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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.48 n.3 Bahía Blanca 2007
On pointed Hopf algebras associated with alternating and dihedral groups
Nicolás Andruskiewitsch and Fernando Fantino
Abstract. We classify finite-dimensional complex pointed Hopf algebras with group of group-like elements isomorphic to . We show that any pointed Hopf algebra with infinitesimal braiding associated with the conjugacy class of
is infinite-dimensional if the order of
is odd except for
in
. We also study pointed Hopf algebras over the dihedral groups.
2000 Mathematics Subject Classification. 16W30; 17B37
This work was partially supported by Agencia Córdoba Ciencia, ANPCyT-Foncyt, CONICET and Secyt (UNC)
Dedicado a María Inés Platzeck en sus # años
In this article, we continue the work of [AZ, AF] on the classification of finite-dimensional complex pointed Hopf algebras with
non-abelian. We follow the Lifting Method - see [AS2] for a general reference; in particular, we focus on the problem of determining when the dimension of the Nichols algebra associated with conjugacy classes of
is infinite. The paper is organized as follows. In Section 1, we review some general facts on Nichols algebras corresponding to finite groups. We discuss the notion of absolutely real element of a finite group in subsection 1.2. We then provide generalizations of [AZ, Lemma 1.3], a basic tool in [AZ, AF], see Lemmata 1.8 and 1.9. Section 2 is devoted to pointed Hopf algebras with coradical
. We prove that any finite-dimensional complex pointed Hopf algebra
with
is isomorphic to the group algebra of
; see Theorem 2.6. This is the first finite non-abelian group
such that all pointed Hopf algebras
with
are known. We also prove that
, for any
in
of odd order, except for
or
in
- see Theorem 2.3. This last case is particularly interesting. It corresponds to a "tetrahedron" rack with constant cocycle
. The technique in the present paper does not provide information on the corresponding Nichols algebra. We also give partial results on pointed Hopf algebras with groups
and
, and on Nichols algebras
, with
even. In Section 3, we apply the technique to conjugacy classes in dihedral groups. It turns out that it is possible to decide when the associated Nichols algebra is finite-dimensional in all cases except for
(if
is odd), or
or
or
or
(if
is even). See below for undefined notations. We finally observe in Section 4 that there is no finite-dimensional Hopf algebra with coradical isomorphic to the Hopf algebra
discovered in [Ni], except for
itself.
For we denote by
the centralizer of
in
. If
is a subgroup of
and
we will denote
the conjugacy class of
in
. Sometimes we will write in rack notations
,
,
. Also, if
is a braided vector space, that is
is a solution of the braid equation, then
denotes its Nichols algebra.
We denote by the group of
-th roots of 1 in
.
1.1. Preliminaries. Let be a finite group,
a conjugacy class of
,
fixed,
an irreducible representation of
,
the corresponding irreducible Yetter-Drinfeld module. Let
, …,
be a numeration of
and let
such that
for all
. Then
. Let
,
,
. If
and
, then the coaction and the action of
are given by
where , for some
and
. The Yetter-Drinfeld module
is a braided vector space with braiding given by
![]() | (1) |
for any ,
, where
for unique
,
and
. Since
, the Schur Lemma says that
![]() | (2) |
Let be a finite non-abelian group. Let
be a conjugacy class of
and let
be an irreducible representation of the centralizer
of a fixed
. Let
be the irreducible Yetter-Drinfeld module corresponding to
and let
be its Nichols algebra. As explained in [AZ, AF, Gñ], we look for a braided subspace
of
of diagonal type such that the dimension of the Nichols algebra
is infinite. This implies that the dimension of
is infinite too.
Lemma 1.1. If is a subspace of
such that
and
, then
.□
Recall that a braided vector space is of diagonal type if there exists a basis
of
and non-zero scalars
,
, such that
, for all
. A braided vector space
is of Cartan type if it is of diagonal type and there exists
,
such that
for all
; by
we mean
if
is not a root of 1, otherwise it means the order of
in the multiplicative group of the units in
. Set
for all
. Then
is a generalized Cartan matrix.
Theorem 1.2. ([H, Th. 4], see also [AS1, Th. 1.1]). Let be a braided vector space of Cartan type. Then
if and only if the Cartan matrix is of finite type. □
We say that is real if it is conjugate to
; if
is real, then the conjugacy class of
is also said to be real. We say that
is real if any
is real.
The next application of Theorem 1.2 was given in [AZ]. Let be a finite group,
,
the conjugacy class of
,
irreducible;
was defined in (2).
Lemma 1.3. Assume that is real. If
then
and
has even order.□
If , this is [AZ, Lemma 2.2]; if
then
but
is excluded by Lemma 1.1.
The class of real groups includes finite Coxeter groups. Indeed, all the characters of a finite Coxeter group are real valued, see subsection 1.2 below, and [BG] for . Therefore, we have:
Theorem 1.4. Let be a finite Coxeter group. If
has odd order, then
, for any
. □
1.2. Absolutely real groups. Let be a finite group. We say that
is absolutely real if there exists an involution
in
such that
. If this happens, any element in the conjugacy class of
is absolutely real and we will say that the conjugacy class of
is absolutely real. We say that
is absolutely real if any
is so. The finite Coxeter groups are absolutely real. Indeed,
- the dihedral groups are absolutely real, by straightforward computations.
- the Weyl groups of semisimple finite dimensional Lie algebras are absolutely real, by [C, Th. C (iii), p. 45].
is absolutely real, by Proposition 1.7 below.
is absolutely real, we have checked it using GAP3, [S].
Remark 1.5. Let ,
be finite groups. We note:
is absolutely real iff both
and
are absolutely real.
is absolutely real iff both
and
are absolutely real.
- Assume
abelian. Then
is absolutely real iff
has exponent 2, i. e.
for some integer
.
- If
is absolutely real and
is abelian of exponent 2 then
is absolutely real.
We first discuss when an element of is absolutely real. Assume that
is of type
. Then
iff
is even.
Lemma 1.6. (a). If , then
is absolutely real in
.
(b). If is even then
is absolutely real in
.
Proof. Let for some
and take
![gjτjgj = τ -1 j](/img/revistas/ruma/v48n3/3a04199x.png)
![g2= id j](/img/revistas/ruma/v48n3/3a04200x.png)
To prove (b), observe that there exists an involution such that
, which is a product of "translations" of the
's. Since the sign of
is
,
iff
is even; (b) follows. We prove (a). By assumption there are at least two points fixed by
, say
,
. By the preceding there exists an involution
such that
. If
we are done, otherwise take
;
is an involution and
. □
Proposition 1.7. The groups and
are absolutely real.
Proof. The type of is either
,
,
or
; in the first two cases
is absolutely real by Lemma 1.6 part (a), in the last two by part (b). Since
(see [Hu, Section 2.13]), then the Coxeter group is absolutely real by Remark 1.5. □
1.3. Generalizations of Lemma 1.3. The next two Lemmata are variations of [AZ, Lemma 2.2]. A result in the same spirit appears in [FGV]. We deal with elements having a power in
, the conjugacy class of
. Clearly, if
is in
, then
is in
, for every
. So,
; this implies that
divides
. Hence
with , recall (2).
Lemma 1.8. Let be a finite group,
,
the conjugacy class of
and
. Assume that there exists an integer
such that
,
and
are distinct elements and
is in
. If
, then
has even order and
.
Proof. We assume that , thus
. It is easy to see that
for every ,
. We will call
,
,
,
,
; so
, for
,
,
. The other relations between
's and
's are obtained from (4). For
and
or
, we define
. Hence,
is a braided vector subspace of
of Cartan type with
![a (l) = a (l) ≡ j |σ|- l + jl mod (N ) 12 21](/img/revistas/ruma/v48n3/3a04277x.png)
![dim 𝔅 (O, ρ) < ∞](/img/revistas/ruma/v48n3/3a04278x.png)
![a12(l) = a21(l) = 0](/img/revistas/ruma/v48n3/3a04279x.png)
![- 1](/img/revistas/ruma/v48n3/3a04280x.png)
(i) Let us suppose that
![a12(1) = a21(1) = 0](/img/revistas/ruma/v48n3/3a04281x.png)
![j|σ|-1 + j ≡ 0 mod (N )](/img/revistas/ruma/v48n3/3a04282x.png)
![N](/img/revistas/ruma/v48n3/3a04283x.png)
![j|σ| - 1](/img/revistas/ruma/v48n3/3a04284x.png)
![N](/img/revistas/ruma/v48n3/3a04285x.png)
![j2 + 1](/img/revistas/ruma/v48n3/3a04286x.png)
- Assume that
. Then
. Since
divides
, we have that
divides
. So,
; hence the result follows.
- Assume that
. Then
. We can see that
divides
. This implies that
divides
, a contradiction.
(ii) Let us suppose that . This implies that
. Since
divides
, we have that
divides
. We consider now two possibilities.
- Assume that
. Then
. So,
divides
. It is easy to see that
divides
. Since
and
are relatively prime,
must be
, a contradiction.
- Assume that
. This means that the subspace
of
is of Cartan type with
, a contradiction.
This concludes the proof. □
Lemma 1.9. Let be a finite group,
,
the conjugacy class of
and
such that
. Assume that there exists an integer
such that
and
is in
.
- If
, then
has even order and
.
- If
, then either
has even order and
, or
.
Proof. We will proceed and use the notation as in the proof of Lemma 1.8. If , then the result follows by Lemma 1.8. Assume that
. This implies that
divides
, so
divides
.
(a) Let and
in
linearly independent and let
- span of
, with
and
. Thus
is a braided vector subspace of
of Cartan type with
![aij = aji](/img/revistas/ruma/v48n3/3a04360x.png)
![i ⁄= j](/img/revistas/ruma/v48n3/3a04361x.png)
![a12 ≡ 2 ≡ a34 mod (N )](/img/revistas/ruma/v48n3/3a04362x.png)
![a13 = a14 = a23 = a24](/img/revistas/ruma/v48n3/3a04363x.png)
![a12 = 0](/img/revistas/ruma/v48n3/3a04365x.png)
![a34 = 0](/img/revistas/ruma/v48n3/3a04366x.png)
![N](/img/revistas/ruma/v48n3/3a04367x.png)
![2](/img/revistas/ruma/v48n3/3a04368x.png)
![a13 = 0](/img/revistas/ruma/v48n3/3a04369x.png)
![j|σ|-1 + j ≡ 0 mod (N )](/img/revistas/ruma/v48n3/3a04370x.png)
![N](/img/revistas/ruma/v48n3/3a04371x.png)
![j2 + 1](/img/revistas/ruma/v48n3/3a04372x.png)
![N](/img/revistas/ruma/v48n3/3a04373x.png)
![aij = - 1](/img/revistas/ruma/v48n3/3a04374x.png)
![i,j](/img/revistas/ruma/v48n3/3a04375x.png)
![A](/img/revistas/ruma/v48n3/3a04376x.png)
![dim 𝔅 (O, ρ) = ∞](/img/revistas/ruma/v48n3/3a04377x.png)
(b) For we define
, with
and
. Hence,
is a braided vector subspace of
of Cartan type with
![|σ|-1 a12 ≡ j + j mod (N )](/img/revistas/ruma/v48n3/3a04385x.png)
![dim 𝔅 (O, ρ) < ∞](/img/revistas/ruma/v48n3/3a04386x.png)
![a12 = 0](/img/revistas/ruma/v48n3/3a04387x.png)
![- 1](/img/revistas/ruma/v48n3/3a04388x.png)
(i) Assume that
![a12 = 0](/img/revistas/ruma/v48n3/3a04389x.png)
![j|σ|-1 + j ≡ 0 mod (N )](/img/revistas/ruma/v48n3/3a04390x.png)
![N](/img/revistas/ruma/v48n3/3a04391x.png)
![j|σ| - 1](/img/revistas/ruma/v48n3/3a04392x.png)
![N](/img/revistas/ruma/v48n3/3a04393x.png)
![j2 + 1](/img/revistas/ruma/v48n3/3a04394x.png)
![N](/img/revistas/ruma/v48n3/3a04395x.png)
![2](/img/revistas/ruma/v48n3/3a04396x.png)
![N = 2](/img/revistas/ruma/v48n3/3a04397x.png)
(ii) Assume that
![a12 = - 1](/img/revistas/ruma/v48n3/3a04398x.png)
![|σ|-1 j + j ≡ - 1 mod (N )](/img/revistas/ruma/v48n3/3a04399x.png)
![N](/img/revistas/ruma/v48n3/3a04400x.png)
![|σ| j - 1](/img/revistas/ruma/v48n3/3a04401x.png)
![N](/img/revistas/ruma/v48n3/3a04402x.png)
![2 j + j + 1](/img/revistas/ruma/v48n3/3a04403x.png)
![N](/img/revistas/ruma/v48n3/3a04404x.png)
![j + 2](/img/revistas/ruma/v48n3/3a04405x.png)
![p](/img/revistas/ruma/v48n3/3a04406x.png)
![N](/img/revistas/ruma/v48n3/3a04407x.png)
![p](/img/revistas/ruma/v48n3/3a04408x.png)
![j - 1](/img/revistas/ruma/v48n3/3a04409x.png)
![j + 1](/img/revistas/ruma/v48n3/3a04410x.png)
![N](/img/revistas/ruma/v48n3/3a04411x.png)
![j2 - 1](/img/revistas/ruma/v48n3/3a04412x.png)
![p](/img/revistas/ruma/v48n3/3a04413x.png)
![j + 1](/img/revistas/ruma/v48n3/3a04414x.png)
![p](/img/revistas/ruma/v48n3/3a04415x.png)
![1](/img/revistas/ruma/v48n3/3a04416x.png)
![N](/img/revistas/ruma/v48n3/3a04417x.png)
![j - 1](/img/revistas/ruma/v48n3/3a04418x.png)
![N](/img/revistas/ruma/v48n3/3a04419x.png)
![3](/img/revistas/ruma/v48n3/3a04420x.png)
![N = 3](/img/revistas/ruma/v48n3/3a04421x.png)
We recall that we will denote or
the conjugacy class of an element
in
, and
in
, a representative of an isomorphism class of irreducible representations of
. We want to determine pairs
, for which
, following the strategy given in [AZ, AF]; see also [Gñ].
The following is a helpful criterion to decide when a conjugacy class of an even permutation in
splits in
.
Proposition 2.1. [JL, Proposition 12.17] Let , with
.
- If
commutes with some odd permutation in
, then
and
.
- If
does not commute with any odd permutation in
, then
splits into two conjugacy classes in
of equal size, with representatives
and
, and
.□
Remarks 2.2. (i) Notice that if satisfies (1) of Proposition 2.1, then
is real. The reciprocal is not true, e.g. consider
in
.
(ii) One can see that if in
is of type
, then
satisfies (2) of Proposition 2.1 if and only if
or
,
and
, for all
. Thus, if
has even order, then
is real.
We state the main Theorem of the section.
Theorem 2.3. Let and
. Assume that
is neither
nor
in
. If
, then
has even order and
.
Proof. If is even the result follows by Lemma 1.3 and Remark 2.2 (ii). Let us suppose that
and odd . If
is in
, then the result follows by Lemma 1.3. Assume that
. We consider two cases.
(i) If , then
is in
, and
because
. Hence, the result follows from Lemma 1.8.
(ii) Assume that . We know that there exist
and
in
, necessarily odd permutations, such that
and
. Then
and
; so
is in
. This implies that
is in
, and
because
is odd. Now, the result follows from Lemma 1.8.
Finally, let us suppose that , with type
. If
or
, then
is real, by Lemma 1.6 (a) and Remark 2.2, respectively. Hence, the result follows by Lemma 1.3. This concludes the proof. □
2.1. Case . Obviously,
; thus
is not real. This case was considered in [AS1, Theorem 1.3].
2.2. Case . It is straightforward to check that
is not real, since
is not real in
. Let
in
; then the type of
may be
,
or
. If the type of
is
, then
, for any
in
, by Lemma 1.1. If the type of
is
, then
is not real; moreover we have
![π 𝔸4 = ⟨π ⟩ ≃ ℤ3](/img/revistas/ruma/v48n3/3a04524x.png)
![^ π ρ ∈ 𝔸 4](/img/revistas/ruma/v48n3/3a04525x.png)
![dim 𝔅 (O π,ρ) = ∞](/img/revistas/ruma/v48n3/3a04526x.png)
The following result is a variation of [AZ, Theorem 2.7].
Proposition 2.4. Let in
of type
. Then
, for every
in
.
Proof. We can assume that . If we call
,
and
, then
and
. If
,
and
, then
,
, and
Let in
and
, where
is the vector space affording
. Thus
is a braided vector space with braiding given by - see (1)-
and
,
and
Clearly, , by Lemma 1.1. If we consider
(resp.
), then
is of Cartan type with matrix of coefficients
given by
![( ) 2 - 1 - 1 A = ( - 1 2 - 1 ) - 1 - 1 2](/img/revistas/ruma/v48n3/3a04561x.png)
![dim 𝔅 (Oπ, ρ) = ∞](/img/revistas/ruma/v48n3/3a04562x.png)
2.3. Case . Here is the key step in the consideration of this case.
Lemma 2.5. Let . Then
, for every
in
.
Proof. Let . If the type of
is either
,
or
, we have that
, by Lemma 1.3 and Proposition 1.7. Let us assume that the type of
is
. For
, let
and
be as in the proof of Proposition 2.4. By Proposition 2.1 and straightforward computations, we have that
and
. Notice that
,
. Let
and
, where
is the vector space affording
; then
is a braided vector subspace of
. Therefore,
, by the same argument as in the proof of Proposition 2.4. □
As an immediate consequence of Lemma 2.5 we have the following result.
Theorem 2.6. Any finite-dimensional complex pointed Hopf algebra with
is necessarily isomorphic to the group algebra of
.
Proof. Let be a complex pointed Hopf algebra with
. Let
be the infinitesimal braiding of
-see [AS2]. Assume that
; thus
. Let
be an irreducible submodule. Then
, by Lemma 2.5. Hence,
and
. □
2.4. Case . Let
be in
. If the type of
is
,
,
,
or
, then
is absolutely real by Lemma 1.6, and if the type of
is
, then
is real because it has even order - see Remark 2.2 (ii). Hence,
is a real group. We summarize our results in the following statement.
Theorem 2.7. Let be an irreducible Yetter-Drinfeld module over
, corresponding to a pair
. If
, then
, with
, and
.
Remark 2.8. In this Theorem we do not claim that the condition is sufficient.
Proof. Let be in
. If the type of
is
Let us suppose that the type of is
; we can assume that
. It is easy to check that
Notice that . It is known that
, where
,
,
,
and
, are the following characters
and is the
-dimensional representation given by
It is clear that ,
,
,
and
. Then
, by Lemma 1.1. Let us consider now that
. We define
,
,
,
,
and
. It is clear that
If we have that
and
. We define
. Then
is a braiding subspace of
of Cartan type with
Since is not of finite type we have that
, by Theorem 1.2.
Finally, let us assume that the type of is
. Then
has
elements and
. We call
, where
,
,
,
,
. It is clear that if
, with
,
or
, then
. This implies that
, by Lemma 1.3. □
Remark 2.9. We can see that every maximal abelian subrack of has two elements. Hence,
is a negative braided space in the sense of [AF].
2.5. Case ,
. Let
, with
even. We now investigate the Nichols algebras associated with
by reduction to the analogous study for the orbit of
in
, [AF]. By Remark 2.2 (ii),
and
. So, we can determinate the irreducible representations of
from those of
. We know that if the type of
is
, then
with
,
.
Some generalities and notation. Let be a finite group,
a subgroup of
of index two, and
a representation of
. It is easy to see that
![G](/img/revistas/ruma/v48n3/3a04718x.png)
![ResGH η = ResGH η′](/img/revistas/ruma/v48n3/3a04719x.png)
![ρ](/img/revistas/ruma/v48n3/3a04720x.png)
![H](/img/revistas/ruma/v48n3/3a04721x.png)
![ρ-](/img/revistas/ruma/v48n3/3a04722x.png)
![H](/img/revistas/ruma/v48n3/3a04723x.png)
![ρ](/img/revistas/ruma/v48n3/3a04724x.png)
![ρ(h ) := ρ(ghg -1)](/img/revistas/ruma/v48n3/3a04725x.png)
![h ∈ H](/img/revistas/ruma/v48n3/3a04726x.png)
![g](/img/revistas/ruma/v48n3/3a04727x.png)
![G \ H](/img/revistas/ruma/v48n3/3a04728x.png)
![g](/img/revistas/ruma/v48n3/3a04729x.png)
![H](/img/revistas/ruma/v48n3/3a04730x.png)
Let such that
; thus
. Let
in
. Then we have two cases:
. If
, then
,
and
.
. We have that
and
.
Moreover, if is an irreducible representation of
, then
is a restriction of some
or is a direct summand of
as in (ii), see [FH, Ch. 5].
Remark 2.10. If and
, it is easy to check that
as braided vector spaces.
We apply these observations to the case and
. We use some notations given in [AF, Section II.D].
Lemma 2.11. Assume that the type of is
, with
and
even. Let
in
, with
.
- If
, then
.
- If
and
, then
- if
, then
.
- Assume that
. If
, then
. Assume that
. If
, with
even or odd, or if
, with
even and
or
, where
or
, then the braiding is negative; otherwise,
.
- if
Proof. (a) follows by Remark 2.2 (ii) and Lemma 1.3. (b). Since ,
, with
,
and
. Notice that
because
. Now, as the racks are the same, i.e.
, we can apply [AF, Theorem 1]. □
Remark 2.12. Keep the notation of the Lemma. If is not isomorphic to its conjugate representation
, then there exists
such that
and
. Clearly,
and
act by scalar
, and we have that
as braided vector spaces. We do not get new information with the techniques available today.
We fix the notation: the dihedral group of order
is generated by
and
with defining relations
and
. Let
be a primitive
-th root of 1 and let
be the character of
,
. If
then we denote the conjugacy class by
or simply
.
Theorem 3.1. Let be the irreducible Yetter-Drinfeld module over
corresponding to a pair
. Assume that its Nichols algebra
is finite-dimensional.
- If
is odd, then
, where
,
.
- If
is even, then
is one of the following:
where
satisfies
.
where
,
and
.
or
, where
,
,
.
or
, where
,
,
.
In the cases (i) and (ii) the dimension is finite. In the cases (iii) and (iv), the braiding is negative in the sense of [AF].
Remark 3.2. There are isomorphisms of braided vector spaces
Remark 3.3. Assume for simplicity that is odd and that
, where
,
are integers
. Then the (indecomposable) rack
is a disjoint union of
racks isomorphic to
; in other words,
is an extension of
by
(and vice versa), see [AG, Section 2]. Thus, there is an epimorphism of braided vector spaces
, as well as an inclusion
. The techniques available today do not allow to compute the Nichols algebra
from the knowledge of the Nichols algebra
.
Remark 3.4. In Theorem 3.1 we do not claim that the conditions are sufficient. See Tables 1, 2. For instance, it is known that when
- see [MS]; for other odd
, this is open.
Table 1. Nichols algebras of irreducible Yetter-Drinfeld modules over ,
odd.
Let us now proceed with the proof of Theorem 3.1.
Proof. If , then
and
, from Lemma 1.1.
We consider now two cases.
CASE 1: odd.
(I) If , with
, it is easy to see that
and
. Then
, where
, with
a primitive
-th root of
. Let us consider
; it is a braided vector space of diagonal type. If
, then
, from Lemma 1.3. Assume
; so we have
This is a contradiction because
is odd.
(II) If , then
and
. Clearly,
. On the other hand,
is a negative braided vector space, since every abelian subrack of
has one element; indeed
,
, if and only if
.
Therefore, the part (a) of the Theorem is proved.
Table 2. Nichols algebras of irreducible Yetter-Drinfeld modules over ,
even.
CASE 2: even. Let us say
.
(I) If , then
and
. Clearly,
, for every
with
. On the other hand, if
is such that
, then it is straightforward to prove that
, the exterior algebra of
; hence
.
(II) If ,
; then
and
. From Lemma 1.3, it is clear that
, for every
such that
, i.e.
. On the other hand, it is easy to see that
, hence
, for every
with
.
(III) If , then
and
. From Lemma 1.1,
.
For the cases or
, we note the following fact.
- If
is odd and
, we have that
- If
is even and
, we have that
The cases (i) and (ii) say that every maximal abelian subrack of has one and two elements, respectively. Hence, in both cases the braiding is negative. Indeed, the result is obvious for the case (i), while in the case (ii) we have that if
and
commute in
, then
and
; thus the braiding is negative.
(IV) If , then
and
. The result follows as in (III) using the isomorphism
,
,
. □
4. On Nichols algebras over semisimple Hopf algebras
Let be a Hopf algebra. Let
be a twist and let
be the corresponding twisted Hopf algebra. If
is a Hopf subalgebra of a Hopf algebra
, then
is a twist for
and
is a Hopf subalgebra of
. Now, if
is semisimple, then this induces a bijection
![]() | (7) |
that preserves standard invariants like dimension, Gelfand-Kirillov dimension, etc. Let now and let
be the non-trivial twist defined in [Ni]. By (7), we conclude immediately from Theorem 2.6.
Theorem 4.1. Let be a finite-dimensional Hopf algebra with coradical isomorphic to
. Then
. □
Again, this is the first classification result we are aware of, for finite-dimensional Hopf algebras with coradical isomorphic to a fixed non-trivial semisimple Hopf algebra. Recently, a semisimple Hopf algebra was discovered in [GN]. This Hopf algebra
is simple, that is it has no non-trivial normal Hopf subalgebra. Since there are finite-dimensional non-semisimple pointed Hopf algebras with group
, there is a finite-dimensional non-semisimple Hopf algebra with coradical isomorphic to
.
Acknowledgement. We are grateful to Professor John Stembridge for information on Coxeter groups, in particular reference [BG]. We thank Matías Graña, Sebastián Freyre and Leandro Vendramín for interesting discussions.
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Nicolás Andruskiewitsch
Facultad de Matemática, Astronomía y Física,
Universidad Nacional de Córdoba. CIEM - CONICET.
Medina Allende s/n
(5000) Ciudad Universitaria, Córdoba, Argentina
andrus@mate.uncor.edu
Fernando Fantino
Facultad de Matemática, Astronomía y Física,
Universidad Nacional de Córdoba. CIEM - CONICET.
Medina Allende s/n
(5000) Ciudad Universitaria, Córdoba, Argentina
fantino@mate.uncor.edu
Recibido: 16 de febrero de 2007
Aceptado: 2 de agosto de 2007