The radicals of Lie algebras

0.26 MB·English

by  Weicai Wu

ON POINTED HOPF ALGEBRAS WITH CLASSICAL WEYL GROUPS
SHOUCHUAN ZHANG
9 Abstract. Many cases of infinite dimensional Nichols algebras of irreducible Yetter-
0
Drinfeld modules over classical Weyl groups are found. It is proved that except a few
0
2 cases Nichols algebras of reducible Yetter-Drinfeld modules over classical Weyl groups

p are infinite dimensional. Some finite dimensional Nichols algebras of Yetter-Drinfeld
e modules over classical Weyl groups are given.
S

4
1

0. Introduction
]
A
Q This article is to contribute to the classification of finite-dimensional complex pointed
. Hopf algebras with Weyl groups of classical type. Weyl groups are very important in
h
t the theories of Lie groups, Lie algebras and algebraic groups. The classification of finite
a
m
dimensional pointed Hopf algebra with finite abelian groups has been finished (see [He06,
[
AS00]). Many cases of infinite dimensional Nichols algebras of irreducible Yetter-Drinfeld

2
(YD) modules over symmetric group were discarded in [AFZ]. All 1-type pointed Hopf
v
−
8 algebras with Weyl groups of exceptional type were found in [ZZWC] and it was showed
4
7 that every non 1-type pointed Hopf algebra is infinite dimensional in [AZ07, ZZWC].
4 −
It was obtained that every Nichols algebra of reducible YD module over simple group
.
2
0 and symmetric group is infinite dimensional in [HS]. Hopf subalgebras of co-path Hopf
9
algebras was studied in [OZ04].
0
: In this paper we discard many cases of infinite dimensional Nichols algebras of irre-
v
i ducible YD modules over classical Weyl groups by mean of co-path Hopf algebras and the
X
r results of [AFZ]. [HS] said that if Nichols algebra of reducible YD module is finite dimen-
a
sional, then their conjugacy classes are square-commutative (see [HS, Theorem 8.6]). We
obtain that except a few cases Nichols algebras of reducible YD modules over classical
Weyl groups are infinite dimensional by applying result of [HS]. We also find some finite
dimensional Nichols algebras of YD modules over classical Weyl groups.
The main results in this paper are summarized in the following statements.
Theorem 1. Let G = A⋊ S be a classical Weyl group with A (C )n and n > 2.
n 2
⊆
Assume that M = M( ,ρ(1)) M( ,ρ(2)) M( ,ρ(m)) is a reducible YD
Oσ1 ⊕ Oσ2 ⊕ ··· ⊕ Oσm
module over kG .
(i) Assume that there exist i = j such that σ , σ / A. If dimB(M) < , then n = 4,
i j
6 ∈ ∞
the type of σ is 22 and the sign of σ is stable for any 1 p m with σ / A.
p p p
≤ ≤ ∈
1
2 SHOUCHUANZHANG
(ii) If σ = a := (g ,g , ,g ) G and ρ(i) = θ := (χ(νi) µ(i)) Ga Ga
i 2 2 ··· 2 ∈ χ(νi),µ(i) ⊗ ↑Gaχ(νi)∈
with odd ν for i = 1,2, ,m, then B(M) is finite dimensional.
i
··· c
Theorem 2. Let 0 ν nandLetG = A⋊S . Letσ S be oftype (1λ1,2λ2,...,nλn)
n n
≤ ≤ ∈
and ρ = ρ′ ρ′′ (S\)σ with ρ′ Sσ and ρ′′ Sσ\ . Assume that B( G,θ )
⊗ ∈ n χ(ν) ∈ ν ∈ {ν+1,···,n} Oσ χ(ν),ρ
is matched with dimB(OσG,θχ(ν),ρ) < ∞. Let µ = ⊗1≤i≤nµi with µi := θχti,ρi as in (2.2)
c
denote ρ′ when σ S and ρ′′ when σ S , respectively. Let λ′ = λ (n ν)
∈ ν ∈ {ν+1,ν+2,···,n} 1 1− −
when σ S ; λ′ = λ ν when σ S . Then some of the following hold:
∈ ν 1 1 − ∈ {ν+1,ν+2,···,n}
(i) (1λ′1,2), µ1 = sgn or ǫ, µ2 = χ(1;2).
(ii) (2,σ ), σ := σ = id, µ = χ , µ = (χ ρ ) (SYj)σj ,
o o 1≤i≤n,1<i is odd i 6 2 (1;2) j (0,...,0;j)⊗ j ↑(SYj)σχj(0,...,0;j)
for all odd j > 1. Q
(iii) (1λ′1,23), µ1 = sgn or ǫ, µ2 = χ(1,1,1;2) ǫ or χ(1,1,1;2) sgn.
⊗ ⊗
Furthermore, if λ′ > 0, then µ = χ sgn.
1 2 (1,1,1;2) ⊗
(iv) (25), µ = χ ǫ or χ sgn.
2 (1,1,1,1,1;2) (1,1,1,1,1;2)
⊗ ⊗
(v) (1λ′1,4), µ1 = sgn or ǫ, µ4 = χ(2;4).
(vi) (1λ′1,42), µ1 = sgn or ǫ, µ4 = χ(1,1;4) sgn or χ(3,3;4) sgn.
⊗ ⊗
(vii) (2,4), µ = χ and µ = ǫ or µ = ǫ and µ = χ .
2 (1;2) 4 2 4 (2;4)
(viii) (2,42), µ = ǫ, µ = χ sgn or χ sgn.
2 4 (1,1;4) (3,3;4)
⊗ ⊗
(ix) (22,4), degµ = 1, µ = χ .
2 4 (2;4)
Furthermore, B( G,θ ) is 1-type under the cases above.
Oσ χ(ν),ρ −
Indeed, Theorem 1 follows Remark 3.8 and Remark 3.12. The proof of Theorem 2 is in
subsection 3.1.
Preliminaries And Conventions
Let k be the complex field and G a finite group. Let Gˆ denote the set of all isomorphic
classes of irreducible representations of group G, Gσ the centralizer of σ, or G the
Oσ Oσ
conjugacy class in G, C the cycle group with order j, g a generator of C and χ a
j j j j
character of C with order j. The Weyl groups of A , B , C and D are called the
j n n n n
classical Weyl groups. Let ρ G denote the induced representation of ρ as in [Sa01].
↑D
Aquiver Q = (Q ,Q ,s,t)isanoriented graph, where Q andQ arethesets ofvertices
0 1 0 1
and arrows, respectively; σ and t are two maps from Q to Q . For any arrow a Q ,
1 0 1
∈
s(a) and t(a) are called its start vertex and end vertex, respectively, and a is called an
arrow from s(a) to t(a). For any n 0, an n-path or a path of length n in the quiver Q is
≥
an ordered sequence of arrows p = a a a with t(a ) = s(a ) for all 1 i n 1.
n n−1 1 i i+1
··· ≤ ≤ −
Note that a 0-path is exactly a vertex and a 1-path is exactly an arrow. In this case, we
define s(p) = s(a ), the start vertex of p, and t(p) = t(a ), the end vertex of p. For a
1 n
0-path x, we have s(x) = t(x) = x. Let Q be the set of n-paths. Let yQx denote the set
n n
of all n-paths from x to y, x,y Q . That is, yQx = p Q s(p) = x,t(p) = y .
∈ 0 n { ∈ n | }
ON POINTED HOPF ALGEBRAS WITH CLASSICAL WEYL GROUPS 3
A quiver Q is finite if Q and Q are finite sets. A quiver Q is locally finite if yQx is a
0 1 1
finite set for any x,y Q .
0
∈
Let G be a group. Let (G) denote the set of conjugate classes in G. A formal sum
K
r = r C of conjugate classes of G with cardinal number coefficients is called a
C∈K(G) C
ramification (or ramification data ) of G, i.e. for any C (G), r is a cardinal number.
P ∈ K C
In particular, a formal sum r = r C of conjugate classes of G with non-negative
C∈K(G) C
integer coefficients is a ramification of G.
P
For any ramification r and a C (G), since r is a cardinal number, we can choice
C
∈ K
a set I (r) such that its cardinal number is r without loss of generality. Let (G) :=
C C r
K
C (G) r = 0 = C (G) I (r) = . If there exists a ramificationr of G such
C C
{ ∈ K | 6 } { ∈ K | 6 ∅}
that the cardinal number of yQx is equal to r for any x,y G with x−1y C (G),
1 C ∈ ∈ ∈ K
then Q is called a Hopf quiver with respect to the ramification data r. In this case, there
is a bijection from I (r) to yQx, and hence we write yQx = a(i) i I (r) for any
C 1 1 { y,x | ∈ C }
x,y G with x−1y C (G).
∈ ∈ ∈ K
deg ρ denotes the dimension of the representation space V for a representation (V,ρ).
Recall the notation RSR in [ZCZ, Def. 1.1]. Let ρ be a representation of Gu(C)
C
(i)
with irreducible decomposition ρ = ρ , where I (r,u) is an index set. Let
⊕i∈IC(r,u) C C
(i)
ρ denote ρ = ρ . (G,r, ρ,u) is called an RSR when
−→ { C}C∈Kr(G) {{ C }i∈IC(r,u)}C∈Kr(G) −→
deg(ρ ) = r for any C (G), written as RSR(G,r, ρ,u). For any RSR(G,r, ρ,u),
C C r →− −→
∈ K
we obtain a co-path Hopf algebra kQc(G, r, ρ,u), a Hopf algebra kG[kQc, G,r, ρ,u] of
−→ 1 −→
one type, a kG-YD module (kQ1,ad(G,r, ρ,u)) and a Nicolas algebra B(G,r, ρ,u) :=
1 −→ →−
B(kQ1,ad(G,r, ρ,u)) (see [ZCZ]).
1 →−
If ramification r = r C and I (r,u) = 1 then we say that RSR(G,r, ρ,u) is of bi-one
C C −→
| |
since r only has one conjugacy class C and I (r,u) = 1. Furthermore, if let σ = u(C),
C
| |
(i)
C = , r = m and ρ = ρ for i I (r,u), then bi-one RSR(G,r, ρ,u) is denoted by
Oσ C C ∈ C −→
RSR(G,m ,ρ) ( or RSR(G, ,ρ)), in short.
σ σ
O O
RSR(G,r, ρ,u) is called 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 (G) and any i I (r,u). In this
C − ∈ Kr ∈ C
case, the Nichols algebra B(G,r, ρ,u) is called to be 1-type.
−→
−
For s G and (ρ,V) Gs, here is a precise description of the YD module M( ,ρ),
s
∈ ∈ O
introduced in [Gr00, AZ07]. Let t = s, ..., t be a numeration of , which is a
1 m s
O
conjugacy class containingcs, and let g G such that g ⊲ s := g sg−1 = t for all
i ∈ i i i i
1 i m. Then M( ,ρ) = g V. Let g v := g v M( ,ρ), 1 i m,
s 1≤i≤m i i i s
≤ ≤ O ⊕ ⊗ ⊗ ∈ O ≤ ≤
v V. If v V and 1 i m, then the action of h G and the coaction are given by
∈ ∈ ≤ ≤ ∈
(0.1) δ(g v) = t g v, h (g v) = g (γ v),
i i i i j
⊗ · ·
where hg = g γ, for some 1 j m and γ Gs. Let B( ,ρ) denote B(M( ,ρ)). By
i j s s
≤ ≤ ∈ O O
[ZZWC, Lemma 1.1], there exists a bi-one arrow Nichols algebra B(G,r, ρ,u) such that
→−
B(Os,ρ) ∼= B(G,r,−→ρ,u) as graded braided Hopf algebras in kkGGYD.
4 SHOUCHUANZHANG
If D is a subgroup of G and C is a congugacy class of D, then C denotes the conjugacy
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
χ Aˆ, 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
∈
(1.1) a = σ a and dσ = σd.
·
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 = σxand x Gσ.
∈ ∈
(cid:3)
Lemma 1.2. Let D be a subgroup of G with σ D and let right coset decompositions
∈
of Dσ in D be
(1.2) D = Dσg .
θ
θ∈Θ
[
Then there exists a set Θ′ with Θ Θ′ such that
⊆
(1.3) G = Gσg
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. (cid:3)
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, Lemma
1
2.5] yields that kG[kQc] is coradically graded. (cid:3)
1
ON POINTED HOPF ALGEBRAS WITH CLASSICAL WEYL GROUPS 5
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 (D), then r is called a subramification of r′, written as
C ≤ CG ∈ Kr
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 (D), i I (r,u), then
CG ∈ Kr ∈ 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 be a subgroup ofG. If σ D, then RSR(D, ,ρ) RSR(G, ,ρ′)
σ σ
∈ O ≤ O
if and only if ρ is isomorphic to subrepresentation of the restriction of ρ′ on Dσ.
Proof. It follows from Definition 1.4. (cid:3)
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 dimensional with finite group G then so is B(D,r, ρ,u).
→−
Proof. (i) For any C (D) and i I (r,u), let X(i) be a representation space of
∈ Kr ∈ 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 (D). Now we show that ψ is a kD-bimodule homomorphism from
r
∈ ∈ K
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 (D),
r
∈ ∈ K
i I (r,u), j J (i), x,y D with x−1y = g−1u(C)g and g h = ζ (h)g , ζ (h) Dσ,
∈ C ∈ C ∈ θ θ θ θ θ′ θ ∈
θ,θ′ Θ Θ (see Lemma 1.2), 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)
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
x′(i,j) Pζ (h) = ψ(i)(x(i,j)) ζ (h) = ψ(i)(xP(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)
6 SHOUCHUANZHANG
(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
algebra map from Tc (kQc) to Tc (kQ′c). By Lemma 1.3, (Tc (kQc)) = kD + kQc.
kD 1 kG 1 P kD 1 1 1
Since the restriction of Tc (φπ ,ψπ ) on (Tc (kQc)) is φ+ψ, we have that Tc (φπ ,ψπ )
kG 0 1 kD 1 1 kG 0 1
is 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. (cid:3)
∼
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 (G) and
r
≤ ∈ K
i I (r,u), then u(C) = u′(C ) = u′′(C ). Let s = u(C) and let X(i), X′(i) and
∈ C G′ G′′ C CG′
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′(i) ,ρ′(i) ) is isomorphic to a subrepresentation of (X′′(i) ,ρ′′(i) ), we have that
CG′ CG′ CG′′ CG′′ |G′s
(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) ). (cid:3)
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, ,ρ) RSR(G, ,ρ Gσ).
Oσ ≤ Oσ ↑Nσ
Proof. Since (X,ρ) = (X 1,ρ′ ) by sending x to x 1 for any x X, the
∼ kNσ kNσ
⊗ | ⊗ ∈
claim holds. (cid:3)
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 ⊗ ↑Gσχ
c c
(i) RSR(G , ,χ ρ) RSR(G, ,θ )
χ σ σ χ,ρ
O ⊗ ≤ O
(ii) RSR(G , ,χ ρ) is 1-type if and only if RSR(G, ,θ ) is 1-type.
χ σ σ χ,ρ
O ⊗ − O −
(iii) RSR(D , ,ρ) RSR(G, ,θ ).
χ σ σ χ,ρ
O ≤ O
(iv) RSR(D , ,ρ) is 1-type if and only if RSR(G, ,θ ) is 1-type.
χ σ σ χ,ρ
O − O −
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 ⊗ · − ⊗ ∈
ON POINTED HOPF ALGEBRAS WITH CLASSICAL WEYL GROUPS 7
(iii) By (i), it is enough to show RSR(D , ,ρ) RSR(G , ,χ ρ). Let P and X
χ σ χ σ
O ≤ O ⊗
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 1.4 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 , ,ρ) is 1-type if and only if
χ σ
O −
RSR(G , ,χ ρ) is 1-type. Since χ (σ) = χ (σ), the claim holds. (cid:3)
χ σ ρ (χ⊗ρ)
O ⊗ −
IfD = D,thenitfollowsfromthepropositionabovethatRSR(D, ,ρ) RSR(G, ,
χ σ σ
O ≤ O
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 (D),
C ∈ G C ⊆ CG ∈ C ∈ Kr
then RSR(D,r, ρ,u) RSR(G,r′,−→ρ′,u′). Furthermore, if I (r,u) = I (r′,u′) for any
−→ ≤ C CG
C (D) and (G) = C C (D) , then RSR(D,r, ρ,u) is 1-type if and
r r′ G r −→
∈ K K { | ∈ K } −
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
(i) Gσ = Gσ Gσ; Gσ = Gσ1 and Gσ = Gσ2. d d
1 × 2 1 1 2 2
(ii) G = G1 G2, where G denotes the conjugacy class containing σ of G.
Oσ Oσ1 ×Oσ2 Oσ
(iii) RSR(G , ,ρ ) RSR(G, ,ρ ρ ) when σ = 1; RSR(G , ,ρ )
1 Oσ1 1 ≤ Oσ1 1 ⊗ 2 2 2 Oσ2 2 ≤
RSR(G, ,ρ ρ ) when σ = 1.
Oσ2 1 ⊗ 2 1
Proof. (i) It is clear Gσ = Gσ1 and Gσ = Gσ2. For any x = (a,h) Gσ, then xσ = 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 onlyshow the first claim. It is clear thatρ isisomorphic toa subrepresentation
1
of the restriction of ρ ρ on the Gσ1. Indeed, assume that X and Y are the represen-
1 ⊗ 2 1
tation spaces of ρ and ρ , respectively. Obviously, Gσ1-module (X,ρ ) is isomorphic to
1 2 1 1
a submodule of the restriction of ρ ρ on Gσ1 under isomorphism ψ form X to X y
1⊗ 2 1 ⊗ 0
by sending x to x y for any x X, where y is a non-zero fixed element in Y. (cid:3)
0 0
⊗ ∈
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, ,ρ ρ ) is 1-type if and only if RSR(G , ,ρ ) is
2 Oσ1 1 ⊗ 2 − 1 Oσ1 1
d d
1-type.
−
(ii) If σ = 1, then RSR(G, ,ρ ρ ) is 1-type if and only if RSR(G , ,ρ ) is
1 Oσ2 1 ⊗ 2 − 2 Oσ2 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. (cid:3)
8 SHOUCHUANZHANG
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. LetP andX betherepresentationspacesofχandρonAσ andDσ,respectively.
χ
((P X) kGσ,θ ) is a one dimensional representation of Gσ = Aσ⋊Dσ if and only
kGσ χ,ρ
⊗ ⊗ χ
if kGσ = kGσ and dimX = 1. However. kGσ = kGσ if and only if Dσ = Dσ. (cid:3)
χ χ χ
Consequently, θ = χ ρ when θ is one dimensional representation.
χ,ρ χ,ρ
⊗
2. Symmetric group S
n
In this section we study the Nichols algebras over symmetric groups.
Without specification, σ S is always of type 1λ12λ2 nλn. g denotes the gen-
n j
∈ ···
erator of cycle group C with order j for natural number j. We keep on the work
j
in [Su78, Page 295-299 ]. Let r := kλ and σ := y ,
j 1≤k≤j−1 k j 1≤l≤λj rj+(l−1)j+1
(cid:16)
y , , y , the multiplPication of cycles of lengthQj in the independent
rj+(l−1)j+2 ··· rj+lj
cycle decomposition of σ, a(cid:17)s well as Y := y s = r +1, ,r . Therefore σ = σ
j s j j+1 i
{ | ··· }
and (S )σ = (S )σi = T T . It follows from [AFZ, subsection 2.2] that T
n Yi 1 × ··· × n Q j
is generated by A ,...,A ,B ,...,B , where A = (y ),...,A = (y ),
Q 1,j λj,j 1,j λj−1,j 1,1 1 λ1,1 λ1
A = (y y ),..., A = (y y ), and so on. More precisely, if
1,2 λ1+1 λ1+2 λ2,2 λ1+2λ2−1 λ1+2λ2
1 < j n, then
≤
A := y , y , , y ,
l,j rj+(l−1)j+1 rj+(l−1)j+2 ··· rj+lj
(cid:16) (cid:17)
B := y , y , y , y y , y ,
h,j rj+(h−1)j+1 rj+hj+1 rj+(h−1)j+2 rj+hj+2 ··· rj+hj rj+(h+1)j
(cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)
l
for all l, h, with 1 l λ , 1 h λ 1. Notice that ϕ(A ) = (1, ,1,gj−1
≤ ≤ j ≤ ≤ j − l,j ··· j
,1··· ,1),1 and ϕ(Bh,j) = 1,(h,h + 1) , where ϕ is an isomorphism(cid:0)frzom G}σ|j to{
(C )λj⋊ S , defined in the proof of [ZWW, Pro. 2.10 ] (also see the below (2.1)). Fur-
j λj(cid:1) (cid:0) (cid:1)
thermore, if Y X 1,2, ,n , then σ is said to be in S .
i>1 i X
∪ ⊆ ⊆ { ··· }
Lemma 2.1. If σ S is the multiplication of m independent cycles with the same
n
∈
length l, i.e. σ is of type lm, then
(i) ϕ(σ) = ((gl−1,gl−1, ,gl−1),(1)), where ϕ is the isomorphism from (S )σ to
l l ··· l n
(C )m⋊ S defined in the proof of [ZWW, Pro. 2.10 ].
l m
(ii) θ (ϕ(σ)) = χ((gl−1,gl−1, ,gl−1)) id for any ρ (\S ) and χ (\C )m, where
χ,ρ l l ··· l ∈ m χ ∈ l
G = (C )m⋊ S , G = (C )m⋊ (S ) and θ = (χ ρ) G .
l m χ l m χ χ,ρ ⊗ ↑Gχ
Proof. (i) Assume that
σ = (a a a )(a a ) (a a ).
10 11 1,l−1 20 2,l−1 m0 m,l−1
··· ··· ··· ···
By [ZWW, Pro. 2.14 ] or [Su78],
ϕ
Sσ = (C )m ⋊S
n ∼ l m
ON POINTED HOPF ALGEBRAS WITH CLASSICAL WEYL GROUPS 9
where the map ϕ is the same as in the proof of [ZWW, Pro. 2.10 ]. Indeed, here is precise
definition of isomorphism ϕ. For any element τ of (S )σ, we will define θ(τ) S and
n m
∈
f (C )m by
τ l
∈
τ−1(a ) = a , j = θ(τ)−1(i), f (i) = gk,
i0 jk τ l
where g is the generator of C , 1 i m. Let
l l
≤ ≤
(2.1) ϕ(τ) = (f ,θ(τ)).
τ
Since σ(a ) = a , we have ϕ(σ) = (f ,θ(σ)) with f = (gl−1,gl−1, ,gl−1) (C )m
i,l−1 i,0 σ σ l l ··· l ∈ l
and θ(σ) = (1) S .
m
∈
(ii) Let P and X be representation spaces of χ and ρ, respectively. For any 0 = p P
6 ∈
and 0 = x X, see
6 ∈
((p x) 1) ϕ(σ) = ((p x) ϕ(σ)) 1
⊗ ⊗kGχ · ⊗ · ⊗kGχ
= χ((gl−1,gl−1, ,gl−1))((p x) 1) (by Part (i)).
l l ··· l ⊗ ⊗kGχ
Since ϕ(σ) is in the center of Cm⋊S and θ is irreducible, we have that Part (ii) holds.
l m χ,ρ
(cid:3)
Obviously, every element in (\C )m can be denoted by χ := χt1,l χt2,l
l (t1,l,t2,l,···,tm,l;l) l ⊗ l ⊗
χtm,l for 0 t l 1. For convenience, we denote χ by χt when it
···⊗ l ≤ j,l ≤ − (t1,l,t2,l,···,tm,l;l)
does not cause mistake.
Lemma 2.2. (i) Every irreducible representation of Sσ = Sσi is isomorphic to
n 1≤i≤n Yi
one of the following list:
Q
(2.2) ⊗1≤i≤n (θχti,ρiϕi),
where ρi ∈ (\SYi)χti and ϕi is the isomorphism from SσYii to (Ci)λi ⋊Sλi as in (2.1).
(ii) Let χ denote the character of ⊗1≤i≤n(θχti,ρiϕi). Then
(2.3) χ(σ) = ( χi(gi)(i−1)tj,i)deg(θχti,ρiϕi).
1≤Yi≤n 1≤Yj≤λi
Proof. (i) By [ZWW, Pro. 2.10], Sσ = Sσi Q=ϕi (C )λi ⋊S . It follows
n 1≤i≤n Yi ∼ 1≤i≤n i λi
from [Se, Pro.25] that every irreducible representation of (C )λi ⋊ S is isomorphic to
Q Q i λi
θχti,ρi. This completes our proof.
(ii) It follows from (i) and Lemma 2.1. (cid:3)
Definition 2.3.
t 1
j,k
(2.4) ξt,σ := +
k 2
1≤k≤Xn,1≤j≤λk
is calledthe distinguishedelementof therepresentation (2.2) orRSR(Sn,Oσ,⊗1≤i≤n(θχti,ρiϕi)),
where 0 t k 1.
j,k
≤ ≤ −
10 SHOUCHUANZHANG
Lemma 2.4. Let χ denote the character of (⊗1≤k≤n(θχtk,ρkϕk)). Then ξt,σ is an integer
if and only if χ(σ) = deg(χ).
−
2πi
Proof. Let ωj := e j , where i := √ 1 and e is the Euler’s constant. For any k with
−
λ = 0, since (k,k 1) = 1, there exists a with (a ,k) = 1 such that ωak(k−1) = ω .
k 6 − k k k k
Choice χ such that χ (g ) = ωak. By formula (2.3), we have
k k k k
(2.5) χ(σ) = e(2πi)P1≤k≤n,1≤j≤λk tjk,kdeg(⊗1≤s≤n(θχts,ρsϕs)).
Using the formula we complete the proof. (cid:3)
Consequently, we have
Proposition2.5. Thedistinguishedelementξt,σ of therepresentation ⊗1≤i≤n(θχti,ρiϕi),
as in (2.2) is an integer and the order of σ is even if and only if B(,OσSn,⊗1≤i≤n(θχti,ρiϕi))
is 1-type.
−
Lemma 2.6. If χ (\C )m, then (S ) = S if and only if χ = χt χt χt for
∈ l m χ m l ⊗ l ⊗···⊗ l
some 0 t l 1.
≤ ≤ −
Proof. NotethatS acts(C )m asfollows: Foranya Cm witha = (ga1,ga2, ,gam)
m l ∈ l l l ··· l
and h S ,
m
∈
(2.6) h a = (gah−1(1),gah−1(2), ,gah−1(m)).
· l l ··· l
Let χ = m χti with 0 t l 1. If (S ) = S and there exist i = j such that t = t .
⊗i=1 l ≤ i ≤ − m χ m 6 i 6 j
Set a = (ga1,ga2, ,gam) Cm with a = 1 and a = 0 when s = i; h = (i,j) S . See
l l ··· l ∈ l i s 6 ∈ m
(h χ)(a) = χ(h−1 a) = χ(gah(1),gah(2), ,gah(m))
· · l l ··· l
= χ (g )tiaj+tjai = χ (g )tj
l l l l
and χ(a) = χ (g )ti. This implies χ = (h χ). We get a contradiction. Conversely, it is
l l
6 ·
clear. (cid:3)
Proposition 2.7. Every one dimensional representation of Sσ is of the following form:
n
(2.7) (χ ρ )ϕ ,
⊗1≤i≤n (ti,···,ti;i) ⊗ i i
where 1 t i 1 and ρ is a one dimensional representation for any 1 i n.
i i
≤ ≤ − ≤ ≤
Proof. By Lemma 2.2, every irreducible representation of Sσ is of form as (2.2).
n
If (2.2) is one dimensional, then it follows from Lemma 1.13 that ⊗1≤i≤n(θχti,ρiϕi) =
⊗1≤i≤n(χti ⊗ρi)ϕi and(Sλi)χti = Sλi. ByLemma 2.5, χti = χtii⊗χtii⊗···⊗χtii = χ(ti,···,ti;i)
for some 0 t i 1. (cid:3)
i
≤ ≤ −
Note that every one dimensional representation of S is χ or ǫ. Therefore every one
m 2
dimensional representation of Sσ can be denoted by (χ χδi) in short,
m ⊗1≤i≤n (ti,···,ti;i) ⊗ 2
where δ = 1 or 0.
i