Equivalences of Yetter-Drinfel’d categories

2.2.1. Left and right modules

In this subsection we discuss the side switch functorT:YD^A_A(C)→^A_AYD(C) for Yetter-Drinfel’d modules. It turns out that, for our purposes, a non-trivial monoidal structure T2:T⊗T → T◦ ⊗ has to be chosen for the switch functor, even in those cases (for Csymmetric) where the identities provide a monoidal structure onT.

Lemma 2.2.1 The isomorphism^AT: Cmd(C)-A→Cmd(C)-A^copof cat-egories from Remark 1.3.16 extends to an isomorphism of catcat-egories

AT:^A_AYD(C)→A^copYD^Acop(C).

The functor ^AT is braided and strict monoidal, considered as a functor between the following monoidal categories:

AT:^A_AYD(C)→_(Acop)^copYD^Acop(C).

Remark 2.2.2 The equality (A^cop)^cop=Aof Hopf algebras from Remark 1.3.9 might suggest the notation

AYD^Acop(C) :=(A^cop)^copYD^Acop(C)

which is not in conflict with other notation used in this thesis. To avoid confusion with the different monoidal category_AYD^Aop(C), we refrain from using this notation.

Proof. Let X = (X, ρ, δ) be in ^A_AYD(C). It follows from Remark 1.3.16 that ^AT(X) = (X, ρ, c⁻¹_X,A◦δ) is an A^cop-comodule and A^cop-module in C. It remains to be shown that^AT(X) obeys the condition of a left-right A^cop-Yetter-Drinfel’d module inC:

A X X A

=

A A X

X

=

A X A X

=

A X X A

.

One finally verifies that the braiding isomorphisms in the Yetter-Drinfel’d categories ^A_AYD(C) and (A^cop)^copYD^Acop(C) coincide as morphisms in the underlying categoryC.

Remark 2.2.3 One can show by similar arguments that the isomorphisms

AT,T^A andTA extend to braided and strict monoidal functors

AT:^A_AYD(C)→^(Aop)opYDA^op(C), T^A:YD^A_A(C)→^AcopYD(A^cop)^cop(C), T_A:YD^A_A(C)→A^opYD^(Aop)op(C).

Theorem 2.2.4 Let A be a Hopf algebra in a braided category C and (X, ρ^r, δ^r) a right Yetter-Drinfel’d module overA. Consider

T(X, ρ^r, δ^r) = (X, ρ⁻◦(S⁻¹⊗idX),(S⊗idX)◦δ⁺), withρ⁻:=ρ^r◦c⁻¹_X,Aandδ⁺:=cX,A◦δ^r. The functor

T= (T_A)^A:YD^A_A(C)→^A_AYD(C)

has a monoidal structureT2(X, Y) :T(X)⊗T(Y)→T(X⊗Y) given by

T2(X, Y) :=

X Y X Y

=c_Y,X^YD ◦c⁻¹_Y,X . (2.6)

The monoidal functor (T,T2) is braided.

Proof. The functorT: YD^A_A(C)→^A_AYD(C) is defined as the composition of the functors in the diagram

YD^A_A(C) ^T //

T_A

A AYD(C)

A^opYD^Aop(C) ^S //_AcopYD^Acop(C)

(^AT)⁻¹

OO

Here S denotes the functor of restriction along S⁻¹: A^cop → A^op and corestriction alongS:A^op→A^cop, thusTis a functor.

Sincec_Y,X^YD is A-linear and A-colinear, we see from the right-hand side of (2.6), that the morphismT2(X, Y) isA-(co)linear, iffc⁻¹_Y,X isA-(co)linear as a morphismTX⊗TY →T(Y ⊗X); this is easily checked. Invertibility

ofT2(X, Y) is clear from (2.6), so it remains to check thatT2is a monoidal structure, i.e. for all Yetter-Drinfel’d modulesX, Y andZ we have

T₂(X⊗Y, Z)◦(T₂(X, Y)⊗id_T(Z))

=T2(X, Y ⊗Z)◦(id_T(X)⊗T2(Y, Z)).

This is a direct consequence of the Yetter-Drinfel’d condition. We conclude that (T,T2) is a monoidal functor.

Finally we show that (T,T2) is a braided monoidal functor, i.e. the equality T(c_X,Y^YD)◦T2(X, Y) =T2(Y, X)◦c_TX,TY^YD

holds. This is evident from drawing the morphisms.

Remark 2.2.5 The functorT= (TA)^A in Theorem 2.2.4 is obtained by first turning the right action into a left action and then the right coaction into a left coaction, which shall be suggested by the notation (TA)^A. There is another braided equivalence between the same braided categories of Yetter-Drinfel’d modules:

T⁰ = (T^A)_A:YD^A_A(C)→^A_AYD(C).

The functorT⁰ is given on objects by

T⁰(X, ρ^r, δ^r) = (X, ρ⁺◦(S⊗idX),(S⁻¹⊗idX)◦δ⁻).

The monoidal structureT₂⁰(X, Y) :T⁰(X)⊗T⁰(Y)→T⁰(X⊗Y) onT⁰ is given by

T⁰₂(X, Y) :=

X Y X Y

=

c_X,Y^YD−1

◦cX,Y .

The two monoidal functors T,T⁰: YD^A_A(C) →^A_AYD(C) are isomorphic as monoidal functors. An isomorphismT→T⁰ is given by the family

θTX :=ρTX◦(S⊗idX)◦δTX =ρ^r_X◦(idX⊗S)◦δ_X^r .

A right Yetter-Drinfel’d module version of Lemma 2.1.5 implies that θis indeed a monoidal isomorphism.

2.2.2. Dually paired Hopf algebras

In this subsection, we prove that for Hopf algebrasAandBthat are related by a non-degenerate Hopf pairing, there is a braided monoidal equivalence between the categories^A_AYD(C) andYD^B_B(C). This equivalence is a strict monoidal functor.

Lemma 2.2.6 Letω: A⊗B →1be a non-degenerate Hopf pairing with inverse copairingω⁰:1→B⊗A. Then

ω⁰D^ω:_AcopYD^A(C)→^BYDB^cop(C)

(X, ρ, δ)7→(X,(id⊗ω)◦(δ⊗id),(id⊗ρ)◦(ω⁰⊗id)), is a strict monoidal braided functor. In particular, the two categories

A^copYD^A(C) and ^BYDB^cop(C) are equivalent as braided monoidal cate-gories.

Proof. Let (X, ρ, δ) be anA-Yetter-Drinfel’d module. From Lemma 1.3.19 it is clear thatD(X, ρ, δ) is aB-module andB-comodule. We have to check the Yetter-Drinfel’d condition. Since X is an A-Yetter-Drinfel’d module, we have the equality

X B

X B ω

ω⁰

=

X B

X B

ω

ω⁰

.

Using thatωis a Hopf pairing,ω⁰is a Hopf copairing and (idB⊗ω)◦(ω⁰⊗ idB) = idB we get the equality

ω

ω⁰

X B

B X

= ω

ω⁰

X B

B X

which is the Yetter-Drinfel’d condition for theB-module andB-comodule structure on D(X). The functor D is strict monoidal, since the functors

ω⁰D: A^cop-Mod(C)→ B-Cmd(C) andD^ω: Cmd(C)-A → Mod(C)-B^cop are strict monoidal. Finally, the braiding is preserved

c_D(X),D(Y^YD ₎=c_X,Y^YD =:D(c_X,Y^YD). This follows from (ω⊗idA)◦(idA⊗ω⁰) = idA.

Corollary 2.2.7 Let ω: A⊗B → 1 be a non-degenerate Hopf pairing with inverse copairingω⁰:1→B⊗A. Then

ω

ω⁰D:^A_AYD(C)→ YD^B_B(C)

(X, ρX, δX)7→(X, ρ_D(X), δ_D(Y)) with

ρ_D(X)= (id⊗ω)◦(c⁻¹_X,A⊗S⁻¹)◦(δ⊗id) δ_D(X)=c_B,X◦(S⊗ρ)◦(ω⁰⊗id)

defines a strict monoidal braided functor.

In particular, the categories ^A_AYD(C) and YD^B_B(C) are equivalent as braided categories.

Proof. Note thatω:A⊗B→1is a Hopf pairing of the two Hopf algebras A^cop and B^opin C. So we have the following composite of braided, strict monoidal functors

A

AYD(C) //

AT

YD^(B_(B^opop⁾)^copcop(C) ^S //YD^B_B(C)

(A^cop)^copYD^Acop(C) ^ω0D

ω //^(Bop)copYDB^op(C)

Bop

T

OO

Here S denotes the functor of restriction along S⁻¹:B → (B^op)^cop and corestriction alongS: (B^op)^cop→B. The top line of the above diagram is the functor^ω_ω0D.

Combining Theorem 2.2.4, Remark 2.2.5 and Corollary 2.2.7, we are now in a position to exhibit explicitly two braided equivalences

Ω,Ω⁰:^A_AYD(C)→^B_BYD(C).

The first functor is the composition Ω :=T◦Dwith monoidal structure Ω2(X, Y) =T(D2(X, Y))◦T2(DX,DY) = id_X⊗Y ◦(c_DY,DX^YD ◦c⁻¹_Y,X)

=c_Y,X^YD ◦c⁻¹_Y,X.

The second to last equal sign uses thatDis a strict braided functor.

The other functor is Ω⁰ :=T⁰◦Dwith monoidal structure Ω⁰₂(X, Y) =T⁰(D2(X, Y))◦T⁰₂(DX,DY) =

c_X,Y^YD−1

◦cX,Y. Graphically the functors and the monoidal structures are

Ω(X, ρX, δX) =















 X,

B X X ω

, X B X

ω⁰

















, Ω2(X, Y) =

X Y

X Y

,

Ω⁰(X, ρX, δX) =















 X,

B X X ω

,

X

B X

ω⁰

















, Ω⁰₂(X, Y) =

X Y

X Y

.

We summarize our findings:

Theorem 2.2.8 Letω:A⊗B→1be a non-degenerate Hopf pairing. The categories^A_AYD(C) and ^B_BYD(C) are braided equivalent via the monoidal functors Ω and Ω⁰ above.

Remark 2.2.9 We end this subsection by relating the equivalence Ω to the equivalence Ω^HS of rational modules overk-Hopf algebras discussed in [HS13].

1. Let k be a field and L_k the category of linearly topologized vector spaces over k. Fix a Hopf algebra E in L_k and two Hopf algebras (R, R^∨) in^E_EYD(L_k) that are related by a non-degenerate Hopf pair-ing. It is then shown in [HS13] that the categories ^R_R^oE

oEYD_rat and

R^∨oE

R^∨oEYDratare equivalent as braided categories. Here, the subscript rat denotes the subcategory of rational modules.

The non-degenerate pairing h,i: R^∨ ⊗R → k and the structural morphisms of the bosonized Hopf algebraRoE are used in [HS13, Theorem 7.1] to construct a functor

(Ω^HS,Ω^HS₂ ) :^R_R^o_o^E_EYDrat →^R_R^∨∨^oo^EEYDrat .

In detail, the functor Ω^HS is constructed as follows: letM be a ratio-nal (RoE)-Yetter-Drinfel’d module and denote the leftR-coaction byδ(m) =m_h−1i⊗m_h0i.

The (R^∨oE)-Yetter-Drinfel’d module Ω^HS(M) is equal toM as an E-Yetter-Drinfel’d module and has the followingR^∨-Yetter-Drinfel’d structure

action: ξm=hξ, m_h−1iim_h0i coaction: δ_ΩHS(M)=

c^YD_M,R∨◦c_R^YD∨,M

(m_[−1]⊗m[0]), where m_[−1]⊗m_[0] is the unique element of R^∨⊗M such that for allr∈Randm∈M we have

rm=

m_[−1], θR(r) m_[0].

The monoidal structure of Ω^HS is given by the family of morphisms Ω^HS₂ (M, N) : Ω^HS(M)⊗Ω^HS(N)→Ω^HS(M⊗N)

m⊗n7→S_R⁻¹

oES_R(n_h−1i)m⊗n_h0i. 2. In this thesis, we started with a non-degenerate Hopf pairingω: A⊗

B →1and constructed an equivalence

Ω^ω:^A_AYD(C)→^B_BYD(C).

Let C be the category of finite dimensional Yetter-Drinfel’d mod-ules over the finite dimensional Hopf algebra E. Set A = R and B = R^∨ and ω: A⊗B → k, such that ω⁻(b⊗a) = hb, ai, cf. Ex-ample 1.3.20. One can show by straight-forward computations, that our functor Ω^ω− coincides with the functor Ω^HS on the full subcat-egory ^R_R^o_o^E_EYDfin⊂^R_R^o_o^E_EYDrat of finite dimensional (Ro E)-Yetter-Drinfel’d modules.

2.2.3. The square of Ω

From a non-degenerate Hopf pairingω: A⊗B→1, we obtained an equiv-alence Ω^ω:^A_AYD(C)→^B_BYD(C). As noted in Example 1.3.20, we also have a non-degenerate Hopf pairingω⁻: B⊗A→1from which we obtain an equivalence Ω^ω−:^B_BYD(C)→^A_AYD(C).

Proposition 2.2.10 The braided monoidal functor Ω^ω−◦Ω^ω:^A_AYD(C)→^A_AYD(C) is isomorphic to the identity functor.

Proof. A direct computation shows that the monoidal functors (Ω^ω,Ω^ω₂)◦

(Ω⁰)^ω−,(Ω⁰)^ω₂⁻

and

(Ω⁰)^ω−,(Ω⁰)^ω₂⁻

◦(Ω^ω,Ω^ω₂) are both equal to the identity functor with identity monoidal structure.

Remark 2.2.5 implies that (Ω⁰)^ω− is monoidally isomorphic to Ω^ω−. Alternatively, a concrete calculation shows that Ω^ω−◦Ω^ωis equal to the monoidal functor that sends the Yetter-Drinfel’d module (X, ρ, δ) to the Yetter-Drinfel’d module

(X, ρ◦(S⁻²⊗idX)◦c⁻¹_A,X◦c⁻¹_X,A, cX,A◦cA,X◦(S²⊗idX)◦δ).

The monoidal structure of Ω^ω−◦Ω^ωis given by the family of isomorphisms c_Y,X^YD ◦c_X,Y^YD ◦c⁻¹_X,Y ◦c⁻¹_Y,X.

From this and Lemma 2.1.5 it is clear that θ_X := ρ_X ◦(S ⊗id_X)◦δ_X defines a monoidal isomorphism

θ: Ω^ω−◦Ω^ω→Id.

3. Graded categories and group

Im Dokument Two constructions in monoidal categories : Equivariantly extended Drinfel’d Centers and Partially dualized Hopf Algebras (Seite 47-55)