes V_2\right)otimes V_3$ must be equivalent and there must be unit object $varepsilon_H$, called the trivial module, such that $varepsilon_Hotimes V$, "V" and $Votimes varepsilon_H$ are equivalent.

This means that for any "v" in $V_1otimes\left(V_2otimes V_3\right)=\left(V_1otimes V_2\right)otimes V_3$ and "h" in "H",:$\left(\left(operatorname\left\{id\right\}otimes Delta\right)Delta h\right)\left(v_\left\{\left(1\right)\right\}otimes v_\left\{\left(2\right)\right\}otimes v_\left\{\left(3\right)\right\}\right)=h_\left\{\left(1\right)\right\}v_\left\{\left(1\right)\right\}otimes h_\left\{\left(2\right)\left(1\right)\right\}v_\left\{\left(2\right)\right\}otimes h_\left\{\left(2\right)\left(2\right)\right\}v_\left\{\left(3\right)\right\}=hv=\left(\left(Deltaotimes operatorname\left\{id\right\}\right)Delta h\right)\left(v_\left\{\left(1\right)\right\}otimes v_\left\{\left(2\right)\right\}otimes v_\left\{\left(3\right)\right\}\right).$This will hold for any three "H"-modules if $Delta$ satisfies $\left(operatorname\left\{id\right\}otimes Delta\right)Delta A=\left(Delta otimes operatorname\left\{id\right\}\right)Delta A$.

The trivial module must be one dimensional, and so an algebra homomorphism $varepsilon:H ightarrow F$ may be defined such that $hv=varepsilon\left(h\right)v$ for all "v" in $varepsilon_H$. The trivial module may be identified with "F", with 1 being the element such that $1otimes v=v=votimes 1$ for all "v". It follows that for any "v" in any "H"-module "V", any "c" in $varepsilon_H$ and any "h" in "H",:$\left(varepsilon\left(h_\left\{\left(1\right)\right\}\right)h_\left\{\left(2\right)\right\}\right)cv=h_\left\{\left(1\right)\right\}cotimes h_\left\{\left(2\right)\right\}v=h\left(cotimes v\right)=h\left(cv\right)=\left(h_\left\{\left(1\right)\right\}varepsilon\left(h_\left\{\left(2\right)\right\}\right)\right)cv.$The existence of an algebra homomorphism &epsilon; satisfying $varepsilon\left(h_\left\{\left(1\right)\right\}\right)h_\left\{\left(2\right)\right\} = h = h_\left\{\left(1\right)\right\}varepsilon\left(h_\left\{\left(2\right)\right\}\right)$ 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 $Delta$ and $varepsilon$ satisfying these conditions. This is the motivation for the definition of a bialgebra, where $Delta$ 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 $S:H ightarrow H$ such that for any "h" in "H", "x" in "V" and "y" in "V"*,

:$langle y, S\left(h\right)x angle = langle hy, x angle.$

where $langlecdot,cdot angle$ is the usual pairing of dual vector spaces. If the map $varphi:Votimes V^* ightarrow varepsilon_H$ induced by the pairing is to be an "H"-homomorphism, then for any "h" in "H", "x" in "V" and "y" in "V"*,:$varphileft\left(h\left(xotimes y\right) ight\right)=varphileft\left(xotimes S\left(h_\left\{\left(1\right)\right\}\right)h_\left\{\left(2\right)\right\}y ight\right)=varphileft\left(S\left(h_\left\{\left(2\right)\right\}\right)h_\left\{\left(1\right)\right\}xotimes y ight\right)=hvarphi\left(xotimes y\right)=varepsilon\left(h\right)varphi\left(xotimes y\right),$which is satisfied if $S\left(h_\left\{\left(1\right)\right\}\right)h_\left\{\left(2\right)\right\}=varepsilon\left(h\right)=h_\left\{\left(1\right)\right\}S\left(h_\left\{\left(2\right)\right\}\right)$ 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 $mu:Aotimes A ightarrow A$, and $ho:Hotimes A ightarrow A$ is a representation of "H" on "A", then "&rho;" is said to be a representation of "H" on an algebra if "&mu;" is "H"-equivariant. As special cases, Lie algebras, Lie superalgebras and groups can also have representations on an algebra.

ee also

*Tannaka-Krein reconstruction theorem

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