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 $\Delta$ 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

* $(V,\boldsymbol)$ is a left ''H''-module, where $\boldsymbol: H\otimes V\to V$ denotes the left action of ''H'' on ''V'',
* $(V,\delta\;)$ is a left ''H''-comodule, where $\delta : V\to H\otimes V$ denotes the left coaction of ''H'' on ''V'',
* the maps $\boldsymbol$ and $\delta$ satisfy the compatibility condition
::$\delta (h\boldsymbolv)=h_v_S(h_) \otimes h_\boldsymbolv_$ for all $h\in H,v\in V$,
:where, using Sweedler notation, $(\Delta \otimes \mathrm)\Delta (h)=h_\otimes h_ \otimes h_ \in H\otimes H\otimes H$ denotes the twofold coproduct of $h\in H$, and $\delta (v)=v_\otimes v_$.

Examples

* Any left ''H''-module over a cocommutative Hopf algebra ''H'' is a Yetter–Drinfeld module with the trivial left coaction $\delta (v)=1\otimes v$.
* The trivial module $V=k\$ with $h\boldsymbolv=\epsilon (h)v$, $\delta (v)=1\otimes v$, is a Yetter–Drinfeld module for all Hopf algebras ''H''.
* If ''H'' is the group algebra ''kG'' of an abelian group ''G'', then Yetter–Drinfeld modules over ''H'' are precisely the ''G''-graded ''G''-modules. This means that
::$V=\bigoplus _V_g$,
:where each $V_g$ is a ''G''-submodule of ''V''.
* More generally, if the group ''G'' is not abelian, then Yetter–Drinfeld modules over ''H=kG'' are ''G''-modules with a ''G''-gradation
::$V=\bigoplus _V_g$, such that $g.V_h\subset V_$.
* Over the base field $k=\mathbb\;$ all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group ''H=kG'' are uniquely given through a conjugacy class subset G\; together with $\chi,X\;$ (character of) an irreducible group representation of the centralizer $Cent(g)\;$ of some representing g\in /math>:
*:$V=\mathcal_^\chi=\mathcal_^\qquad V=\bigoplus_V_=\bigoplus_X$
** As ''G''-module take $\mathcal_^\chi$ to be the induced module of $\chi,X\;$:
*::$Ind_^G(\chi)=kG\otimes_X$
*:(this can be proven easily not to depend on the choice of ''g'')
** To define the ''G''-graduation (comodule) assign any element $t\otimes v\in kG\otimes_X=V$ to the graduation layer:
*::$t\otimes v\in V_$
** It is very custom to directly construct $V\;$ as direct sum of ''X''´s and write down the ''G''-action by choice of a specific set of representatives $t_i\;$ for the $Cent(g)\;$-cosets. From this approach, one often writes
*::h\otimes v\subset times X \;\; \leftrightarrow \;\; t_i\otimes v\in kG\otimes_X \qquad\text\;\;h=t_igt_i^
*:(this notation emphasizes the graduation$h\otimes v\in V_h$, rather than the module structure)

Braiding

Let ''H'' be a Hopf algebra with invertible antipode ''S'', and let ''V'', ''W'' be Yetter–Drinfeld modules over ''H''. Then the map $c_:V\otimes W\to W\otimes V$,
::$c(v\otimes w):=v_\boldsymbolw\otimes v_,$
:is invertible with inverse
::$c_^(w\otimes v):=v_\otimes S^(v_)\boldsymbolw.$
:Further, for any three Yetter–Drinfeld modules ''U'', ''V'', ''W'' the map ''c'' satisfies the braid relation
::$(c_\otimes \mathrm_U)(\mathrm_V\otimes c_)(c_\otimes \mathrm_W)=(\mathrm_W\otimes c_) (c_\otimes \mathrm_V) (\mathrm_U\otimes c_):U\otimes V\otimes W\to W\otimes V\otimes U.$

A monoidal category $\mathcal$ consisting of Yetter–Drinfeld modules over a Hopf algebra ''H'' with bijective antipode is called a Yetter–Drinfeld category. It is a braided monoidal category with the braiding ''c'' above. The category of Yetter–Drinfeld modules over a Hopf algebra ''H'' with bijective antipode is denoted by $^H_H\mathcal$.

References

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{{DEFAULTSORT:Yetter-Drinfeld category
Hopf algebras
Quantum groups
Monoidal categories