Yetter–Drinfeld category

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.

Definition

Let H be a Hopf algebra over a field k. Let

Δ

{\displaystyle \Delta }

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

,

.

)

{\displaystyle (V,{\boldsymbol {.}})}

is a left H-module, where

.

:

H

⊗

V

→

V

{\displaystyle {\boldsymbol {.}}:H\otimes V\to V}

denotes the left action of H on V,

(

V

,

δ

)

{\displaystyle (V,\delta \;)}

is a left H-comodule, where

δ

:

V

→

H

⊗

V

{\displaystyle \delta :V\to H\otimes V}

denotes the left coaction of H on V,

the maps

.

{\displaystyle {\boldsymbol {.}}}

and

δ

{\displaystyle \delta }

satisfy the compatibility condition

δ

(

h

.

v

)

=

h

(

1

)

v

(

−

1

)

S

(

h

(

3

)

)

⊗

h

(

2

)

.

v

(

0

)

{\displaystyle \delta (h{\boldsymbol {.}}v)=h_{(1)}v_{(-1)}S(h_{(3)})\otimes h_{(2)}{\boldsymbol {.}}v_{(0)}}

for all

h

∈

H

,

v

∈

V

{\displaystyle h\in H,v\in V}

,

where, using Sweedler notation,

(

Δ

⊗

i

d

)

Δ

(

h

)

=

h

(

1

)

⊗

h

(

2

)

⊗

h

(

3

)

∈

H

⊗

H

⊗

H

{\displaystyle (\Delta \otimes \mathrm {id} )\Delta (h)=h_{(1)}\otimes h_{(2)}\otimes h_{(3)}\in H\otimes H\otimes H}

denotes the twofold coproduct of

h

∈

H

{\displaystyle h\in H}

, and

δ

(

v

)

=

v

(

−

1

)

⊗

v

(

0

)

{\displaystyle \delta (v)=v_{(-1)}\otimes v_{(0)}}

.