Yetter-Drinfeld category

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In mathematics a Yetter-Drinfel'd category is a special type of braided monoidal category. It consists of modules over a Hopf algebra which satisfy some additional axioms.

[edit] Definition

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-Drinfel'd module over H if

$(V,\boldsymbol{.})$ is a left H-module, where $\boldsymbol{.}: H\otimes V\to V$ denotes the left action of H on V and ⊗ denotes a tensor product,
$(V,δ)$ is a left H-comodule, where $\delta : V\to H\otimes V$ denotes the left coaction of H on V,
the maps $\boldsymbol{.}$ and $δ$ satisfy the compatibility condition

$\delta (h\boldsymbol{.}v)=h_{(1)}v_{(-1)}S(h_{(3)}) \otimes h_{(2)}\boldsymbol{.}v_{(0)}$ for all $h\in H,v\in V$ ,

where, using Sweedler notation, $(\Delta \otimes \mathrm{id})\Delta (h)=h_{(1)}\otimes h_{(2)} \otimes h_{(3)} \in H\otimes H\otimes H$ denotes the twofold coproduct of $h\in H$ , and $\delta (v)=v_{(-1)}\otimes v_{(0)}$ .

[edit] Examples

Any left H-module over a cocommutative Hopf algebra H is a Yetter-Drinfel'd module with the trivial left coaction $\delta (v)=1\otimes v$ .
The trivial module $V = k {v}$ with $h\boldsymbol{.}v=\epsilon (h)v$ , $\delta (v)=1\otimes v$ , is a Yetter-Drinfel'd module for all Hopf algebras H.
If H is the group algebra kG of an abelian group G, then Yetter-Drinfel'd modules over H are precisely the G-graded G-modules. This means that

$V=\bigoplus _{g\in G}V_g$ ,

where each

V g

is a G-submodule of V.

More generally, if the group G is not abelian, then Yetter-Drinfel'd modules over H=kG are G-modules with a G-gradation

$V=\bigoplus _{g\in G}V_g$ , such that $g.V_h\subset V_{ghg^{-1}}$ .

[edit] Braiding

Let H be a Hopf algebra with invertible antipode S, and let V, W be Yetter-Drinfel'd modules over H. Then the map $c_{V,W}:V\otimes W\to W\otimes V$ ,

$c(v\otimes w):=v_{(-1)}\boldsymbol{.}w\otimes v_{(0)}$ ,

is invertible with inverse

$c_{V,W}^{-1}(w\otimes v):=v_{(0)}\otimes S^{-1}(v_{(-1)})\boldsymbol{.}w$ .

Further, for any three Yetter-Drinfel'd modules U, V, W the map c satisfies the braid relation

$(c_{V,W}\otimes \mathrm{id}_U)(\mathrm{id}_V\otimes c_{U,W})(c_{U,V}\otimes \mathrm{id}_W)=(\mathrm{id}_W\otimes c_{U,V}) (c_{U,W}\otimes \mathrm{id}_V) (\mathrm{id}_U\otimes c_{V,W}):U\otimes V\otimes W\to W\otimes V\otimes U.$

[edit] Yetter-Drinfel'd category

A monoidal category $\mathcal{C}$ consisting of Yetter-Drinfel'd modules over a Hopf algebra H with bijective antipode is called a Yetter-Drinfel'd category. It is a braided monoidal category with the braiding c above. The category of Yetter-Drinfel'd modules over a Hopf algebra H with bijective antipode is denoted by ${}^H_H\mathcal{YD}$ .

[edit] References

S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Lecture Notes vol 82, American Math Society, Providence, RI, 1993. ISBN 0821807382

Yetter-Drinfeld category

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Examples

[edit] Braiding

[edit] Yetter-Drinfel'd category

[edit] References

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