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–Drinfeld category of a Hopf algebra H, particularly the Nichols algebra of a braided vector space in that category.
The notion should not be confused with quasitriangular Hopf algebra.
Let H be a Hopf algebra over a field k, and assume that the antipode of H is bijective. A Yetter–Drinfeld module R over H is called a braided bialgebra in the Yetter–Drinfeld category
if
is a unital associative algebra, where the multiplication map
and the unit
are maps of Yetter–Drinfeld modules,
is a coassociative coalgebra with counit
, and both
and
are maps of Yetter–Drinfeld modules,
- the maps
and
are algebra maps in the category
, where the algebra structure of
is determined by the unit
and the multiplication map
![{\displaystyle (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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7f4237976ec0c3c2e9121662dc1e8144532e936)
- Here c is the canonical braiding in the Yetter–Drinfeld category
.
A braided bialgebra in
is called a braided Hopf algebra, if there is a morphism
of Yetter–Drinfeld modules such that
for all ![{\displaystyle r\in R,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/552e90ad66ce281fa21e771e83f192364fe1ffc5)
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
![{\displaystyle H=k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/856757c08dc8c16791588e9af7dd24493451dd91)
- A super Hopf algebra is nothing but a braided Hopf algebra over the group algebra
.
- The tensor algebra
of 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
![{\displaystyle \Delta (v)=1\otimes v+v\otimes 1\quad {\text{for all}}\quad v\in V.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/734933e60fa80964f30abc90dcdf6035e6f25abe)
- The counit
then satisfies the equation
for all ![{\displaystyle v\in V.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13375e380c5699070639c00dcc62c3d91b05c7cd)
- The universal quotient of
, that is still a braided Hopf algebra containing
as primitive elements is called the Nichols algebra. They take the role of quantum Borel algebras in the classification of pointed Hopf algebras, analogously to the classical Lie algebra case.
Radford's biproduct
[edit]
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
![{\displaystyle (r\#h)(r'\#h')=r(h_{(1)}{\boldsymbol {.}}r')\#h_{(2)}h',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebc24fe8975f4106f549556c849b586f67fa4ee)
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
![{\displaystyle \Delta (r\#h)=(r^{(1)}\#r^{(2)}{}_{(-1)}h_{(1)})\otimes (r^{(2)}{}_{(0)}\#h_{(2)}),\quad r\in R,h\in H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68abc956fe5ef8a6affeacd18bb12226b3844306)
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.