Pointed Hopf Algebras with classical Weyl Groups

0.31 MB·English

by  Shouchuan Zhang

Pointed Hopf algebras with classical Weyl groups
Shouchuan Zhang a,b, Yao-Zhong Zhang b,
1
1 a. Department of Mathematics, Hunan University
0
2
Changsha 410082, P.R. China

r
a
M b. School of Mathematics and Physics, The University of Queensland
Brisbane 4072, Australia
2

] Emails: [email protected] (SZ); [email protected] (YZZ)
A
Q
March 3, 2011
.
h
t
a
m
Abstract
[

2 We prove that except in three cases Nichols algebras of irreducible Yetter-
v Drinfeld modules over classical Weyl groups A⋊ S supported by S are infinite
0 n n
8 dimensional. We give the necessary and sufficient conditions for Nichols algebras of
2
1 Yetter-Drinfeld modules over classical Weyl groups A⋊Sn supported by A to be
.
0 finite dimensional.
1
9 2000 Mathematics Subject Classification: 16W30, 16G10
0
keywords: Quiver, Hopf algebra, Weyl group.
:
v
i
X
r 0 Introduction
a
The framework of the paper is the classification of finite dimensional complex pointed
Hopf algebras with non abelian group G. It is known that the first necessary step is to
consider the Nichols algebras associated to the irreducible Yetter-Drinfeld modules over G
anddecide if theyarefinitedimensional or not. These irreducible Yetter-Drinfeldmodules
are easy to classify: they are in one-to -one correspondence with pairs (C,ρ) where C is a
conjugacy class of G and ρ is an irreducible representation of the centralizer of an element
in C. In short, it is necessary to see if the dimension of the corresponding Nichols algebra
B(C,ρ) is finite or not. As pointed out early by Grana, one may start attacking this
problem by looking at Nichols subalgebras of B(C,ρ).
This paper deal specifically with the case when G is the Weyl group of a simple Lie
algebra. The most prominent example is the symmetric group S . For the group, a fair
n
1
complete analysis is presented in [AFGV08], as a culmination of the series papers of
[AZ07, AFZ]. The outcome is that all B(C,ρ) have infinite dimension, except for a small
list of examples when n ≤ 6 and remarkable cases corresponding to C = the class of
transpositions, and ρ = the restriction of representation sign or a closely related one. It
is natural to guess that other Weyl group would also be distinguished.
The main results in this paper are summarized in the following statements.
Theorem 0.1. Let G = A⋊S with n > 2. Let σ ∈ S be of type (1λ1,2λ2,...,nλn).
n n
If ρ ∈ Gσ, then ρ may be presented as follows: ρ = θ = (χ⊗µ) ↑Gσ ∈ A\σ ⋊Sσ with
χ,µ (Gσ)χ n
χ = (χb1 ⊗···χbn) ∈ Aσ and µ ∈ ([Sσ) , where µ = ⊗ µ with µ ∈ Sσi. Furthermore,
2 2 n χ 1≤i≤n i i Yi
if dim B(OG,ρ) < ∞, then some of the following hold with Y ⊆ W or with Y ∩W = ∅.
σ i χ i χ
c c
(i) (1λ1,2), µ = sgn or ǫ, µ = χ .
1 2 (1;2)
(ii) (21,31), µ = χ , µ = χ .
2 (1;2) 3 (0;3)
(iii) (23), µ = χ ⊗ǫ or χ ⊗sgn.
2 (1,1,1;2) (1,1,1;2)
(iv) (4), µ = χ or χ .
1 (1;4) (3;4)
Theorem 0.2. Let G = A⋊S be a classical Weyl group with A ⊆ (C )n and n > 2.
n 2
Assume that M = M(O ,ρ(1)) ⊕ M(O ,ρ(2)) ⊕ ··· ⊕ M(O ,ρ(m)) is a reducible YD
σ1 σ2 σm
module over kG .
(i) Assume that there exist i 6= j such that σ , σ ∈/ A. If dimB(M) < ∞, then n = 4,
i j
the type of σ is 22 and the sign of σ is stable when σ ∈/ A.
p p p
(ii) Assume that there exists σ ∈/ A. If dimB(M) < ∞, then σ = (a,τ) with τ2 = 1.
i i
(iii) Assume that σ ∈/ A for 1 ≤ i ≤ m. If dimB(M) < ∞, then there is at most one
i
σ ∈/ Z(G).
i
(iv) If σ = α =: (g ,g ,··· ,g ) ∈ G and ρ(i) = θ := (χ(i)⊗µ(i)) ↑Gα ∈ Gα, 1 ≤
i 2 2 2 χ(i),µ(i) Gα
χ(i)
i ≤ m. Then B(M) is finite dimensional if and only if χ(i)(α) = −1 for i = 1,2,··· ,m.
c
Indeed, Theorem 0.1 follows Theorem 2.7. Theorem 0.2(i ) follows Remark 4.12.
Theorem 0.2(ii ) follows Theorem 4.13. Theorem 0.2(iii ) follows Theorem 4.13. Theorem
0.2(iv ) follows Remark 3.9.
Now we present these results by means of the following table. In this table Nichols
algebras B(O ,ρ) of irreducible Yetter-Drinfeld modules over G = A⋊S ( n > 2) have
σ n
infinite dimensions.
2
case σ Representation Reference
1 σ = e, the unity of G any [AZ07]
2 σ ∈ A ,n ≥ 5,n 6= 6 any Th. 2.5
n
3 σ ∈ S ,n ≥ 3,n 6= 4,5,6,
n
σ is not a transposition any Th. 2.7
4 the type of σ is 23, σ ∈ S , n = 6 µ 6= χ ⊗ǫ,
6 2 (1,1,1;2)
µ 6= χ ⊗sgn. Th. 2.7
2 (1,1,1;2)
5 the type of σ is 2131, σ ∈ S , n = 5 µ 6= χ , µ 6= χ Th. 2.7
5 2 (1;2) 3 (0;3)
6 the type of σ is 41, σ ∈ S , n = 4 µ 6= χ , χ Th. 2.7
4 1 (1;4) (3;4)
7 the type of σ is not 23, 2131, 41,
1λ121 any Th. 2.7
8 σ = (α,τ), α = (g ,g ,··· ,g ), χ(α) = 1,
2 2 2
τ ∈ S , τ satisfies the case 1–7 µ satisfies the case 1–7 Pro. 2.10
n
9 σ = (a,τ), the type of τ is 22,
a = (ga1,ga2,ga3,ga4),
2 2 2 2
a +a ≡ a +a ≡ 0 (mod 2) any Pro. 2.9
1 2 3 4
Table 1
For general case σ = (a,τ) ∈ G = A⋊ S , we have not considered because we have
n
not known the relation between Nichols algebras over group G = A⋊S and group S .
n n
This paper is organized as follows. In section 1 we give the relation between Nichols
algebras over group A⋊D and group D. In section 2 we prove that except in three cases
Nichols algebras of irreducible YD modules supported by S are infinite dimensional. In
n
section 3 we give the relationship between Nichols algebrasover a groupG anda subgroup
D of G. In section 4, we give a necessary and sufficient condition for a Nichols algebra
of a YD module supported by A to be finite dimensional. It is proved that if M is a
reducible YD module over kG supported by S with n ≥ 3, then dimB(M) = ∞ and if
n
M is a YD module over kG supported by A with n ≥ 5 and n 6= 6, then dimB(M) = ∞.
n
In section 5 we establish the relationship between Nichols algebras over the Weyl groups
of B and D . In the Appendix, the conjugacy classes of the Weyl groups of B and D
n n n n
are presented.
Preliminaries and Conventions
Let k be the complex field and G a finite group. Let G denote the set of all isomorphism
classes of irreducible representations of the group G, Gσ be the centralizer of σ, O or OG
σ σ
b
be the conjugacy class in G, C the cyclic group with order j, g be a generator of C and
j j j
χ be a character of C with order j. The Weyl groups of A , B , C and D are called the
j j n n n n
3
classical Weyl groups, written asW(A ), W(B ), W(C ) and W(D ), respectively. Given
n n n n
a representation ρ of the subgroup D of G, let ρ ↑G denote the induced representation of
D
G as in [Sa01].
Let deg(ρ) denote the dimension of the representation space V for a representation
(ρ,V).
Recall the notation RSR in [ZCZ, Definition 1.1]. Let ρ be a representation of Gu(C)
C
(i)
with irreducible decomposition ρ = ⊕ ρ , where u is a map from K(G) to G with
C i∈IC(r,u) C
−→
u(C) ∈ C for any C ∈ K(G) and I (r,u) is an index set. Let ρ denote {ρ } =
C C C∈Kr(G)
(i) −→
{{ρ } } . Then (G,r, ρ,u) is called an RSR if deg(ρ ) = r for any C ∈
C i∈IC(r,u) C∈Kr(G) C C
−→ −→
K (G), written as RSR(G,r, ρ,u). For any RSR(G,r, ρ,u), we obtain a co-path Hopf
r
−→ −→
algebra kQc(G, r, ρ,u), a Hopf algebra kG[kQc, G,r, ρ,u] of one type, a kG-YD module
1
(kQ1,ad(G,r,→−ρ,u)) and a Nichols algebra B(G,r,→−ρ,u) := B(kQ1,ad(G,r,→−ρ,u)) (see
1 1
[ZCZ]).
→−
If a ramification r = r C and |I (r,u)|= 1 then we say that RSR(G,r, ρ,u) is of bi-
C C
one since r only has one conjugacy class C and |I (r,u)|= 1. Furthermore, if σ = u(C),
C
(i) −→
C = O , r = m and ρ = ρ for i ∈ I (r,u), then bi-one RSR(G,r, ρ,u) is denoted by
σ C C C
RSR(G,mO ,ρ), or RSR(G,O ,ρ) in short.
σ σ
→−
RSR(G,r, ρ,u) is said to be −1-type, if u(C) is real (i.e. u(C)−1 ∈ C) and the order
(i)
of u(C) is even with ρ (u(C)) = −id for any C ∈ K (G) and any i ∈ I (r,u). In this
C r C
case, the Nichols algebra B(G,r,−→ρ,u) is said to be −1-type.
For s ∈ G and (ρ,V) ∈ Gs, here is a precise description of the YD module M(O ,ρ),
s
see [Gr00, AZ07]. Let t = s, ..., t be a numeration of conjugacy class O containing
1 m s
s, and choose h ∈ G succh that h (cid:3) s := h sh−1 = t for all 1 ≤ i ≤ m. Then
i i i i i
M(O ,ρ) = ⊕ h ⊗ V. Let h v := h ⊗ v ∈ M(O ,ρ), 1 ≤ i ≤ m, v ∈ V. If
s 1≤i≤m i i i s
v ∈ V and 1 ≤ i ≤ m, then the action of h ∈ G and the coaction are given by
δ(h v) = t ⊗h v, h·(h v) = h (γ ·v), (0.1)
i i i i j
where hh = h γ, for some 1 ≤ j ≤ m and γ ∈ Gs. Let B(O ,ρ) denote B(M(O ,ρ)).
i j s s
By [ZZWCY, Lemma 1.1], there exists a bi-one arrow Nichols algebra B(G,r,−→ρ,u) such
that B(O ,ρ) ∼= B(G,r,→−ρ,u) as graded braided Hopf algebras in kGYD. This gives the
s kG
relation of the two Nichols algebras. A YD module is said to be reducible if it has a
non-trivial YD submodule. If ρ = ρ(1) ⊕ρ(2) ⊕···⊕ρ(m) with ρ(i) ∈ Gσ for 1 ≤ i ≤ m,
then M(OG,ρ(1)) ⊕ M(OG,ρ(2)) ⊕ ··· ⊕ M(OG,ρ(m)) is called a YD module of ρ, also
σ σ σ
c
written as M(OG,ρ).
σ
Let V be a braided vector space with a basis {x | i = 1,2,··· ,n} and B(x ⊗x ) =
i i j
q (x ⊗x ). If there exists a generalized Cartan matrix (a ) such that satisfy
ij j i ij
q q = qaij (0.2)
ij ji jj
4
for any i,j = 1,2,··· ,n, then braiding B ( or V, or B(V)) is called a braiding of the
Cartan type. We assume that we choose a such that they maximally satisfy (0.2). That
ij
is, a isthemaximal non-positiveintegersatisfying (0.2). Thus B(V)isfinitedimensional
ij
if and only if A is of finite type (see [He06b, Theorem 4]).
IfD isasubgroup ofG andC isa conjugacyclass ofD, then C denotes theconjugacy
G
class of G containing C.
1 G = A ⋊ D
In this section we give the relation between Nichols algebras over group A⋊D and group
D.
Let G = A ⋊ D be a semidirect product of abelian group A and group D. For any
ˆ
χ ∈ A, let D := {h ∈ D | h·χ = χ}; G := A⋊D . For an irreducible representation
χ χ χ
ρ of D , let θ := (χ ⊗ ρ) ↑G , the induced representation of χ ⊗ ρ on G. By [Se,
χ χ,ρ Gχ
ˆ
Pro.25], every irreducible representation of G is of the following form: θ . Let ǫ ∈ A
χ,ρ
with ǫ(a) = 1 for any a ∈ A. Thus D = D and θ is an irreducible representation of G.
ǫ ǫ,ρ
Lemma 1.1. Let G = A⋊D and σ ∈ D. Then Gσ = Aσ ⋊Dσ.
Proof. If x = (a,d) ∈ Gσ, then xσ = σx. Thus
a = σ ·a and dσ = σd. (1.1)
This implies d ∈ Dσ and a ∈ Aσ since σ ·a = σaσ−1.
Conversely, if x = (a,d) ∈ Aσ ⋊ Dσ, then (1.1) holds. This implies xσ = σx and
x ∈ Gσ. 2
Lemma 1.2. Let D be a subgroup of G with σ ∈ D and let right coset decompositions
of Dσ in D be
D = Dσg . (1.2)
θ
θ∈Θ
[
Then there exists a set Θ′ with Θ ⊆ Θ′ such that
G = Gσg (1.3)
θ
θ∈Θ′
[
is a right coset decompositions of Gσ in G.
Proof. For any h,g ∈ D, It is clear that hg−1 ∈ Dσ if and only if hg−1 ∈ Gσ, which
prove the claim. 2
5
→−
Lemma 1.3. If kQc(G,r, ρ,u) is a co-path Hopf algebra (see [ZZC, ZCZ]), then kG+
−→
kQ = (kG[kQc]) , where kG[kQc] := kG[kQc,G,r, ρ,u] and (kG[kQc]) denotes the
1 1 1 1 1 1 1
second term of the coradical filtration of kG[kQc].
1
Proof. By [ZCZ, Lemma 2.2], R := diag(kG[kQc]) is a Nichols algebra. [AS98,
1
Lemma 2.5] yields that kG[kQc] is coradically graded. 2
1
Definition 1.4. Let D be a subgroup of G; r and r′ ramifications of D and G, respec-
tively. If r ≤ r′ for any C ∈ K (D), then r is called a subramification of r′, written as
C CG r
r ≤ r′. Furthermore, if u(C) = u′(C ), I (r,u) ⊆ I (r′,u′) and ρ(i) is isomorphic to a
G C CG C
subrepresentation of the restriction of ρ′(i) on Du(C) for any C ∈ K (D), i ∈ I (r,u), then
CG −→ r C
−→ →−
RSR(D,r, ρ,u) is called a sub-RSR of RSR(G,r′, ρ′,u′), written as RSR(D,r, ρ,u) ≤
→−
RSR(G,r′, ρ′,u′).
Lemma 1.5. LetD bea subgroup ofG. Ifσ ∈ D, then RSR(D,O ,ρ) ≤ RSR(G,O ,ρ′)
σ σ
if and only if ρ is isomorphic to subrepresentation of the restriction of ρ′ on Dσ.
Proof. It follows from Definition 3.7. 2
−→
→−
Proposition 1.6. Let D be a subgroup of G. If RSR(D,r, ρ,u) ≤ RSR(G,r′, ρ′,u′),
then
→−
→−
(i) kQc(D,r, ρ,u) is a Hopf subalgebra kQ′c(G,r′, ρ′,u′).
−→
→−
(ii) kD[kQc,D,r, ρ,u] is a Hopf subalgebra kG[kQ′c,G,r′, ρ′,u′].
−→
(iii) If B(G,r′, ρ′,u′) is finite dimensionalwith finite group G then so is B(D,r,→−ρ,u).
Proof. (i) For any C ∈ K (D) and i ∈ I (r,u), let X(i) be a representation space of
r C C
ρ(i) with a basis {x(i,j) | j ∈ J (i)} and X′(i) a representation space of ρ′(i) with a basis
C C C CG CG
{x′(i,j) | j ∈ J (i)} and J (i) ⊆ J (i). ψ(i) is a kDu(C)-module monomorphism from
CG CG C CG C
X(i) to X′(i) with x′(i,j) = ψ(i)(x(i,j)) for i ∈ I (r,u), j ∈ J (i).
C CG CG C C C C
→−
Let φ be an inclusion map from kD to kG and ψ is a map from kQc(D,r, ρ,u) to
−→ 1
kQ′c(G,r′, ρ′,u′) by sending a(i,j) to a′(i,j) for any y,x ∈ D, i ∈ I (r,u), j ∈ J (i)
1 y,x y,x C C
with x−1y ∈ C ∈ K (D). Now we show that ψ is a kD-bimodule homomorphism from
r
kQc to (kQ′c) and a kG-bicomodule homomorphism from φ(kQc)φ to kQ′c. We only
1 φ 1 φ 1 1
show this about right modules since the others are similar. For any h ∈ D, C ∈ K (D),
r
i ∈ IC(r,u), j ∈ JC(i), x,y ∈ D with x−1y = gθ−1u(C)gθ and gθh = ζθ(h)gθ′, ζθ(h) ∈ Dσ,
θ,θ′ ∈ Θ ⊆ Θ (see Lemma 3.3), see
C CG
ψ(a(i,j) ·h) = ψ( k(i,j,s)a(i,s) ) (by [ZCZ, Pro.1.2])
y,x C,h yh,xh
s∈XJC(i)
= k(i,j,s)a′(i,s) and
C,h yh,xh
s∈XJC(i)
ψ(a(i,j))·h = a′(i,j) ·h
y,x y,x
= k(i,j,s)a′(i,s) (by [ZCZ, Pro.1.2]),
CG,h yh,xh
s∈XJCG(i)
6
where x(i,j) ·ζ (h) = k(i,j,s)x(i,s), x′(i,j) ·ζ (h) = k(i,j,s)x′(i,s). Since
C θ s∈JC(i) C,h C CG θ s∈JCG(i) CG,h CG
P P
x′(i,j) ·ζ (h) = ψ(i)(x(i,j))·ζ (h) = ψ(i)(x(i,j) ·ζ (h))
CG θ C C θ C C θ
= ψ(i)( k(i,j,s)x(i,s)) = k(i,j,s)x′(i,s),
C C,h C C,h CG
s∈XJC(i) s∈XJC(i)
(i,j) (i,j)
which implies ψ(a ·h) = ψ(a )·h.
y,x y,x
By [ZZC, Lemma 1.5], Tc (φπ ,ψπ ) := φπ + Tc(ψπ )∆ is a graded Hopf
kG 0 1 0 n>0 n 1 n−1
algebramapfromTc (kQc)toTc (kQ′c). ByLemma3.5,(Tc (kQc)) = kD+kQc. Since
kD 1 kG 1 P kD 1 1 1
the restriction of Tc (φπ ,ψπ ) on (Tc (kQc)) is φ+ ψ, we have that Tc (φπ ,ψπ ) is
kG 0 1 kD 1 1 kG 0 1
injective by [Mo93, Theorem 5.3.1].
(ii) It follows from Part (i).
(iii). By[ZCZ,Lemma2.1]kD[kQc,D,r,−→ρ,u] ∼= B(D,r,−→ρ,u)#kDandkG[kQ′c,G,r′,
−→ −→
ρ′,u′] ∼= B(G,r′, ρ′,u′)#kG. Applying Part (ii) we complete the proof. 2
The relation “ ≤ ” has the transitivity, i.e.
Lemma 1.7. Assume that G is a subgroup of G′ and G′ is a subgroup of G′′. If
−→ −→ −→
−→
RSR(G,r, ρ,u) ≤ RSR(G′,r′, ρ′,u′) and RSR(G′,r′, ρ′,u′) ≤ RSR(G′′,r′′,ρ′′,u′′), then
−→
−→
RSR(G,r, ρ,u) ≤ RSR(G′′,r,ρ′′,u′′).
Proof. Obviously, G is a subgroup of G′′ and r ≤ r′′. For any C ∈ K (G) and
r
i ∈ IC(r,u), then u(C) = u′(CG′) = u′′(CG′′). Let s = u(C) and let XC(i), X′(CiG)′ and
X′′(i) be representation spaces of ρ(i), ρ′(i) and ρ′′(i) , respectively. Let (X(i),ρ(i))
CG′′ C CG′ CG′′ C C
be isomorphic to a subrepresentation (N,ρ′(i) | ) of (X′(i) ,ρ′(i) | ). Considering
CG′ Gs CG′ CG′ Gs
(X′(CiG)′,ρ′(CiG)′) is isomorphic to a subrepresentation of (X′′(CiG)′′,ρ′′(CiG)′′ |G′s), we have that
(N,ρ′(i) | ) is isomorphic to a subrepresentation of (X′′(i) ,ρ′′(i) | ). Consequently,
CG′ Gs CG′′ CG′′ Gs
(X(i),ρ(i)) is isomorphic to a subrepresentation of (X′′(i) ,ρ′′(i) | ). 2
C C CG′′ CG′′ Gs
Lemma 1.8. Let N be a subgroup of G and (X,ρ) be an irreducible representation of
Nσ with σ ∈ N. If the induced representation ρ′ := ρ ↑Gσ is an irreducible representation
Nσ
of Gσ, then RSR(N,O ,ρ) ≤ RSR(G,O ,ρ ↑Gσ).
σ σ Nσ
Proof. Since (X,ρ) ∼= (X ⊗ 1,ρ′ | ) by sending x to x⊗1 for any x ∈ X, the
kNσ kNσ
claim holds. 2
Proposition 1.9. Let G = A⋊ D with σ ∈ D, χ ∈ Aσ, G = A⋊D , ρ ∈ Dσ and
χ χ χ
θ := (χ⊗ρ) ↑Gσ . Then
χ,ρ Gσ
χ c c
(i) RSR(G ,O ,χ⊗ρ) ≤ RSR(G,O ,θ )
χ σ σ χ,ρ
(ii) RSR(G ,O ,χ⊗ρ) is −1-type if and only if RSR(G,O ,θ ) is −1-type.
χ σ σ χ,ρ
(iii) RSR(D ,O ,ρ) ≤ RSR(G,O ,θ ).
χ σ σ χ,ρ
(iv) RSR(D ,O ,ρ) is −1-type if and only if RSR(G,O ,θ ) is −1-type.
χ σ σ χ,ρ
7
Proof. (i) It follows from Lemma 1.8.
(ii) Let P and X be representation spaces of χ and ρ, respectively. Then (P⊗X)⊗
kGσ
χ
kGσ is a representation space of ρ′ := θ . If ρ(σ) = −id, then for any g ∈ Gσ, x ∈ X,
χ,ρ
p ∈ P, we have ((p⊗x)⊗ g)·σ = ((p⊗x)·σ)⊗ g = −(p⊗x)⊗ g. Therefore
kGσ kGσ kGσ
χ χ χ
ρ′(σ) = −id. Conversely, if ρ′(σ) = −id, then ((p⊗x)⊗ 1)·σ = ((p⊗x)·σ)⊗ 1 =
kGσ kGσ
χ χ
−(p⊗x)⊗ 1. Therefore (p⊗x)·σ = −p⊗x for any x ∈ X.
kGσ
χ
(iii) By (i), it is enough to show RSR(D ,O ,ρ) ≤ RSR(G ,O ,χ⊗ρ). Let P and X
χ σ χ σ
be the representation spaces of χ and ρ on Aσ and Dσ, respectively. Thus (P ⊗X,χ⊗ρ)
χ
is an irreducible representation of Gσ := Aσ ⋊(Dσ) . Considering Definition 3.7 we only
χ χ
need to show that ρ is isomorphic to a submodule of the restriction of χ⊗ρ on Dσ. Fix
χ
a nonzero p ∈ P and define a map ψ from X to P ⊗ X by sending x to p ⊗ x for any
x ∈ X. It is clear that ψ is a kDσ-module isomorphism.
χ
(iv) Considering Part (ii), we only show that RSR(D ,O ,ρ) is −1-type if and only
χ σ
if RSR(G ,O ,χ⊗ρ) is −1-type. Since χ (σ) = χ (σ), the claim holds. 2
χ σ ρ (χ⊗ρ)
IfD = D,thenitfollowsfromthepropositionabovethatRSR(D,O ,ρ) ≤ RSR(G,O ,
χ σ σ
θ ). Therefore we have
χ,ρ
Corollary 1.10. Let G = A ⋊ D. If r ≤ r′ and ρ′(i) = θ with D = D,
CG χ(Ci),ρ(Ci) χ(Ci)
χ(i) ∈ A[u(C), u(C) = u′(C ) and I (r,u) ⊆ I (r′,u′) for any i ∈ I (r,u), C ∈ K (D),
C G C→− CG C r
−→
then RSR(D,r, ρ,u) ≤ RSR(G,r′, ρ′,u′). Furthermore, if I (r,u) = I (r′,u′) for any
C CG
−→
C ∈ Kr(D) and Kr′(G) = {CG | C ∈ Kr(D)}, then RSR(D,r, ρ,u) is −1-type if and
−→
only if RSR(G,r′, ρ′,u′) is −1-type.
Lemma 1.11. Let G = G × G . If σ = (σ ,σ ) ∈ G with ρ ∈ Gσ1 and ρ ∈ Gσ2,
1 2 1 2 1 1 2 2
then
d d
(i) Gσ = Gσ ×Gσ; Gσ = Gσ1 and Gσ = Gσ2.
1 2 1 1 2 2
(ii) OG = OG1 ×OG2, where OG denotes the conjugacy class containing σ of G.
σ σ1 σ2 σ
(iii) RSR(G ,O ,ρ ) ≤ RSR(G,O ,ρ ⊗ ρ ) when σ = 1; RSR(G ,O ,ρ ) ≤
1 σ1 1 σ1 1 2 2 2 σ2 2
RSR(G,O ,ρ ⊗ρ ) when σ = 1.
σ2 1 2 1
Proof. (i)It isclear Gσ = Gσ1 andGσ = Gσ2. Foranyx = (a,h) ∈ Gσ, thenxσ = σx,
1 1 2 2
which implies that aσ = σ a and hσ = σ h. Thus x ∈ Gσ × Gσ and Gσ ⊆ Gσ × Gσ.
1 1 2 2 1 2 1 2
Similarly, we have Gσ ×Gσ ⊆ Gσ.
1 2
(ii) It is clear.
(iii) We only show the first claim. It is clear that ρ is isomorphic to a subrepresen-
1
tation of the restriction of ρ ⊗ ρ on the Gσ1. Indeed, assume that X and Y are the
1 2 1
representation spaces of ρ and ρ , respectively. Obviously, Gσ1-module (X,ρ ) is isomor-
1 2 1 1
phic to a submodule of the restriction of ρ ⊗ρ on Gσ1 under isomorphism ψ form X to
1 2 1
X ⊗y by sending x to x⊗y for any x ∈ X, where y is a non-zero fixed element in Y.
0 0 0
2
8
Lemma 1.12. Let G = G ×G and σ = (σ ,σ ) ∈ G with ρ ∈ Gσ1 and ρ ∈ Gσ2.
1 2 1 2 1 1 2 2
(i) If σ = 1, then RSR(G,O ,ρ ⊗ρ ) is −1-type if and only if RSR(G ,O ,ρ ) is
2 σ1 1 2 1 σ1 1
d d
−1-type.
(ii) If σ = 1, then RSR(G,O ,ρ ⊗ρ ) is −1-type if and only if RSR(G ,O ,ρ ) is
1 σ2 1 2 2 σ2 2
−1-type.
Proof. (i) Considering χ (σ ) = χ (σ )deg(ρ ), we can complete the proof.
ρ1⊗ρ2 1 ρ1 1 2
(ii) It is similar. 2
Lemma 1.13. θ is a one dimensional representation of Gσ = Aσ ⋊Dσ if and only
χ,ρ
if Dσ = Dσ and degρ = 1
χ
Proof. Let P and X be the representation spaces of χ and ρ on Aσ and Dσ, respec-
χ
tively. ((P ⊗X)⊗ kGσ,θ ) is a one dimensional representation of Gσ = Aσ ⋊Dσ if
kGσ χ,ρ
χ
and only if kGσ = kGσ and dimX = 1. However. kGσ = kGσ if and only if Dσ = Dσ. 2
χ χ χ
Consequently, θ = χ⊗ρ when θ is one dimensional representation.
χ,ρ χ,ρ
2 Classical Weyl groups
In this section we give a necessary and sufficient condition for a Nichols algebra of ir-
reducible YD module supported by A to be finite dimensional, and show that except
in three cases Nichols algebras of irreducible YD modules supported by S are infinite
n
dimensional.
By Section 6 (i.e. the Appendix), [Ca72], [ZWW, Definition 2.5], [Su78, page 272],
(C )n ⋊ S is isomorphic to the Weyl groups W(B ) and W(C ) of B and C , where
2 n n n n n
n > 2. If A = {a ∈ (C )n | a = (ga1,ga2,··· ,gan) with a +a +···+a ≡ 0 ( mod 2)},
2 2 2 2 1 2 n
then A ⋊ S is isomorphic to the Weyl group W(D ) of D , where n > 3. Obviously,
n n n
when A = {a ∈ (C )n | a = (ga1,ga2,··· ,gan) with all a = 0}, A⋊S are isomorphic to
2 2 2 2 i n
the Weyl group of A , where n > 1. Note that S acts on A as follows: for any a ∈ A
n−1 n
with a = (ga1,ga2,··· ,gan) and h ∈ S , define
2 2 2 n
h·a := (gah−1(1),gah−1(2),··· ,gah−1(n)). (2.1)
2 2 2
Without specification, G = A⋊S is a subgroup of (C )n ⋊S .
n 2 n
2.1 σ ∈ A
Lemma 2.1. If σ ∈ A, then (A⋊S )σ = A⋊Sσ.
n n
Proof. It is clear that (a,τ) ∈ (A⋊S )σ if and only if τ ∈ Sσ. 2
n n
9
Let σ = (ga1,ga2,··· ,ga2) ∈ A. If ρ is a representation of Gσ, then ρ = θ =
2 2 2 (χ⊗µ)
(χ ⊗ µ) ↑Gσ with χ = χb1 ⊗ χb2 ⊗ ··· ⊗ χbn ∈ Aˆ, (Gσ) = A ⋊ (Sσ) and (Sσ) :=
(Gσ)χ 2 2 2 χ n χ n χ
c
{τ ∈ Sσn | τ · χ = χ} and µ ∈ ([Sσn)χ, see [Se, Proposition 2.5]. Let fτ·σ,χ := b1aτ−1(1) +
b2aτ−1(2) + ...bnaτ−1(n), Wσ := {i | ai 6= 0} and Wχ = {i | bi 6= 0}. It is clear that
fτ·σ,χ =| Wτ·σ ∩ Wχ |, where Wτ·σ = {i | aτ−1(i) 6= 0} = τWσ. fτ·σ,χ is written as fτ·σ in
short. Note that S acts on Aˆ as follows: for any χ ∈ Aˆ with χ = χb1 ⊗χb2 ⊗···χbn and
n 2 2 2
h ∈ S , define
n
h·χ := (χbh−1(1),χbh−1(2),··· ,χbh−1(n)).
2 2 2
Lemma 2.2. Under the notations above, we have
(i) ρ(σ) = q id with q = (−1)fσ.
σ,σ σ,σ
(ii) If W = W , then (Sσ) = Sσ.
σ χ n χ n
(iii) If Wσ = Wχ, then fg·σ = fg−1·σ for any g ∈ Sn.
Proof. Let P and V be the representation spaces of χ and µ, respectively. Then the
representation space of ρ is kGσ ⊗ (P ⊗V). For any p ∈ P, v ∈ V and τ ∈ S , we
k(Gσ)χ n
have
ρ(τ ·σ)(1⊗ (p⊗v)) = 1⊗ (((τ ·σ)·p)⊗x) (2.2)
k(Gσ)χ k(Gσ)χ
= χ(τ ·σ)1⊗ (p⊗x) = (−1)fτ·σ1⊗ (p⊗x).
k(Gσ)χ k(Gσ)χ
(i) It follows from (2.2) that q = χ(σ) = (−1)fσ.
σ,σ
(ii) If τ ∈ Sσ, then τ(W ) = W , i.e. τ(W ) = W . Consequently, τ ∈ (Sσ) .
n σ σ χ χ n χ
(iii) Let g(W )∩W = {l ,l ,··· ,l } with g(m ) = l for 1 ≤ i ≤ r. Then g−1(l ) = m
σ σ 1 2 r i i i i
for 1 ≤ i ≤ r and g−1(Wσ)∩Wσ = {m1,m2,··· ,mr}. This implies fg·σ = fg−1·σ. 2
Theorem 2.3. Let G = A⋊S with σ ∈ A and n > 2. Then dim B(OG,ρ) < ∞ if
n σ
and only if f is odd and either W = W or |W | = n or |W | = n.
σ χ σ σ χ
Proof. Let P and V be the representation spaces of χ and µ, respectively. Thus
the representation space of ρ is kGσ ⊗ (P ⊗ V). Let ξ = 1 ⊗ (p ⊗ v ) with
k(Gσ)χ k(Gσ)χ 0 0
0 6= p ∈ P and 0 6= v ∈ V.
0 0
We show this by following seven steps.
(i) If f is even, then ρ(σ) = (−1)fσid = id by Lemma 2.2 (see also [AZ07, Remark
σ
1.1]) and dim B(OG,ρ) = ∞.
σ
From now on we assume that f is odd.
σ
(ii) Assume that i ∈ W ∩W , i ∈ W \W andi ∈/ W . Let h = 1, h = (i ,i ,i ),
0 χ σ 1 σ χ 2 σ 1 2 0 1 2
h = h−1, t = h ·σ and γ = h−1h ·σ for i,j = 1,2,3. By simple computation we have
3 2 i i ij j i
γ = h−1 · σ, γ = h · σ, γ = h · σ, γ = h−1 · σ, γ = h−1 · σ, and γ = h · σ.
12 2 21 2 13 2 31 2 23 2 32 2
10