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Yetter–Drinfeld category

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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 Δ {\displaystyle \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 , . ) {\displaystyle (V,{\boldsymbol {.}})} is a left H-module, where . : H ⊗ V → V {\displaystyle {\boldsymbol {.}}:H\otimes V\to V} denotes the left action of H on V,
  • ( V , δ ) {\displaystyle (V,\delta \;)} is a left H-comodule, where δ : V → H ⊗ V {\displaystyle \delta :V\to H\otimes V} denotes the left coaction of H on V,
  • the maps . {\displaystyle {\boldsymbol {.}}} and δ {\displaystyle \delta } satisfy the compatibility condition
δ ( h . v ) = h ( 1 ) v ( − 1 ) S ( h ( 3 ) ) ⊗ h ( 2 ) . v ( 0 ) {\displaystyle \delta (h{\boldsymbol {.}}v)=h_{(1)}v_{(-1)}S(h_{(3)})\otimes h_{(2)}{\boldsymbol {.}}v_{(0)}} for all h ∈ H , v ∈ V {\displaystyle h\in H,v\in V} ,
where, using Sweedler notation, ( Δ ⊗ i d ) Δ ( h ) = h ( 1 ) ⊗ h ( 2 ) ⊗ h ( 3 ) ∈ H ⊗ H ⊗ H {\displaystyle (\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 ∈ H {\displaystyle h\in H} , and δ ( v ) = v ( − 1 ) ⊗ v ( 0 ) {\displaystyle \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 δ ( v ) = 1 ⊗ v {\displaystyle \delta (v)=1\otimes v} .
  • The trivial module V = k { v } {\displaystyle V=k\{v\}} with h . v = ϵ ( h ) v {\displaystyle h{\boldsymbol {.}}v=\epsilon (h)v} , δ ( v ) = 1 ⊗ 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
V = ⨁ g ∈ G V g {\displaystyle V=\bigoplus _{g\in G}V_{g}} ,
where each V g {\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
V = ⨁ g ∈ G V g {\displaystyle V=\bigoplus _{g\in G}V_{g}} , such that g . V h ⊂ V g h g − 1 {\displaystyle g.V_{h}\subset V_{ghg^{-1}}} .
  • Over the base field k = C {\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 [ g ] ⊂ G {\displaystyle \subset G\;} together with χ , X {\displaystyle \chi ,X\;} (character of) an irreducible group representation of the centralizer C e n t ( g ) {\displaystyle Cent(g)\;} of some representing g ∈ [ g ] {\displaystyle g\in } :
    V = O [ g ] χ = O [ g ] X V = ⨁ h ∈ [ g ] V h = ⨁ h ∈ [ g ] X {\displaystyle V={\mathcal {O}}_{}^{\chi }={\mathcal {O}}_{}^{X}\qquad V=\bigoplus _{h\in }V_{h}=\bigoplus _{h\in }X}
    • As G-module take O [ g ] χ {\displaystyle {\mathcal {O}}_{}^{\chi }} to be the induced module of χ , X {\displaystyle \chi ,X\;} :
    I n d C e n t ( g ) G ( χ ) = k G ⊗ k C e n t ( g ) X {\displaystyle 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 ⊗ v ∈ k G ⊗ k C e n t ( g ) X = V {\displaystyle t\otimes v\in kG\otimes _{kCent(g)}X=V} to the graduation layer:
    t ⊗ v ∈ V t g t − 1 {\displaystyle t\otimes v\in V_{tgt^{-1}}}
    • It is very custom to directly construct V {\displaystyle V\;} as direct sum of X´s and write down the G-action by choice of a specific set of representatives t i {\displaystyle t_{i}\;} for the C e n t ( g ) {\displaystyle Cent(g)\;} -cosets. From this approach, one often writes
    h ⊗ v ⊂ [ g ] × X ↔ t i ⊗ v ∈ k G ⊗ k C e n t ( g ) X with uniquely h = t i g t i − 1 {\displaystyle h\otimes v\subset \times X\;\;\leftrightarrow \;\;t_{i}\otimes v\in kG\otimes _{kCent(g)}X\qquad {\text{with uniquely}}\;\;h=t_{i}gt_{i}^{-1}}
    (this notation emphasizes the graduation h ⊗ v ∈ V h {\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 c V , W : V ⊗ W → W ⊗ V {\displaystyle c_{V,W}:V\otimes W\to W\otimes V} ,

c ( v ⊗ w ) := v ( − 1 ) . w ⊗ v ( 0 ) , {\displaystyle c(v\otimes w):=v_{(-1)}{\boldsymbol {.}}w\otimes v_{(0)},}
is invertible with inverse
c V , W − 1 ( w ⊗ v ) := v ( 0 ) ⊗ S − 1 ( v ( − 1 ) ) . w . {\displaystyle 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 ⊗ i d U ) ( i d V ⊗ c U , W ) ( c U , V ⊗ i d W ) = ( i d W ⊗ c U , V ) ( c U , W ⊗ i d V ) ( i d U ⊗ c V , W ) : U ⊗ V ⊗ W → W ⊗ V ⊗ U . {\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.}

A monoidal category C {\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 H H Y D {\displaystyle {}_{H}^{H}{\mathcal {YD}}} .

References

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