Yetter–Drinfeld category

Examples

• Any left H-module over a cocommutative Hopf algebra H is a Yetter–Drinfeld module with the trivial left coaction ${\displaystyle \delta (v)=1\otimes v}$.
• The trivial module ${\displaystyle V=k\{v\}}$ with ${\displaystyle h\mathbf{.}v=ϵ(h)v}$, ${\displaystyle \delta (v)=1\otimes v}$, 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

${\displaystyle V=\underset{g\in G}{⨁}{V}_g}$,

where each ${\displaystyle V_g}$ 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

${\displaystyle V=\underset{g\in G}{⨁}{V}_g}$, such that ${\displaystyle g.V_h\subset V_{gh{g}^{-1}}}$.

• Over the base field ${\displaystyle k=\mathbb{C}}$ all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given through a conjugacy class ${\displaystyle [g]\subset G}$ together with ${\displaystyle \chi ,X}$ (character of) an irreducible group representation of the centralizer ${\displaystyle Cent(g)}$ of some representing ${\displaystyle g\in [g]}$:

  ${\displaystyle V={\mathcal{O}}_{[g]}^{\chi }={\mathcal{O}}_{[g]}^{X}V=\underset{h\in [g]}{⨁}{V}_h=\underset{h\in [g]}{⨁}X}$

  • As G-module take ${\displaystyle {\mathcal{O}}_{[g]}^{\chi }}$ to be the induced module of ${\displaystyle \chi ,X}$:

  ${\displaystyle In{d}_{Cent(g)}^G(\chi )=kG{\otimes }_{kCent(g)}X}$

  (this can be proven easily not to depend on the choice of g)

  • To define the G-graduation (comodule) assign any element ${\displaystyle t\otimes v\in kG{\otimes }_{kCent(g)}X=V}$ to the graduation layer:

  ${\displaystyle t\otimes v\in V_{tg{t}^{-1}}}$

  • It is very custom to directly construct ${\displaystyle V}$ as direct sum of X´s and write down the G-action by choice of a specific set of representatives ${\displaystyle t_i}$ for the ${\displaystyle Cent(g)}$-cosets. From this approach, one often writes

  ${\displaystyle h\otimes v\subset [g]\times X\leftrightarrow t_i\otimes v\in kG{\otimes }_{kCent(g)}X\text{with uniquely}h=t_{i}g{t}_i^{-1}}$

  (this notation emphasizes the graduation ${\displaystyle h\otimes v\in V_h}$, 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 ${\displaystyle c_{V,W}:V\otimes W\to W\otimes V}$,

${\displaystyle c(v\otimes w):=v_{(-1)}\mathbf{.}w\otimes v_{(0)},}$

is invertible with inverse

${\displaystyle c_{V,W}^{-1}(w\otimes v):=v_{(0)}\otimes S^{-1}(v_{(-1)})\mathbf{.}w.}$

Further, for any three Yetter–Drinfeld modules U, V, W the map c satisfies the braid relation

${\displaystyle (c_{V,W}\otimes {\mathrm{i}\mathrm{d}}_U)({\mathrm{i}\mathrm{d}}_V\otimes c_{U,W})(c_{U,V}\otimes {\mathrm{i}\mathrm{d}}_W)=({\mathrm{i}\mathrm{d}}_W\otimes c_{U,V})(c_{U,W}\otimes {\mathrm{i}\mathrm{d}}_V)({\mathrm{i}\mathrm{d}}_U\otimes c_{V,W}):U\otimes V\otimes W\to W\otimes V\otimes U.}$

A monoidal category ${\displaystyle \mathcal{C}}$ 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 ${\displaystyle {}_H^H\mathcal{Y}\mathcal{D}}$.