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 $\mathrm{\Delta }$ 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

• $(V,\mathit{.})$ is a left H-module, where $\mathit{.}:H\otimes V\to V$ 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 $\mathit{.}$ and $\delta$ satisfy the compatibility condition

$\delta (h\mathit{.}v)=h_{(1)}{v}_{(-1)}S(h_{(3)})\otimes h_{(2)}\mathit{.}{v}_{(0)}$ for all $h\in H,v\in V$,

where, using Sweedler notation, $(\mathrm{\Delta }\otimes \mathrm{i}\mathrm{d})\mathrm{\Delta }(h)=h_{(1)}\otimes h_{(2)}\otimes h_{(3)}\in H\otimes H\otimes H$ 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 $h\mathit{.}v=ϵ(h)v$, $\delta (v)=1\otimes v$, 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

$V=\underset{g\in G}{⨁}{V}_g$,

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

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

• Over the base field $k=ℂ$ 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 $g\in [g]$:

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

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

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

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

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

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

  • 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

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

  (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 $c_{V,W}:V\otimes W\to W\otimes V$,

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

is invertible with inverse

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

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

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

A monoidal category $𝒞$ 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 ${}_H^H𝒴𝒟$.

References

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