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 [math]\displaystyle{ \Delta }[/math] 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

• [math]\displaystyle{ (V,\boldsymbol{.}) }[/math] is a left H-module, where [math]\displaystyle{ \boldsymbol{.}: H\otimes V\to V }[/math] denotes the left action of H on V,
• [math]\displaystyle{ (V,\delta\;) }[/math] is a left H-comodule, where [math]\displaystyle{ \delta : V\to H\otimes V }[/math] denotes the left coaction of H on V,
• the maps [math]\displaystyle{ \boldsymbol{.} }[/math] and [math]\displaystyle{ \delta }[/math] satisfy the compatibility condition

[math]\displaystyle{ \delta (h\boldsymbol{.}v)=h_{(1)}v_{(-1)}S(h_{(3)}) \otimes h_{(2)}\boldsymbol{.}v_{(0)} }[/math] for all [math]\displaystyle{ h\in H,v\in V }[/math],

where, using Sweedler notation, [math]\displaystyle{ (\Delta \otimes \mathrm{id})\Delta (h)=h_{(1)}\otimes h_{(2)} \otimes h_{(3)} \in H\otimes H\otimes H }[/math] denotes the twofold coproduct of [math]\displaystyle{ h\in H }[/math], and [math]\displaystyle{ \delta (v)=v_{(-1)}\otimes v_{(0)} }[/math].

Examples

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

[math]\displaystyle{ V=\bigoplus _{g\in G}V_g }[/math],

where each [math]\displaystyle{ V_g }[/math] 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

[math]\displaystyle{ V=\bigoplus _{g\in G}V_g }[/math], such that [math]\displaystyle{ g.V_h\subset V_{ghg^{-1}} }[/math].

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

  [math]\displaystyle{ V=\mathcal{O}_{[g]}^\chi=\mathcal{O}_{[g]}^{X}\qquad V=\bigoplus_{h\in[g]}V_{h}=\bigoplus_{h\in[g]}X }[/math]

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

  [math]\displaystyle{ Ind_{Cent(g)}^G(\chi)=kG\otimes_{kCent(g)}X }[/math]

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

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

  [math]\displaystyle{ t\otimes v\in V_{tgt^{-1}} }[/math]

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

  [math]\displaystyle{ 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} }[/math]

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

[math]\displaystyle{ c(v\otimes w):=v_{(-1)}\boldsymbol{.}w\otimes v_{(0)}, }[/math]

is invertible with inverse

[math]\displaystyle{ c_{V,W}^{-1}(w\otimes v):=v_{(0)}\otimes S^{-1}(v_{(-1)})\boldsymbol{.}w. }[/math]

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

[math]\displaystyle{ (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. }[/math]

A monoidal category [math]\displaystyle{ \mathcal{C} }[/math] 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 [math]\displaystyle{ {}^H_H\mathcal{YD} }[/math].

References

• Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2.