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


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