Let H be a Hopf algebra over a
fieldk. Let denote the
coproduct and S the
antipode of H. Let V be a
vector space over k. Then V is called a (left left) Yetter–Drinfeld module overH if
is a left H-
module, where denotes the left action of H on V,
is a left H-
comodule, where denotes the left coaction of H on V,
the maps and satisfy the compatibility condition
for all ,
where, using
Sweedler notation, denotes the twofold coproduct of , and .
Examples
Any left H-module over a cocommutative Hopf algebra H is a Yetter–Drinfeld module with the trivial left coaction .
The trivial module with , , is a Yetter–Drinfeld module for all Hopf algebras H.
If H is the
group algebrakG of an
abelian groupG, then Yetter–Drinfeld modules over H are precisely the G-graded G-modules. This means that
,
where each is a G-submodule of V.
More generally, if the group G is not abelian, then Yetter–Drinfeld modules over H=kG are G-modules with a G-gradation
, such that .
Over the base field all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given[1] through a
conjugacy class together with (character of) an irreducible group representation of the
centralizer of some representing :
(this can be proven easily not to depend on the choice of g)
To define the G-graduation (comodule) assign any element to the graduation layer:
It is very custom to directly construct as direct sum of X´s and write down the G-action by choice of a specific set of representatives for the -
cosets. From this approach, one often writes
(this notation emphasizes the graduation , rather than the module structure)
Braiding
Let H be a Hopf algebra with invertible antipode S, and let V, W be Yetter–Drinfeld modules over H. Then the map ,
is invertible with inverse
Further, for any three Yetter–Drinfeld modules U, V, W the map c satisfies the braid relation
A
monoidal category consisting of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode is called a Yetter–Drinfeld category. It is a braided monoidal category with the braiding c above. The category of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode is denoted by .
References
^Andruskiewitsch, N.; Grana, M. (1999). "Braided Hopf algebras over non abelian groups". Bol. Acad. Ciencias (Cordoba). 63: 658–691.
arXiv:math/9802074.
CiteSeerX10.1.1.237.5330.
Let H be a Hopf algebra over a
fieldk. Let denote the
coproduct and S the
antipode of H. Let V be a
vector space over k. Then V is called a (left left) Yetter–Drinfeld module overH if
is a left H-
module, where denotes the left action of H on V,
is a left H-
comodule, where denotes the left coaction of H on V,
the maps and satisfy the compatibility condition
for all ,
where, using
Sweedler notation, denotes the twofold coproduct of , and .
Examples
Any left H-module over a cocommutative Hopf algebra H is a Yetter–Drinfeld module with the trivial left coaction .
The trivial module with , , is a Yetter–Drinfeld module for all Hopf algebras H.
If H is the
group algebrakG of an
abelian groupG, then Yetter–Drinfeld modules over H are precisely the G-graded G-modules. This means that
,
where each is a G-submodule of V.
More generally, if the group G is not abelian, then Yetter–Drinfeld modules over H=kG are G-modules with a G-gradation
, such that .
Over the base field all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given[1] through a
conjugacy class together with (character of) an irreducible group representation of the
centralizer of some representing :
(this can be proven easily not to depend on the choice of g)
To define the G-graduation (comodule) assign any element to the graduation layer:
It is very custom to directly construct as direct sum of X´s and write down the G-action by choice of a specific set of representatives for the -
cosets. From this approach, one often writes
(this notation emphasizes the graduation , rather than the module structure)
Braiding
Let H be a Hopf algebra with invertible antipode S, and let V, W be Yetter–Drinfeld modules over H. Then the map ,
is invertible with inverse
Further, for any three Yetter–Drinfeld modules U, V, W the map c satisfies the braid relation
A
monoidal category consisting of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode is called a Yetter–Drinfeld category. It is a braided monoidal category with the braiding c above. The category of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode is denoted by .
References
^Andruskiewitsch, N.; Grana, M. (1999). "Braided Hopf algebras over non abelian groups". Bol. Acad. Ciencias (Cordoba). 63: 658–691.
arXiv:math/9802074.
CiteSeerX10.1.1.237.5330.