Yetter–Drinfeld category explained

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 ⊗ V\to V denotes the left action of H on V,(V,\delta ) is a left H-comodule, where \delta : V\to H ⊗ 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)} ) ⊗ h _{(2)} \boldsymbol{.}v _{(0)} for all h\in H,v\in V ,

where, using Sweedler notation,

(\Delta ⊗ id )\Delta (h) = h _{(1)} ⊗ h _{(2)} ⊗ h _{(3)} \in H ⊗ H ⊗ H denotes the twofold coproduct of h\in H , and \delta (v) = v _{(-1)} ⊗ 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 ⊗ v .

• The trivial module

V = k\{v\} with h\boldsymbol{.}v = \epsilon (h)v , \delta (v) =1 ⊗ 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 = o plus _{g\in} 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 = o plus _{g\in} V _{g} , such that g.V _{h\subset} V _{ghg ^{-1}} .

• Over the base field

k = C all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given^[1] 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 = l{O} ^{X} _{[g]} V = o plus _{h\in[g]} V _{h} = o plus _{h\in[g]} X

• • As G-module take

l{O} ^{\chi} _{[g]} to be the induced module of \chi,X : Ind ^{G(\chi) = kG ⊗} _{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 ⊗ v\in kG ⊗ _{kCent(g)} X = V to the graduation layer:t ⊗ 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 ⊗ v\subset[g] x X \leftrightarrow t _{i ⊗} v\in kG ⊗ _{kCent(g)} X withuniquely h = t _{igt} ^{-1} ^{i}

(this notation emphasizes the graduation

h ⊗ 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 ⊗ W\to W ⊗ V ,c(v ⊗ w): = v _{(-1)} \boldsymbol{.}w ⊗ v _{(0)} ,

is invertible with inverse

c ^{-1} _{V,W} (w ⊗ v): = v _{(0)} ⊗ 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} ⊗ id _{U)(id} _{V ⊗} c _{U,W} )(c _{U,V} ⊗ id _{W)=(id} _{W ⊗} c _{U,V} ) (c _{U,W} ⊗ id _{V)} ( id _{U ⊗} c _{V,W} ):U ⊗ V ⊗ W\to W ⊗ V ⊗ U. l{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 l{YD}} .

References

• Book: Montgomery, Susan . Susan Montgomery

. Susan Montgomery . Hopf algebras and their actions on rings . Regional Conference Series in Mathematics . 82 . Providence, RI . . 1993 . 0-8218-0738-2 . 0793.16029 .

Notes and References

1. N. . Andruskiewitsch . M. . Grana . Braided Hopf algebras over non abelian groups . Bol. Acad. Ciencias (Cordoba) . 63 . 1999 . 658–691 . math/9802074 . 10.1.1.237.5330 .