- Braided Hopf algebra
mathematicsa braided Hopf algebra is a Hopf algebrain a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter-Drinfel'd category of a Hopf algebra "H".
Let "H" be a Hopf algebra over a field "k", and assume that the antipode of "H" is bijective. A Yetter-Drinfel'd module "R" over "H" is called a braided bialgebra in the Yetter-Drinfel'd category if
* is a unital
associative algebra, where the multiplication map and the unit are maps of Yetter-Drinfel'd modules,
* is a coassociative
coalgebrawith counit , and both and are maps of Yetter-Drinfel'd modules,
* the maps and are algebra maps in the category , where the algebra structure of is determined by the unit and the multiplication
.:Here "c" is the canonical braiding in the Yetter-Drinfel'd category .
A braided bialgebra in is called a braided Hopf algebra, if there is a morphism of Yetter-Drinfel'd modules such that:: for all ,
where in slightly modified Sweedler notation --- a change of notation is performed in order to avoid confusion in Radford's biproduct below.
* Any Hopf algebra is also a braided Hopf algebra over
* A super Hopf algebra is nothing but a braided Hopf algebra over the group algebra .
tensor algebraof a Yetter-Drinfeld module is always a braided Hopf algebra. The coproduct of is defined in such a way that the elements of "V" are primitive, that is:: :The counit then satisfies the equation for all
* Let . There exists a largest ideal of "TV" with the following properties.:: :::One has , and the quotient "TV/I" is a braided Hopf algebra in . It is called the Nichols algebra of "V", named after the mathematician Warren Nichols, and is denoted by .
For any braided Hopf algebra "R" in there exists a natural Hopf algebra which contains "R" as a subalgebra and "H" as a Hopf subalgebra. It is called Radford's biproduct, named after its discoverer, the Hopf algebraist David Radford. It was rediscovered by Shahn Majid, who called it bosonization.
As a vector space, is just . The algebra structure of is given by:: ,
where , (Sweedler notation) is the coproduct of , and is the left action of "H" on "R". Further, the coproduct of is determined by the formula::
Here denotes the coproduct of "r" in "R", and is the left coaction of "H" on .
Andruskiewitsch, Nicolás and Schneider, Hans-Jürgen, "Pointed Hopf algebras", New directions in Hopf algebras, 1--68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002.
Wikimedia Foundation. 2010.