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.

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[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 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 Vg 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-10: 0821807382