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\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)} $$.

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 $$V=k\{v\}$$ with $$h\boldsymbol{.}v=\epsilon (h)v$$, $$ \delta (v)=1\otimes v$$, 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
 * $$ V=\bigoplus _{g\in G}V_g$$,
 * where each $$V_g$$ is a G-submodule of V.


 * More generally, if the group G is not abelian, then Yetter–Drinfeld modules over H=kG are G-modules with a G-gradation
 * $$ V=\bigoplus _{g\in G}V_g$$, such that $$g.V_h\subset V_{ghg^{-1}}$$.


 * Over the base field $$k=\mathbb{C}\;$$ all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given through a conjugacy class $$[g]\subset G\;$$ together with $$\chi,X\;$$ (character of) an irreducible group representation of the centralizer $$Cent(g)\;$$ of some representing $$g\in[g]$$:
 * $$V=\mathcal{O}_{[g]}^\chi=\mathcal{O}_{[g]}^{X}\qquad V=\bigoplus_{h\in[g]}V_{h}=\bigoplus_{h\in[g]}X$$
 * As G-module take $$\mathcal{O}_{[g]}^\chi$$ to be the induced module of $$\chi,X\;$$:
 * $$Ind_{Cent(g)}^G(\chi)=kG\otimes_{kCent(g)}X$$
 * (this can be proven easily not to depend on the choice of g)
 * To define the G-graduation (comodule) assign any element $$t\otimes v\in kG\otimes_{kCent(g)}X=V$$ to the graduation layer:
 * $$t\otimes v\in V_{tgt^{-1}}$$
 * 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
 * $$h\otimes v\subset[g]\times X \;\; \leftrightarrow \;\; t_i\otimes v\in kG\otimes_{kCent(g)}X \qquad\text{with uniquely}\;\;h=t_igt_i^{-1}$$
 * (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 $$ c_{V,W}:V\otimes W\to W\otimes V$$,
 * $$c(v\otimes w):=v_{(-1)}\boldsymbol{.}w\otimes v_{(0)},$$
 * 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–Drinfeld modules U, V, W the map c satisfies the braid relation
 * $$(c_{V,W}\otimes \mathrm{id}_U)(\mathrm{id}_V\otimes c_{U,W})(c_{U,V}\otimes \mathrm{id}_W)=(\mathrm{id}_W\otimes c_{U,V}) (c_{U,W}\otimes \mathrm{id}_V) (\mathrm{id}_U\otimes c_{V,W}):U\otimes V\otimes W\to W\otimes V\otimes U.$$

A monoidal category $$ \mathcal{C}$$ 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 $$ {}^H_H\mathcal{YD}$$.