Representation theory of Hopf algebras
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.
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
invariantunder "H" if for all "h" in "H", , where ε is the counitof "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 productof 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 such that for any "v" in and any "h" in "H",
and for any "v" in and "a" and "b" in "H",
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 for all "a" and "b" in "H".
For the category of "H"-modules to be a strict
monoidal categorywith respect to , and must be equivalent and there must be unit object , called the trivial module, such that , "V" and are equivalent.
This means that for any "v" in and "h" in "H",:This will hold for any three "H"-modules if satisfies .
The trivial module must be one dimensional, and so an
algebra homomorphismmay be defined such that for all "v" in . The trivial module may be identified with "F", with 1 being the element such that for all "v". It follows that for any "v" in any "H"-module "V", any "c" in and any "h" in "H",:The existence of an algebra homomorphism ε satisfying 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 and satisfying these conditions. This is the motivation for the definition of a
bialgebra, where is called the comultiplicationand 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 such that for any "h" in "H", "x" in "V" and "y" in "V"*,
where is the usual
pairingof dual vector spaces. If the map induced by the pairing is to be an "H"-homomorphism, then for any "h" in "H", "x" in "V" and "y" in "V"*,:which is satisfied if 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 , and 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.
Tannaka-Krein reconstruction theorem
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