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", $hv=varepsilon(h)v$, 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 $V\_1otimes\; V\_2$ 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",

:$hv=Delta\; h(v\_\{(1)\}otimes\; v\_\{(2)\})=h\_\{(1)\}v\_\{(1)\}otimes\; h\_\{(2)\}v\_\{(2)\},,$

and for any "v" in $V\_1otimes\; V\_2$ 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 $Delta(ab)=Delta(a)Delta(b)$ for all "a" and "b" in "H".

For the category of "H"-modules to be a strict monoidal category with respect to $otimes$, $V\_1otimes(V\_2otimes\; V\_3)$ and $(V\_1otimes\; V\_2)otimes\; V\_3$ 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 $V\_1otimes(V\_2otimes\; V\_3)=(V\_1otimes\; V\_2)otimes\; V\_3$ and "h" in "H",:$((operatorname\{id\}otimes\; Delta)Delta\; h)(v\_\{(1)\}otimes\; v\_\{(2)\}otimes\; v\_\{(3)\})=h\_\{(1)\}v\_\{(1)\}otimes\; h\_\{(2)(1)\}v\_\{(2)\}otimes\; h\_\{(2)(2)\}v\_\{(3)\}=hv=((Deltaotimes\; operatorname\{id\})Delta\; h)(v\_\{(1)\}otimes\; v\_\{(2)\}otimes\; v\_\{(3)\}).$This will hold for any three "H"-modules if $Delta$ 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 $varepsilon:H\; ightarrow\; F$ 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 $1otimes\; v=v=votimes\; 1$ for all "v". It follows that for any "v" in any "H"-module "V", any "c" in $varepsilon\_H$ and any "h" in "H",:$(varepsilon(h\_\{(1)\})h\_\{(2)\})cv=h\_\{(1)\}cotimes\; h\_\{(2)\}v=h(cotimes\; v)=h(cv)=(h\_\{(1)\}varepsilon(h\_\{(2)\}))cv.$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 $Delta$ 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"^*,

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

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"^*,:$varphileft(h(xotimes\; y)\; ight)=varphileft(xotimes\; S(h\_\{(1)\})h\_\{(2)\}y\; ight)=varphileft(S(h\_\{(2)\})h\_\{(1)\}xotimes\; y\; ight)=hvarphi(xotimes\; y)=varepsilon(h)varphi(xotimes\; y),$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.

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 "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.

ee also

*Tannaka-Krein reconstruction theorem