Braided Hopf algebra
Encyclopedia
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, particurlarly the Nichols algebra of a braided vectorspace in that category.
The notion should not be confused with quasitriangular Hopf algebra.
if
A braided bialgebra in
is called a braided Hopf algebra, if there is a morphism 
of Yetter–Drinfel'd modules such that
where
in slightly modified Sweedler notation
– a change of notation is performed in order to avoid confusion in Radford's biproduct below.
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 
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
a braided Hopf algebra is a Hopf algebra
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfel'd category of a Hopf algebra H, particurlarly the Nichols algebra of a braided vectorspace in that category.
The notion should not be confused with quasitriangular Hopf algebra.
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 if
-
is a unital associative algebraAssociative algebraIn mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
, where the multiplication map
and the unit 
are maps of Yetter–Drinfel'd modules, -
is a coassociative coalgebraCoalgebraIn mathematics, coalgebras or cogebras are structures that are dual to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...
with 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 map
-
-
- 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 notationCoalgebra
In mathematics, coalgebras or cogebras are structures that are dual to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...
– a change of notation is performed in order to avoid confusion in Radford's biproduct below.
Examples
- Any Hopf algebra is also a braided Hopf algebra over
- A super Hopf algebra is nothing but a braided Hopf algebra over the group algebraGroup ringIn algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
. - The tensor algebraTensor algebraIn mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
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
-
- The counit
then satisfies the equation 
for all 
- 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.
- The universal quotient of
Radford's biproduct
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 MajidShahn Majid
Born in 1960 in Patna, Bihar, India Shahn Majid is an English pure mathematician and theoretical physicist, trained at Cambridge University and Harvard and since 2001 a Professor of Mathematics at the School of Mathematical Sciences, Queen Mary, University of London.Majid is best known for his...
, who called it bosonization.
As a vector space,
is just . The algebra structure of is given by
where
, (Sweedler notationCoproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
) 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