Braided Hopf algebra

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In mathematics a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter-Drinfel'd category of a Hopf algebra H.

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[edit] Definition

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 {}^H_H\mathcal{YD} if

  • (R,\cdot ,\eta ) is a unital associative algebra, where the multiplication map \cdot :R\times R\to R and the unit \eta :k\to R are maps of Yetter-Drinfel'd modules,
  • (R,\Delta ,\varepsilon ) is a coassociative coalgebra with counit \varepsilon , and both Δ and \varepsilon are maps of Yetter-Drinfel'd modules,
  • the maps \Delta :R\to R\otimes R and \varepsilon :R\to k are algebra maps in the category {}^H_H\mathcal{YD}, where the algebra structure of R\otimes R is determined by the unit \eta \otimes \eta(1) : k\to R\otimes R and the multiplication map
(R\otimes R)\times (R\otimes R)\to R\otimes R,\quad (r\otimes s,t\otimes u) \mapsto \sum _i rt_i\otimes s_i u, \quad \text{and}\quad c(s\otimes t)=\sum _i t_i\otimes s_i .
Here c is the canonical braiding in the Yetter-Drinfel'd category {}^H_H\mathcal{YD}.

A braided bialgebra in {}^H_H\mathcal{YD} is called a braided Hopf algebra, if there is a morphism S:R\to R of Yetter-Drinfel'd modules such that

S(r^{(1)})r^{(2)}=r^{(1)}S(r^{(2)})=\eta(\varepsilon (r)) for all r\in R,

where \Delta _R(r)=r^{(1)}\otimes r^{(2)} in slightly modified Sweedler notation --- a change of notation is performed in order to avoid confusion in Radford's biproduct below.

[edit] Examples

  • Any Hopf algebra is also a braided Hopf algebra over H = k
  • A super Hopf algebra is nothing but a braided Hopf algebra over the group algebra H=k(\mathbb{Z}/2\mathbb{Z}) .
  • The tensor algebra TV of a Yetter-Drinfeld module V\in {}^H_H\mathcal{YD} is always a braided Hopf algebra. The coproduct Δ of TV is defined in such a way that the elements of V are primitive, that is
\Delta (v)=1\otimes v+v\otimes 1 \quad \text{for all}\quad v\in V.
The counit \varepsilon :TV\to k then satisfies the equation \varepsilon (v)=0 for all v\in V .
  • Let V\in {}^H_H\mathcal{YD}. There exists a largest ideal of TV with the following properties.
I\subset \bigoplus _{n=2}^\infty T^nV,
\Delta (I)\subset I\otimes TV+TV\otimes I.
One has I\in {}^H_H\mathcal{YD}, and the quotient TV/I is a braided Hopf algebra in {}^H_H\mathcal{YD}. It is called the Nichols algebra of V, named after the mathematician Warren Nichols, and is denoted by \mathfrak{B}(V).

[edit] Radford's biproduct

For any braided Hopf algebra R in {}^H_H\mathcal{YD} there exists a natural Hopf algebra R\# H 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, R\# H is just R\otimes H . The algebra structure of R\# H is given by

(r\# h)(r'\#h')=r(h_{(1)}\boldsymbol{.}r')\#h_{(2)}h' ,

where r,r'\in R,\quad h,h'\in H, \Delta (h)=h_{(1)}\otimes h_{(2)} (Sweedler notation) is the coproduct of h\in H , and \boldsymbol{.}:H\otimes R\to R is the left action of H on R. Further, the coproduct of R\# H is determined by the formula

\Delta (r\#h)=(r^{(1)}\#r^{(2)}{}_{(-1)}h_{(1)})\otimes (r^{(2)}{}_{(0)}\#h_{(2)}), \quad r\in R,h\in H.

Here \Delta _R(r)=r^{(1)}\otimes r^{(2)} denotes the coproduct of r in R, and \delta (r^{(2)})=r^{(2)}{}_{(-1)}\otimes r^{(2)}{}_{(0)} is the left coaction of H on r^{(2)}\in R .

[edit] References

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.