Yetter-Drinfeld category

Yetter-Drinfeld category

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

Definition

Let "H" be a Hopf algebra over a field "k". Let Delta denote the coproduct and "S" the antipode of "H". Let "V" bea vector space over "k". Then "V" is called aYetter-Drinfel'd module over "H" if

* (V,oldsymbol{.}) is a left "H"-module, where oldsymbol{.}: Hotimes V o V denotes the left action of "H" on "V" and ⊗ denotes a tensor product,
* (V,delta ) is a left "H"-comodule, where delta : V o Hotimes V denotes the left coaction of "H" on "V",
* the maps oldsymbol{.} and delta satisfy the compatibility condition:: delta (holdsymbol{.}v)=h_{(1)}v_{(-1)}S(h_{(3)})otimes h_{(2)}oldsymbol{.}v_{(0)} for all hin H,vin V,:where, using Sweedler notation, (Delta otimes mathrm{id})Delta (h)=h_{(1)}otimes h_{(2)}otimes h_{(3)} in Hotimes Hotimes H denotes the twofold coproduct of hin H , and delta (v)=v_{(-1)}otimes v_{(0)} .

Examples

* Any left "H"-module over a cocommutative Hopf algebra "H" is a Yetter-Drinfel'd module with the trivial left coaction delta (v)=1otimes v.
* The trivial module V=k{v} with holdsymbol{.}v=epsilon (h)v, delta (v)=1otimes 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=igoplus _{gin 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=igoplus _{gin G}V_g, such that g.V_hsubset V_{ghg^{-1.

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}:Votimes W o Wotimes V,::c(votimes w):=v_{(-1)}oldsymbol{.}wotimes v_{(0)},:is invertible with inverse::c_{V,W}^{-1}(wotimes v):=v_{(0)}otimes S^{-1}(v_{(-1)})oldsymbol{.}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}_Votimes c_{U,W})(c_{U,V}otimes mathrm{id}_W)=(mathrm{id}_Wotimes c_{U,V}) (c_{U,W}otimes mathrm{id}_V) (mathrm{id}_Uotimes c_{V,W}):Uotimes Votimes W o Wotimes Votimes U.

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_Hmathcal{YD}.

References

* S. Montgomery, "Hopf Algebras and Their Actions on Rings", CBMS Lecture Notes vol 82, American Math Society, Providence, RI, 1993. ISBN-10: 0821807382


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