Normal Hopf subalgebras of semisimple Drinfeld doubles

0.26 MB·English

by  Sebastian Burciu

KERNELS OF REPRESENTATIONS OF SEMISIMPLE
DRINFELD DOUBLES
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1 SEBASTIAN BURCIU
0
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Abstract. Adescriptionforkernelsofrepresentationsofsemisim-
v
o ple Drinfeld doubles D(A) is given. Using this we also obtain a
N description of all normal Hopf subalgebras of D(A). As an appli-
cation,kernelsofirreduciblerepresentationsofD(G)arecomputed
4
and shown that they are all normal Hopf subalgebras of D(G).

]
A
Q
Introduction
.
h
t Let A be a semisimple Hopf algebra and D(A) be its Drinfeld dou-
a
m ble. It is well known that the category Rep(D(A)) of representations
[ D(A) is a modular category [1] and is braided equivalent to the cen-

ter Z(Rep(A)) of the tensor category Rep(A). The Drinfeld doubles
3
v D(A)playaveryimportantroleintheclassification ofsemisimple Hopf
2
algebras.
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8 In this paper we will give a description of the kernels of the repre-
2 sentations of D(A). This will allow us to obtain a description of the
.
5 normal Hopf subalgebras of D(A) in terms of normal Hopf subalgebras
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of A and A∗.
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1 Recently the author introduced in [3] the notion of kernel of a rep-
:
v resentation of a semisimple Hopf algebra. It was proven that if the
i character of the representation is central in the dual Hopf algebra then
X
the kernel is a normal Hopf subalgebra. It is not known whether the
r
a kernel is in general a normal Hopf subalgebra. In this paper we prove
that in the case D(G), of a Drinfeld double of a finite group G, all
representation kernels are normal Hopf subalgebras of D(G).
Fusion subcategories of the category Rep(D(G)) of a finite group G
were recently studied in [13]. They are parameterized in terms of a pair
of commuting subgroups of G and a G-invariant bicharacter defined
on their product. The main ingredient used in [13] is the notion of
centralizer for a fusion subcategory introduced in [10]. In this paper
we will identify from the above mentioned parametrization all fusion
2000 Mathematics Subject Classification. Primary 16W30, 18D10.
Key words and phrases. Drinfeld doubles; Fusion subcategories; Kernels of
representations.
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2 SEBASTIANBURCIU
subcategory of Rep(D(G)) that are of the form Rep(D(G)//L) where
L is a normal Hopf subalgebra of D(G). They correspond to those
bicharacters from [13] satisfying a stronger condition than that of G-
invariance.
The paper is organized as follows. The first section recalls few basic
results on Hopf algebras and the kernels of their characters that are
needed in the paper.
The second section is concerned with fusion subcategories of Rep(A)
for a semisimple Hopf algebra A. Theorem 2.3 gives a necessary and
sufficient condition for such a category to be normal (i.e of the form
Rep(A//L) for some normal Hopf subalgebra L of A). A description of
the commutator subcategory Rep(A//L)co is given in Proposition 2.4.
Next section studies kernels of representations of Drinfeld doubles
D(A). The key idea is to use a lemma from [4] concerning fusion
subcategories of a products of two fusion categories. This lemma can
be regarded as a quantum analogue of Goursat’s lemma for groups. A
description of normal Hopf subalgebras of D(A) is also given in terms
of normal Hopf subalgebras of A and A∗. Two general examples are
also considered in this section.
The last two sections are concerned with the special case D(G),
of Drinfeld doubles of finite groups. Some basic results from group
representations that are needed are recalled in the first section. A basis
for central characters in D(G) is also given here. In the last section
the parametrization from [13] of fusion subcategories of Rep(D(G)) is
recalled. ThissectiongivesadescriptionofallnormalHopfsubalgebras
of D(G) and shows that the kernel of any character of D(G) is normal.
We work over the algebraic closed field C. For a vector space V
by |V| is denoted the dimension dimCV. We use Sweedler’s notation
∆(x) = x ⊗ x for comultiplication. All the other Hopf notations
P 1 2
are those used in [11].
1. Preliminaries
1.1. Notations. Let A be a finite dimensional semisimple Hopf alge-
bra over C. Then A is also cosemisimple [8]. The character ring C(A)
of A is a semisimple subalgebra of A∗ [17] and it has a vector space
basis given by the set Irr(A) of irreducible characters of A. Moreover,
C(A) = Cocom(A∗), the space of cocommutative elements of A∗. By
duality, the character ring of A∗ is a semisimple subalgebra of A and
C(A∗) = Cocom(A). If M is an A-module with character χ then M∗ is
also anA-modulewithcharacter χ∗ = χ◦S. This induces aninvolution
“ ∗ ” : C(A) → C(A) on C(A). Let m (χ, µ) be the multiplicity form
A
SEMISIMPLE HOPF ALGEBRAS 3
on C(A). For d ∈ Irr(A∗) denote by C the simple subcoalgebra of A
d
whose character as A∗-module equals d [7]. Denote by t the integral
A
in A∗ with t (1) = |A|. It is known that t is also the regular character
A A
of A [11].
1.2. Kernels of characters for semisimple Hopf algebras. Let M
be a representation of A which affords the character χ. Define ker (χ)
A
as the set of all irreducible characters d ∈ Irr(A∗) which act as the
scalar ǫ(d) on M. Then Proposition 1. 2 of [3] implies that
ker (χ) = {d ∈ Irr(A∗)| χ(d) = ǫ(d)χ(1)}.
A
Similarly let z (χ) be the the set of all irreducible characters d ∈
A
Irr(A∗) which act as a scalar αǫ(d) on M, where α is a root of unity.
Then from the same proposition it follows
z (χ) = {d ∈ Irr(A∗)| |χ(d)| = ǫ(d)χ(1)}.
A
Clearly ker (χ) ⊂ z (χ).
A A
Since the sets ker (χ) and z (χ) are closed under multiplication and
A A
“ ∗ ” they generate Hopf subalgebras of A denoted by A and Z (χ),
χ A
respectively (see [3]).
Remark 1.1.
Suppose that K is a Hopf subalgebra of a semisimple Hopf algebra A
via i : K ֒→ A. The restriction functor from A-modules to K-modules
induces a map res : C(A) → C(K). It is easy to see that res = i∗| ,
C(A)
the restriction of the dual map i∗ : A∗ → K∗ to the subalgebra of
characters C(A) ⊂ A∗.
2. Fusion subcategories of Rep(A)
2.1. Fusion categories and their universal gradings. In this sub-
section we recall few facts on fusion categories from [6] and [4].
Let C be a fusion subcategory. Let O(C) be its set of simple objects
considered up to isomorphism. Recall that C is defined as the fusion
ad
subcategory of C containing X ⊗X∗ for each simple object X of C.
A grading ofa fusion category C by a groupG isa map deg : O(C) →
G with the following property: for any simple objects X,Y,Z ∈ C
such that X ⊗Y contains Z one has deg(Z) = deg(X)deg(Y) . This
corresponds to a decomposition C = ⊕ C , where C ⊂ C is the full
g∈G g g
additive subcategory generated by simple objects of degree g. The
subcategory C corresponding to g = 1 is a fusion subcategory; it is
1
called the trivial component of the grading. A grading is said to be
trivial if C = C. It is said to be faithful if the map deg : O(C) → G
1
is surjective. For any fusion category C, as explained in Sect. 3.2 of
4 SEBASTIANBURCIU
[6], there is a notion of universal grading whose group is called the
universal grading group and it is denoted by U . Its trivial component
C
is the fusion subcategory C . The following Lemma appears in [4].
ad
Lemma 2.1. Let C be a fusion category and
C = ⊕ C
g∈U g
C
be its universal grading. There is a one-to-one correspondence between
fusion subcategories D ⊂ C containing C and subgroups G ⊂ U ,
ad C
namely
D 7→ G := {g ∈ U |D ∩C 6= 0}
D C g
and
G 7→ D := ⊕ C .
G g∈G g
Let A be a semisimple Hopf algebra. It is known that Rep(A) is a
fusion category. Moreover there is a maximal central Hopf subalgebra
K(A) of A such that Rep(A) = Rep(A//K(A)), see [6]. Since K(A)
ad
iscommutative it follows that K(A) = k[U ]∗ where U is theuniversal
A A
grading group of Rep(A). For example if A = kG then K(A) = kZ(G)
ˆ
and U = Z(G), the linear dual group of the center Z(G) of G.
A
Let D be a fusion subcategory of Rep(A) and O(D) be its set of
objects. Then I := ∩ Ann (V) is a Hopf ideal in A [14] and
D V∈O(D) A
D = Rep(A/I ). For a fusion category D ⊂ Rep(A) define its regular
D
character as rD := PX∈Irr(D)dimC(X)χX where Irr(D) is the set of
irreducible objects of D and χ is the character of X as A-module.
X
Thus r ∈ C(A).
D
Remark 2.2. Let B be a normal Hopf subalgebra of a semisimple Hopf
A and χ a character of A affording the representation M . Then M
χ χ
is a representation of A//B if and only if A ⊃ B.
χ
Theorem 2.3. Let A be a finite dimensional semisimple Hopf algebra
and D be a fusion subcategory of Rep(A). Then D = Rep(A//L) for
some normal Hopf subalgebra L of A if and only if the regular character
r of D is central in A∗. In this case L = A .
D rD
Proof. If D = Rep(A//L) then by Theorem 2.4 of [3] it follows that r
D
is an integral of (A//L)∗. Since this is a normal Hopf subalgebra of A∗,
it follows from Lemma 1 of [9] that r is central in A∗. Conversely, if
D
r is central in A∗ then by Proposition 3.3 of [3] it follows that A is a
D rD
normal Hopf subalgebra of A. Since r2 = |D|r the same Proposition
implies that D = Rep(A//A ) D D (cid:3)
r
D
SEMISIMPLE HOPF ALGEBRAS 5
2.2. The commutator subcategory. Recall that the notion of com-
mutator subcategory from [6]. If D is a fusion subcategory of C then
Dco is the full abelian subcategory of C generated by those objects X
such that X ⊗X∗ ∈ O(D).
In this subsection we will describe the category Rep(A//L)co for L
a normal Hopf subalgebra of A. For a,b ∈ A define [a,b] = ab−ba the
usual commutator. Then for L a Hopf subalgebra of A let [A, L] be
the ideal generated by [a,l] with a ∈ A and l ∈ L.
Proposition 2.4. Let L be a normal Hopf subalgebra of A. Then [A,L]
is a Hopf ideal of A and
(2.5) Rep(A//L)co = Rep(A/[A,L])
Proof. To see that [A,L] is a Hopf ideal note that
∆([a, l]) = Xa1l1 ⊗a2l2 −Xl1a1 ⊗l2a2
= X(a1l1 −l1a1)⊗a2l2 +Xl1a1 ⊗(a2l2 −l2a2)
Nowconsider M anirreducibleA/[A,L]-moduleaffordingthecharacter
χ ∈ C(A). Since (la).m = (al).m for all a ∈ A, l ∈ L and m ∈
M it follows that left multiplication by l on M is a morphism of A-
module. Schur’s Lemma implies that each l ∈ L acts by a scalar
on M. Thus χ ↓A= χ(1)ψ for some linear character ψ of L. Then
L
(χχ∗) ↓A= χ(1)2ψψ−1 = χ(1)2ǫ .
L L
Conversely suppose that M ⊗M∗ is a trivial L-module for an irre-
ducible A-module. Let A = I ⊕Ann (M) be the decomposition of A
M A
in two-sided ideals where I is the minimal ideal in A corresponding of
M
M. It is well known (see [16] for example) that the minimal ideal I in
A corresponding of M satisfies I ∼= M ⊗M∗ where I is regardeMd as
M M
A-moduleby theadjointaction. Therefore l xSl = ǫ(l)xforall x ∈ I
1 2 M
and l ∈ L. Then lx = (l xS(l ))l = xl for all x ∈ I and l ∈ L. Then
1 2 3 M
(la−al)m = 0 for all a ∈ A. Indeed if a ∈ Ann (M) this is clear since
A
(la).m = (al).m = 0. On the other hand if a ∈ I then al − la = 0
M
and thus (al−la).m = 0. This shows that M ∈ Rep(A/[A, L]). (cid:3)
Definition 2.6. Let L be a normal Hopf subalgebra of A. An ir-
reducible character α of L is called A-stable if there is a character
χ ∈ Rep(A) such that χ ↓A= χ(1)α. Such a character χ is said to seat
L α(1)
over α.
Denote by G (L) the set of all A-stable linear characters of L.
st
Lemma 2.7. Let M be an irreducible module of A affording a character
χ. Then M ∈ O(Rep(A//L)co) if and only if L acts trivially on some
6 SEBASTIANBURCIU
tensor power M⊗n of M. In these conditions χ ↓A= χ(1)ψ for some
L
A-stable linear character ψ of L.
Proof. Suppose that L acts trivially on some tensor power M⊗n of M.
This means that χn ↓L= χ(1)nǫ . Let
A L
A
χ ↓ = X mαα
L
α∈Irr(L)
for some nonnegative integers m and
α
χn−1 ↓A= X nαα
L
α∈Irr(L)
for some nonnegative integers n . Recall that m (ǫ , αβ) > 0 if and
α L L
only if α = β∗. Since χn ↓A= χ(1)nǫ this implies that χ ↓A= χ(1)α
L L L α(1)
and χn−1 ↓A= χ(1)n−1α∗ for a fixed character α. It follows by counting
L α(1)
the multiplicity of ǫ in χn ↓A that α(1) = 1. Thus α is an A-stable
L L
linear character of L. .
Conversely suppose that M ∈ O(Rep(A//L)co) and let
χ ↓A= m α. Since m (ǫ , αβ∗) = δ , counting the mul-
L Pα∈Irr(K) α L L α,β
tiplicity of ǫ in χ ↓A χ∗ ↓A implies that χ ↓A= χ(1)α for a L-linear
L L L L
character α. If nis the order of α inG(L∗) then clearly χn ↓A= χ(1)nǫ
which shows that L acts trivially on M⊗n L (cid:3)L
By duality it follows that the subcategory Rep(L∗)co of Rep(A∗) has
a similar description as in Proposition 2.4.
3. Kernels of characters of representations of Drinfeld
doubles
The Drinfeld double D(A) of A is defined as follows: D(A) ∼= A∗cop⊗
A as coalgebras and multiplication is given by
(g ⊲⊳ h)(f ⊲⊳ l) = Xg(h1 ⇀ f ↼ S−1h3) ⊲⊳ h2l.
Its antipode is given by S(f ⊲⊳ h) = S−1(h)S(f).
It is known that Rep(D(A)∗) = Rep(A∗)rev ⊠Rep(A) since D(A) is
a cocycle twist of A∗cop⊗A. It is also known that D(A) is a semisimple
Hopf algebra if and only if A is a semisimple Hopf algebra [11].
3.1. Fusion subcategories of direct products of categories. Let
C1 and C2 be two fusion categories. Identify them with the correspond-
ingfusionsubcategoriesofC1⊠C2 givenbyC1⊠1and1⊠C2 respectively.
Then every simple object of C1 ⊠C2 is of the form X ⊠X where X
1 2 i
is a simple object of Ci.
SEMISIMPLE HOPF ALGEBRAS 7
Let D ⊂ C1 ⊠ C2 be a fusion subcategory. Define Li(D) := D ∩
Ci, i = 1,2. Let also K1(D) be the fusion subcategory generated by
all simple objects X of C1 such that X ⊠ X ∈ D for some simple
1 1 2
objectX ofC2. SimilarlydefinethefusionsubcategoryK2(D). Clearly
2
L1(D)⊠L2(D) ⊂ D ⊂ K1(D)⊠K1(D).
The following Lemma is taken from [4].
Lemma 3.1. Let D ⊂ C1 ⊠C2 be a fusion subcategory. Then there is
a group X and faithful X-gradings K (D) = ⊕ Ki(D) with trivial
i x∈X x
components Li(D), i = 1,2 such that
(3.2) D = ⊕ K1(D) ⊠K2(D) .
x∈X x x
Proof. First, let us show that
(3.3) D ⊃ K1(D) ⊠K2(D) = (K1(D)⊠K2(D)) ,
ad ad ad
where Ki(D) denotes the adjoint subcategory of Ki(D). To prove
ad
this, note that if X ⊠X is an object of C ⊠C that is also in O(D)
1 2 1 2
then (X ⊗ X∗) ⊠ (X ⊗ X∗) ∈ D. Since X ⊗ X∗ contains the unit
1 1 2 2 i i
object it follows that (X ⊗ X∗) ⊠ 1 ∈ D and 1 ⊠ (X ⊗ X∗) ∈ D.
1 1 2 2
Therefore
(3.4) Ki(D) ⊂ Li(D) for i = 1,2.
ad
But L1(D)⊠L2(D) ⊂ D, so Equation 3.4 implies equation 3.3. Now
let U be the universal grading group of Ki(D). By Equation 3.3
Ki(D)
and Lemma 2.1,
D = ⊕ (K1(D)⊠K2(D))
γ∈Γ γ
for some subgroup Γ ∈ U = U ×U .
K1(D)⊠K2(D) K1(D) K2(D)
By the definition of Ki(D), the subgroup Γ ∈ U × U has
K1(D) K2(D)
the following property: the maps Γ → U and Γ → U are
K1(D) K2(D)
surjective. Goursat’s lemma for groups (see [15]) implies that Γ equals
the fiber product U × U for some group X equipped with
K1(D) X K2(D)
epimorphisms U → X, i = 1,2. These epimorphisms definefaithful
Ki(D)
X-gradingsof Ki(D) such that Equation 3.2 holds. Formula 3.2 implies
that Li(D) := D∩Ci equals the trivial component of the X-grading of
Ki(D). (cid:3)
The following remark is straightforward.
Remark 3.5. Let C be a fusion category and C = ⊕ C be a faithful
g∈G g
grading of C. For any object X ∈ C one has X⊗n ∈ C where n ≥ 1
g g g 1
is the order of g ∈ G. Indeed C⊗n ⊂ C = C .
g gn 1
Remark 3.6.
8 SEBASTIANBURCIU
If α is an irreducible A-stable character of L then formulae from [2]
shows that in this situation
χ(1)2
α ↑A= Pχ∈Aα X χ(1)χ
L α ↑A (1)
L χ∈Aα
where A is the set of all irreducible characters χ of A that seat over
α
α. Also if t ∈ (A//L)∗ is the integral on A//L with t (1) = |A|
A//L A//L |L|
it follows from Theorem 4.3 of [2] that
α ↑A χt
(3.7) L = A//L
α(1) χ(1)
for any χ ∈ A .
α
Let T ⊂ Irr(A) be a set of irreducible characters of of A and χ an-
other character of A. Denote by Tχ the set all irreducible constituents
of µχ where µ ∈ T. Also let Irr(χ) be the set of all irreducible con-
stituents of χ.
Lemma 3.8. Suppose A is a semisimple Hopf algebra and L is a
normal Hopf subalgebra. Suppose that there is a finite group G and
faithful grading on Rep(A) = ⊕ Rep(A) with the trivial component
g∈G g
Rep(A) = Rep(A//L). Then for any χ ∈ Rep(A) it follows that
1 g g
χ ↓A= χ(1)ψ where ψ is a A-stable linear character of L. Moreover
g g g
L
A
Rep(A) = Rep(A//L)χ = Irr(ψ ↑ )
g g g
L
Proof. By Remark 3.5 it follows that χn ∈ Rep(A) = Rep(A//L).
g 1
Then Lemma 2.7 implies that χ ↓A= χ(1)ψ where ψ is a A-stable
g g g
linear character of L. The last equalLities follows from Remark 3.6. (cid:3)
3.2. Hopf subalgebras of Drinfeld doubles D(A). If H is a Hopf
subalgebra of D(A) then
Rep(H∗) ⊂ Rep(D(A)∗) = Rep(A)⊠Rep(A∗)rev
Then Lemma 3.1 implies that any Hopf subalgebra H of D(A) is
completely determined by four fusion subcategories satisfying the fol-
lowing properties:
(3.9) (K1) ⊂ L1 ⊂ K1 ⊂ Rep(A)
ad
(3.10) (K2) ⊂ L2 ⊂ K2 ⊂ Rep(A∗)
ad
with faithful gradings
SEMISIMPLE HOPF ALGEBRAS 9
(3.11) Ki = ⊕ (Ki)
x∈X x
by a given group X such that Ki = L for i = 1,2. In this situation
1 i
(3.12) Rep(H∗) = ⊕ (K1) ⊠(K2) .
x∈X x x
3.3. Kernels of representations of Drinfeld doubles D(A).
Theorem 3.13. Let M be a D(A)-module with character χ. Then
there is a finite group X and irreducible characters η and d of A and
x x
A∗ respectively with η = ǫ and d = 1 such that
1 A 1
kerD(A)(χ) = G(kerA∗(χ ↓A∗)ηx ×kerA (χ ↓A)dx).
x∈X
Moreover ηn ∈ ker (χ ↓ ) and dn ∈ ker (χ ↓ ) for some n ≥ 1 and
x A∗ A∗ x A A
all x ∈ X.
Proof. Since ker (χ) is the set of simple objects of a fusion sub-
D(A)
category Rep(A∗)rev ⊠ Rep(A) one can apply Lemma 3.1. Note that
L (ker (χ)) has as set of simple objects the set ker (χ ↓D(A)) and
1 D(A) A∗ A∗
L (ker (χ)) has as set of simple objects the set ker (χ ↓D(A)).
2 D(A) A A
Let L := A∗ ⊂ A∗ and L := A ⊂ A be the Hopf
A∗ (χ↓DA∗(A)) A (χ↓DA(A))
subalgebras determined by ker (χ ↓D(A)) and ker (χ ↓D(A)) respec-
A∗ A∗ A A
tively. Similarly define K as the Hopf subalgebra of A∗ determined
A∗
by K (ker χ) and K ⊂ A as the Hopf subalgebra of A generated
1 D(A) A
by K (ker χ). Then by Lemma 3.1 one has that (K ) ⊂ L
2 D(A) A ad A
and (K ) ⊂ L . Applying Lemma 3.8 it follows that there are
A∗ ad A∗
irreducible characters χ and d of A respectively A∗ such that
x x
D(A) D(A)
kerD(A)(χ) = G(kerA∗(χ ↓A∗ )ηx ×kerA (χ ↓A )dx).
x∈X
Remark 3.5 implies that ηn ∈ ker (χ ↓ ) and dn ∈ ker (χ ↓ ) for
some n ≥ 1. x A∗ A∗ x A A (cid:3)
Remark 3.14.
If η ⊲⊳ d ∈ ker (χ) one can prove something a little stronger than
D(A)
ηn ∈ ker (χ ↓ ) and dn ∈ ker (χ ↓ ).
A∗ A∗ A A
Clearly C ⊲⊳ C ⊂ D(A) . Then from 3.10 above it follows that
η d χ
C C ⊂ L . Thus for all m ∈ M and x ∈ C one has d∗xm =
d∗ d A d
ǫ(d)ǫ(x)m. Multiplying this equality by η∗ and noticing that η∗d∗ ∈
ker χ one obtains: ǫ(d)ǫ(x)(χ∗m) = (χ∗d∗)xm = χ(1)ǫ(d)xm.
D(A)
10 SEBASTIANBURCIU
A basis of the simple coalgebra C is given by the comatrix entries
d
xd with 1 ≤ i,j ≤ ǫ(d). Therefore if x = xd with i 6= j then xdm = 0
ij ij ij
for all m ∈ M. If i = j then the above formula implies that xdm =
ii
1 η∗m. This shows that xdm does not depend on i.
χ(1) ii
Recall that the exponent of A is the smallest positive number n > 0
such that h[n] = ǫ(h)1 for all h ∈ A. The generalized power h[n] is
defined by h[n] = h h ...h . The exponent of a finite dimensional
P(h) 1 2 n
semisimple Hopf algebra is always finite and divides the third power of
the dimension of A [5]. If h = d one has
d[n] = X xij1xj1j2···xjn−1i
i,j1,j2,···jn−1
and therefore d[n]m = ǫ(d)xn.m = ǫ(d)xn m. If n is divisible by the
Pi=1 ii 11
exponent of A then (xd)nm = m for all m ∈ M. Thus one obtains that
ii
(xd)n ∈ ker (χ ↓D(A)).
ii A A
3.4. Normal Hopf subalgebras of Drinfeld doubles D(A). Inthis
subsection we give a description of normal Hopf subalgebras ofDrinfeld
doubles D(A).
Remark 3.15.
Suppose that K isa normal Hopf subalgebra of Aandlet H = A//K
with the natural projection π : A → H. Then H∗ ⊂ A∗ via π∗. Thus
K K
Remark 1.1 shows that π | is the restriction map of A∗-characters
K C(A∗)
to H∗. This implies that an irreducible character d of A∗ seats over
an irreducible character x of H∗ if and only if m (π (d), x) > 0.
H∗ K
Suppose now that x is an A∗-stable character of H∗. In this situation
we denote by C the subcoalgebra of A generated by the set A
πK−1(x) x
of all characters d ∈ Irr(A∗) that seat over x, i.e with the property
π (d) = ǫ(d)x. Also Remark 3.6 shows that the set A is precisely the
K x
set of irreducible constituents of Λ d for any A∗-character d seating
K
over x. Here Λ ∈ K is the idempotent integral of K.
K
Theorem 3.16. Any normal Hopf subalgebra H of D(A) is of the
following form:
H = ⊕ C ⊲⊳ C
x∈X x↑AL1 πL2−1(ψ(x))
where L and L are normal Hopf subalgebras of A, X ⊂ G (L ) is a
1 2 st 1
subgroupof A-stable linearcharactersof L and ψ : X → G ((A//L )∗)
1 st 2
is a group monomorphism into the group of A∗-stable linear characters
of (A//L )∗.
2