In mathematics a Yetter–Drinfeld category is a special type of braided monoidal category. It consists of modules over a Hopf algebra which satisfy some additional axioms.
Let H be a Hopf algebra over a field k. Let
Δ
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 over H if
(
V
,
.
)
is a left H-module, where
.
:
H
⊗
V
→
V
denotes the left action of H on V and ⊗ denotes a tensor product,
(
V
,
δ
)
is a left H-comodule, where
δ
:
V
→
H
⊗
V
denotes the left coaction of H on V,
the maps
.
and
δ
satisfy the compatibility condition
where, using Sweedler notation,
(
Δ
⊗
i
d
)
Δ
(
h
)
=
h
(
1
)
⊗
h
(
2
)
⊗
h
(
3
)
∈
H
⊗
H
⊗
H
denotes the twofold coproduct of
h
∈
H
, and
δ
(
v
)
=
v
(
−
1
)
⊗
v
(
0
)
.
Any left H-module over a cocommutative Hopf algebra H is a Yetter–Drinfeld module with the trivial left coaction
δ
(
v
)
=
1
⊗
v
.
The trivial module
V
=
k
{
v
}
with
h
.
v
=
ϵ
(
h
)
v
,
δ
(
v
)
=
1
⊗
v
, is a Yetter–Drinfeld module for all Hopf algebras H.
If H is the group algebra kG of an abelian group G, then Yetter–Drinfeld modules over H are precisely the G-graded G-modules. This means that
where each
V
g
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
Over the base field
k
=
C
all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given through a conjugacy class
[
g
]
⊂
G
together with
χ
,
X
(character of) an irreducible group representation of the centralizer
C
e
n
t
(
g
)
of some representing
g
∈
[
g
]
:
V
=
O
[
g
]
χ
=
O
[
g
]
X
V
=
⨁
h
∈
[
g
]
V
h
=
⨁
h
∈
[
g
]
X
As G-module take
O
[
g
]
χ
to be the induced module of
χ
,
X
:
(this can be proven easily not to depend on the choice of
g)
To define the G-graduation (comodule) assign any element
t
⊗
v
∈
k
G
⊗
k
C
e
n
t
(
g
)
X
=
V
to the graduation layer:
It is very custom to directly construct
V
as direct sum of X´s and write down the G-action by choice of a specific set of representatives
t
i
for the
C
e
n
t
(
g
)
-cosets. From this approach, one often writes
(this notation emphasizes the graduation
h
⊗
v
∈
V
h
, rather than the module structure)
Let H be a Hopf algebra with invertible antipode S, and let V, W be Yetter–Drinfeld modules over H. Then the map
c
V
,
W
:
V
⊗
W
→
W
⊗
V
,
is invertible with inverse
Further, for any three Yetter–Drinfeld modules
U,
V,
W the map
c satisfies the braid relation
(
c
V
,
W
⊗
i
d
U
)
(
i
d
V
⊗
c
U
,
W
)
(
c
U
,
V
⊗
i
d
W
)
=
(
i
d
W
⊗
c
U
,
V
)
(
c
U
,
W
⊗
i
d
V
)
(
i
d
U
⊗
c
V
,
W
)
:
U
⊗
V
⊗
W
→
W
⊗
V
⊗
U
.
A monoidal category
C
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
H
H
Y
D
.
