and Smash Products
8.4.1 Invariants and Coinvariants
Definition. LetAbe a rightH-comodule algebra andM ∈ MH
A orM ∈ AMH
with costructure δ. We define the set of coinvariantsof M as
McoH :={m∈M|δ(m)∈M⊗Ht}. (8.20)
IfAis a leftH-comodule algebra andM ∈ HM
AorM ∈ HAMwith costructure
δ, we define the set of coinvariants of M as
coHM :={m∈M|δ(m)∈H
8.4 A Structure Theorem for Weak Hopf Modules 81
Lemma 8.4.1. Let A be a rightH-comodule algebra. If M ∈ MH
A, then McoH ={m∈M|m(0)⊗m(1)=m1A(0)⊗1 A (1)}. If M ∈ AMH, then McoH ={m∈M|m(0)⊗m(1)= 1A(0)m⊗1A(1)}.
Let nowA be a left H-comodule algebra. IfM ∈ HM
A, then coHM ={m∈M|m (−1)⊗m(0)= 1A(−1)⊗m1 A (0)}. If M ∈ H AM, then coHM ={m∈M|m (−1)⊗m(0)= 1A(−1)⊗1A(0)m}.
Proof. The first equation is [ZZ04, Lemma 2.1], the second and third follow
from Lemma 8.1.3, and the last equation is the opcop version of [ZZ04, Lemma 2.1].
Corollary 8.4.2. If A is an H-comodule algebra, the following holds for a
coinvariant elementa∈AcoH:
ρ(a) = 1A(0)a⊗1(1)A =a1A(0)⊗1A(1).
Example 8.4.3. Let M be a rightA-module. Then
(M = AH)coH ={m1A (0)⊗1A(1)|m∈M}. In fact, ifP imi⊗hi∈(M = AH)coH, then X i mi⊗hi(1)⊗hi(2)= X i mi1A(0)⊗hi1A(1)⊗1A(2). By applying id⊗ε⊗id we obtain X i mi⊗hi = X i mi1A(0)⊗ε(hi1A(1))1A(2) (7.15) = X i mi1A(0)⊗ε(hiεt(1A(1)))1A(2) (8.5) = X i mi1A(0)1A(0) 0 ⊗ε(hi1A(1))1A(1) 0 =X i miε(hi)1A(0)⊗1A(1).
In the special case whenA=His the weak Hopf module algebra itself then, as for Hopf algebras, there exists a projection E:M →McoH:
Lemma and Definition 8.4.4. LetM ∈ MH
H and define
E :M →M
m7→m(0)S(m(1)).
82 8. Weak Hopf Modules
Proof. E(M) ⊂ McoH is shown in the proof of [ZZ04, Theorem 2.2]. If m ∈
McoH, thenE(m) =m1(1)S(1(2)) =m by Lemma 8.4.1.
The dual version of the coinvariants of a weak Hopf module are the invariants of objects in (HM)A:
Lemma and Definition 8.4.5. Let A be a left H-module algebra and M ∈
(HM)A. Theinvariants ofM are defined as
MH :={m∈M|h·m=εt(h)·m∀h∈H}. (8.22)
Then McoH =MH∗. In fact,
{m∈M|ϕ * m=εt(ϕ)* m∀ϕ∈H∗}
={m∈M|m(0)ϕ(m(1)) =m(0)ε(1(1)m(1))ϕ(1(2))∀ϕ∈H∗}
={m∈M|m(0)⊗m(1) =m(0)⊗εt(m(1))}.
8.4.2 A Structure Theorem
The structure theorem for Hopf modules over ordinary Hopf algebras states a category equivalence between vector spaces and Hopf modules. It implies in particular that every Hopf module over a Hopf algebra H is a free H-module. For weak Hopf modules, the freeness has to be replaced by projectivity. B¨ohm, Nill, and Szlach´anyi [BNS99, Theorem 3.9] proved a structure Theorem for weak Hopf modules in MH
H which was generalized by Zhang and Zhu [ZZ04] to
H-comodule algebras A which allow an H-comodule algebra mapγ :H → A. In this case, M ∈ MH
A impliesM ∈ MHH via γ.
Theorem 8.4.6. [ZZ04, Theorem 2.2 ] Let A be a right H-comodule algebra and γ :H→A anH-comodule algebra map. Let M ∈ MH
A. Then
McoH⊗AcoH A ∼= M inMHA m⊗a7→ma
m(0)γ(S(m(1)))⊗m(2)←[m.
Here, theH-comodule structure and the theA-module structure ofMcoH⊗
AcoHA
are those of A.
Corollary 8.4.7. If M ∈ MH
H, then M is a projective rightH-module and it
is a free right H-module if McoH is a free right Ht-module.
Proof. By the theorem,M ∼=McoH⊗HcoHH whereHcoH =Htis a semisimple
algebra and therefore McoH is a direct summand of a freeHt-module. Hence,
8.4 A Structure Theorem for Weak Hopf Modules 83
Lemma 8.4.8.LetAandγ be as in the theorem. LetV be a rightAcoH-module,
thenV ⊗AcoH A∈ MHA and
V ∼= (V ⊗AcoH A)coH v7→v⊗1A.
Proof. LetEA and E be the map from 8.4.4 for Aand V ⊗AcoH Aregarded as
objects inMH
H via γ, respectively. We know E(V ⊗AcoH A) = (V ⊗AcoH A)coH
andEA(A) =AcoH and we haveE(V ⊗AcoHA) =V ⊗AcoH1A. In fact, ifv ∈V
anda∈A, then
E(v⊗AcoH a) =v⊗AcoH a(0)γ(S(a(1)))
=v⊗AcoH EA(a) =vEA(a)⊗AcoH 1A.
Hence, together with this lemma the theorem above implies a category equivalence between the category of rightAcoH-modules and the categoryMH
A:
Theorem 8.4.9. Let A be a right H-comodule algebra and γ : H → A an H-comodule algebra map. Then
MAcoH ≈ MHA V 7→ V ⊗AcoH A McoH ←[ M .
8.4.3 Weak Smash Products
As for ordinary Hopf algebras, a special case of H-comodule algebras which allow an H-comodule algebra map γ : H → A, are smash products (in fact, such algebras are always smash products, see [Zha10]), and thus the structure theorem can be applied to smash products.
Lemma and Definition 8.4.10. [Nik00] LetR be a rightH-module algebra, then the smash product R#H is R⊗Ht H as a vector space where R is a right Ht-module as in 8.2.2. R#H is an H-comodule algebra with unit 1R#1
and with multiplication and costructure given by
(r#h)(s#g) =r(h(1)·s)#h(2)g, (8.23)
ρ(r#h) =r#h(1)⊗h(2), (8.24) forr, r0 ∈R, h, h0 ∈ H. The map H → R#H, h 7→ 1R#h is an H-comodule
algebra map. R → R#H, r 7→ r#1 is an algebra inclusion and (R#H)coH =
R#1∼=R.
In the following we will associate elements ofR#1 with elements of R and elements of 1#H will be associated to elements ofH, and we will writerh for
84 8. Weak Hopf Modules
r#h and thenhr= (h(1)·r)h(2).
LetA:=R#H, and V ∈ MR, then V#H:=V ⊗RAis an object in MH A,
where the module and comodule structure are the one of A. Again we write (vr)h forv⊗Rr#h=vr⊗R1R#h. Then
E(vh) =vh(1)S(h(2)) =vεt(h) =v(1R/ εt(h))∈V,
for allv∈V andh∈H, where/is theHt-action onRas defined in (8.19); and
therefore (V#H)coH = E(V#H) ∼= V. With the structure theorem for weak Hopf modules 8.4.6 we obtain the following:
Corollary 8.4.11. If R is a leftH-module algebra and A:=R#H, then MR ≈ MHR#H
V 7→ V#H McoH ←[ M are quasi-inverse equivalences, where
M ∼=McoH#H inMH R#H
m7→E(m(0))m(1) ma←[m⊗Ra.
As for Hopf algebras [MS99] one can use this equivalence to show that the
H-stable ideals of R correspond to the H-costable ideals ofA:= R#H and in particular,R is H-simple if and only ifA isH-simple.
Proposition 8.4.12. Let R be an H-module algebra andA:=R#H, then {H-stable ideals of R} Φ −−→ ←− Ψ {H-costable ideals of A} I 7−→ I#H JcoH ←−[ J are well-defined mutually inverse bijections.
Proof. If I is an H-stable ideal of R, then it is straightforward to check that
I#H is an H-costable ideal of R#H, since if r or r0 ∈ I, then (rh)(r0h0) = (r(h(1) ·r0))h(2)h0 ∈ I#H. On the other hand, let J be an H-costable ideal of A. It is clear that JcoH = J ∩(R#H)coH = J ∩R is an ideal of R. It is
H-stable since forr∈J∩R=JcoH and h∈H one hash·r∈R and:
h·r= (h(1)·r)(εt(h(2))·1R)
= (h(1)·r)/ εt(h(2))
= (h(1)·r)εt(h(2))