# 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

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

Examples

* Any left "H"-module over a cocommutative Hopf algebra "H" is a Yetter-Drinfel'd module with the trivial left coaction $delta \left(v\right)=1otimes v$.
* The trivial module $V=k\left\{v\right\}$ with , $delta \left(v\right)=1otimes v$, 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::,: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::, such that $g.V_hsubset V_\left\{ghg^\left\{-1$.

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 $c_\left\{V,W\right\}:Votimes W o Wotimes V$,::,:is invertible with inverse::.:Further, for any three Yetter-Drinfel'd modules "U", "V", "W" the map "c" satisfies the braid relation::$\left(c_\left\{V,W\right\}otimes mathrm\left\{id\right\}_U\right)\left(mathrm\left\{id\right\}_Votimes c_\left\{U,W\right\}\right)\left(c_\left\{U,V\right\}otimes mathrm\left\{id\right\}_W\right)=\left(mathrm\left\{id\right\}_Wotimes c_\left\{U,V\right\}\right) \left(c_\left\{U,W\right\}otimes mathrm\left\{id\right\}_V\right) \left(mathrm\left\{id\right\}_Uotimes c_\left\{V,W\right\}\right):Uotimes Votimes W o Wotimes Votimes U.$

Yetter-Drinfel'd category

A monoidal category $mathcal\left\{C\right\}$ 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 $\left\{\right\}^H_Hmathcal\left\{YD\right\}$.

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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