Canonical bases for the quantum extended Kac-Moody algebras

0.42 MB·English

by  Guanglian Zhang

CANONICAL BASES OF SINGULARITY RINGEL-HALL ALGEBRAS AND
HALL POLYNOMIALS
GUANGLIANZHANG
9 Abstract. In this paper, the singularity Ringel-Hall algebras are defined. A new class of
0 perverse sheaves are shown to havepurity property. The canonical bases of singularity Ringel-
0 Hallalgebrasareconstructed. Asanapplication, theexistenceofHallpolynomials inthetame
2
quiveralgebras is proved.

n
a
J

6
1
0. Introduction

]
T
0.1SinceG.Lusztig[L1,L2,L3]andM.Kashiwara[K]haveprovedthatthereexistthecanonical
R
bases in thequantized universal enveloping algebras, the canonical bases is playing an extremely
.
h
vital role in the research of Lie algebra, Heck algebra and the quantized Schur algebra. The
t
a
elements in the canonical bases are not only characteristic as perverse sheaves of quiver variety
m
[L1, L2, L3], butalso they are characteristic as the elements in Ringel-Hall algebra (see [L4] and
[
[LXZ]).

2 Because of the deep relations between Ringel-Hall algebra and the many profound results in
v
therepresentationtheoryoffinitedimensionalalgebra, itisworthtostudytherepresentationsof
0
2 quiver algebras using the canonical bases and study the canonical bases and the representation
7
of the other algebras (e.g., q Schur algebra )using the representations of quivers.
1 −
. M. Varagnolo and E. Vasserot [VV] proved the decomposition conjecture for quantized Schur
1
algebra of type A.
0
9 A natural question is that how one can prove the decomposition conjecture for quantized
0
Schur algebra of the other types.
:
v Using the Ringel-Hall algebra model in the study of canonical bases is an important skill in
i
X [VV]. Let U+(Q) (resp. U+(Q)) be the generic Hall algebra, where Q = A (resp. Q = A ).
n ∞ n ∞
r Based on the proof in [VV], we had to construct q Fock spaces of the other affinetypes in order
a − e
to prove the decomposition conjecture for quantized Schur algebra of the general type.
According toProposition 5in [VV], we knowthat U+(Q)/I = ∞,whereI is thesubmodule
∞ ∼
of U+∞(Q) generated by bO, where the O’s are unstable orbits inVNakajima variety of type A∞.
We conjecture that the q Fock spaces of the other affine types should be the quotient of some
−
subalgebraofRingel-Hallalgebra(e.g., SingularityRingel-Hallalgebras)modulounstableorbits.
Inthispaper,weprovethatthequotientofthesingularity Ringel-Hall algebras modulounstable
orbits is isomorphic to q Fock space in the case of type A.
−
G.Lusztig [L4] proved that the perverse sheaves corresponding to the canonical bases have
purityproperty. Inthispaper,wefindanewclass of perversesheaves whichhaspurityproperty.
The research was supported in part byNSFgrant 10771112.
The research was also supported in part by Research Institute for Mathematical Sciences, Kyoto University,
Kyotoand Chern Instituteof Mathematics, NankaiUniversity,Tianjin,China.
1
2 GUANGLIANZHANG
Thereby, we prove that the singularity Ringel-Hall algebras have the canonical bases. As an
application of the singularity Ringel-Hall algebras , we prove that the tame quiver algebras have
Hall polynomials.
0.2 The paper is organized as follows. In 1 we give a quick review of the definitions of Ringel-
§
Hall algebras and Double Ringel-Hall algebras. In 2 we define the singularity Ringel-Hall
§
algebras s(Λ) and study the rations between the singularity Ringel-Hall algebras and the
H
quantized universal enveloping algebras. We prove Proposition 2.1.2 and fromthis, we pointout
that the quotient of the singularity Ringel-Hall algebras modulo unstable orbits is isomorphic
to q Fock space in the case of type A. In 3 we construct the PBW basis of s(Λ). In
− § − H
4 we prove that the closure of semi-simple objects in have purity property. In 5 we study
i
§ T §
the fibres of p . We also give a new class of perverse sheaves which have purity property. In
3
6 we prove that the singularity Ringel-Hall algebras have the canonical bases. In 7 A.Hubery
§ §
[H] have proved the existence of Hall polynomials on the tame quivers for Segre classes. In this
subsection, by using the extension algebras of singularity Ringel-Hall algebras, we give a simple
and direct proof for the existence of Hall polynomials on the tame quivers.
Acknowledgments. I would like to express my sincere gratitude to H.Nakajima for a number
of interesting discussions. I am also grateful to J. Xiao for a number of interesting discussions.
1. Ringel-Hall algebras
1.1AquiverQ = (I,H,s,t)consistsofavertexsetI,anarrowsetH,andtwomapss,t : H I
→
such that an arrow ρ H starts at s(ρ) and terminates at t(ρ).
∈
Throughout the paper, F denotes a finite field with q elements, k = F the algebraic closed
q q
field ,Q = (I,H,s,t) is a fixed connected quiver, and Λ = F Q is the path algebra of Q over
q
F . By modΛ we denote the category of all finite dimensional left Λ-modules, or equivalently
q
finite modules. It is well-known that modΛ is equivalent to the category of finite dimensional
representations of Q over F . We shall simply identify Λ-modules with representations of Q.
q
The set of isomorphism classes of (nilpotent) simple Λ-modules is naturally indexed by the
set I of vertices of Q. Thenthe Grothendieck group G(Λ) of modΛ is the freeAbelian groupZI.
For each nilpotent Λ-module M, the dimension vector dimM = (dimM )i is an element
i∈I i
of G(Λ). The Ringel-Hall algebra (Λ) is graded by NI, more prPecisely, by dimension vectors
H
of modules.
The Euler form , on G(Λ) = ZI is defined by
h− −i
α,β = a b a b
i i s(ρ) t(ρ)
h i −
Xi∈I ρX∈H
for α = a i and β = b i in ZI. For any nilpotent Λ-modules M and N one has
i∈I i i∈I i
P P
dimM,dimN = dim Hom (M,N) dim Ext (M,N).
h i Fq Λ − Fq Λ
The symmetric Euler form is defined as
(α,β) = α,β + β,α for α,β ZI.
h i h i ∈
This gives rise to a symmetric generalized Cartan matrix C = (a ) with a = (i,j). It is
ij i,j∈I ij
easy to see that C is independent of the field F and the orientation of Q.
q
Throughout the paper, we concentrate on tame quiver Q. The symmetric Euler forms give
rise to the Cartan matrices of types A,D and E.
CANONICAL BASES 3
1.2 Ringel-Hall algebra. Given threemodules L,M,N in modΛ,let gL denote the number
MN
of Λ-submodules W of L such that W N and L/W M in modΛ. More generally, for
≃ ≃
M , ,M ,L modΛ, let gL denote the number of the filtrations 0 = L L
1 ··· t ∈ M1···Mt 0 ⊆ 1 ⊆ ··· ⊆
Lt = L of Λ-submodules such that Li/Li−1 Mi for i = 1, ,t. Let vq = √q C and
≃ ··· ∈ P
be the set of isomorphism classes of finite dimensional nilpotent Λ-modules. Then the twisted
Ringel-Hall algebra ∗(Λ) is defined by setting ∗(Λ) = (Λ) as Q(v )-vector space, but the
q
H H H
multiplication is defined by
u u = vhdimM,dimNi gL u .
[M]∗ [N] q MN L
X
[L]∈P
Following [R3], for any Λ-module M, we denote M = v−dimM+dimEndΛ(M)u . Note that
[M]
h i
M M a Q(v )-basis of ∗(Λ).
q
{h i | ∈ P} H
The Q(v )-algebras ∗(Λ) depends on q. We will use ∗(Λ) to indicate the dependence on q
q H Hq
when such a need arises.
1.3 A construction by Lusztig. For any finite dimensional I-graded k-vector space V =
V with a given F rational structure by Frobenius map F, let E be the subset of
i∈I i q− V
Pρ∈HHom(Vs(ρ),Vt(ρ))definingnilpotentrepresentationsofQ.NotethatEV = ρ∈HHom(Vs(ρ),Vt(ρ))
⊕ ⊕
when Q has no oriented cycles. The space of F rational points of E is the fixed point set EF.
q− V V
Let G = GL(V ), its subgroup of F rational points is GF. Then the group G =
V i∈I i q− V V
i∈IGL(Vi) aQcts naturally on EV by
Q
(g,x) g x= x′ where x′ = g x g−1 for all ρ H.
7→ • ρ t(ρ) ρ s(ρ) ∈
The restriction of the action of G on E gives an action of the finite group GF on EF.
V V V V
For γ NI, we fix a I-graded k-vector space V with dimV = γ. We set E = E and
∈ γ γ γ Vγ
G = G . For α,β NI and γ = α+β, we consider the diagram
γ Vγ ∈
E E p1 E′ p2 E′′ p3 E .
α β γ
× ←− −→ −→
Here E′′ is the set of all pairs (x,W), consisting of x E and an x-stable I-graded subspace W
γ
∈
ofV withdimW = β,andE′ isthesetofallquadruples(x,W,R′,R′′),consistingof(x,W) E′′
γ
∈
and two invertible linear maps R′ : Fβ W and R′′ : Fα Fγ/W. The maps are defined in
q → q → q
an obvious way: p (x,W,R′,R′′)= (x,W), p (x,W) = x, and p (x,W,R′,R′′)= (x′,x′′), where
2 3 1
x R′ = R′ x′ and x R′′ = R′′ x′′ for all ρ H.
ρ s(ρ) t(ρ) ρ ρ s(ρ) t(ρ) ρ ∈
If M E ,N E and L E , we define
α β α+β
∈ ∈ ∈
Z = p p−1( ),Z = Z p−1.
2 1 OM ×ON L,M,N ∩ 3
The varieties and morphisms in above diagram are naturally defined over F . So we have
q
EF EF p1 E′F p2 E′′F p3 E .
α × β ←− −→ −→ γ
For anymapp : X Y offinitesets, p∗ : C(Y) C(X)isdefinedbyp (f)(x)= f(p(x))and
→ → ∗
p : C(X) C(Y) is defined by p(h)(y) = h(x), on the integration along the fibers.
! → ! x∈p−1(y)
Let C (EF) be the space of GF-invariant fPunctions EF C( or Q .) Given f C (EF) and
GF V V V → l ∈ GF α
g C (EF), there is a unique h C (E′′F) such that p∗(h) = p∗(f g). Then define f g by
∈ GF β ∈ G 2 1 × ◦
f g = (p )(h) C (EF).
◦ 3 ! ∈ GF γ
4 GUANGLIANZHANG
Let
m(α,β) = a b + a b .
i i s(ρ) t(ρ)
Xi∈I ρX∈H
We again define the multiplication in the C-space K = C (EF) by
⊕α∈NI GF α
f g = v−m(α,β)f g
∗ q ◦
for all f C (EF) and g C (EF). Then (K, ) becomes an associative C-algebra.
∈ GF α ∈ GF β ∗
For M EF, let E be the G -orbit of M. We take 1 C (EF) to be the
∈ α OM ⊂ α α [M] ∈ GF α
characteristic function of F , and set f = v−dimOM1 . We consider the subalgebra (L, )
OM [M] q [M] ∗
of(K, )generatedbyf overQ(v ),forallM EF andallα NI.InfactLhasaQ(v )-basis
∗ [M] q ∈ α ∈ q
f M EF,α NI . Since 1 1 (W)= gW for any W EF, we have
{ [M]| ∈ α ∈ } [M]◦ [N] MN ∈ γ
Proposition 1.3.1 [LXZ] The linear map ϕ :(L, ) ∗(Λ) defined by
∗ −→ H
ϕ(f ) = M , for all [M]
[M]
h i ∈ P
is an isomorphism of the associative Q(v )-algebras.
q
1.4 Double Ringel-Hall algebra D(Λ). First, we define a Hopf algebra +(Λ) which is a
H
Q(v) vector space with the basis K µ Z[I],α , whose Hopf algebra structure is given
µ
− { | ∈ ∈ P}
as
(a) Multiplication ([R1])
u+ u+ = vhα,βi gλ u+, for all ,α,β ,
α ∗ β αβ λ ∈P
λX∈P
K u+ = v(µ,α,)u+ K , for all ,α ,µ N[I],
µ∗ α α ∗ µ ∈ P ∈
K K = K K = K , for all ,µ,ν N[I].
µ ν ν µ µ+ν
∗ ∗ ∈
(b)Comultiplication ([G])
a a
(u+) = vhα,βi α βgλ u+K u+, for all ,λ ,
△ λ a αβ α β ⊗ β ∈ P
αX,β∈P λ
(K ) = K K , for all ,µ N[I].
µ µ µ
△ ⊗ ∈
with counite ǫ(u+) = 0, for all λ = 0 , and ǫ(K ) = 1. Here a denotes the cardinality of
λ 6 ∈ P µ λ
finite set Aut (M) with dimM = λ.
Λ
(c)Antipode([X])
S(u+)= δ + ( 1)m
λ λ0 − ×
X X
π∈P,λ1,···,λm∈P\{0}
a a
v2Pi<jhλi,λji λ1·a·· λmgλλ1···λmgλπ1···λmK−λu+π.
λ
for all λ , andS(K ) = K for all µ Z[I]. Ithas the subalgebragenerated by u λ ,
µ −µ λ
∈ P ∈ { | ∈ P}
which is isomorphic to ∗(Λ).
H
Dually, we can define a Hopf algebra −(Λ). Following Ringel, we have a bilinear form
H
ϕ : +(Λ) −(Λ) Q(v) defined by
H ×H −→
V
ϕ(K u+,K u−)= v−(µ,ν)−(α,ν)+(µ,β)| α|δ
µ α ν β a αβ
α
CANONICAL BASES 5
for all µ,ν Z[I] and all α,β . Thanks to [X], we can obtain the reduced Drinfeld double
∈ ∈ P
D(Λ) of Ringel-Hall algebra of Λ, and the triangular decomposition
D(Λ) = − +,
H ⊗T ⊗H
where denotes the torus subalgebra generated by K :µ Z[I] .
µ
T { ∈ }
The subalgebra of D(Λ) generated by u±,K i I is called the composition algebra of Λ
{ i ±| ∈ }
denoted by (Λ). It is also a Hopf algebra and admits a triangular decomposition
C
(Λ) = −(Λ) +(Λ),
C C ⊗T ⊗C
where +(Λ) is the composition algebra generated by u+ : i I, and −(Λ) is defined dually.
C i ∈ C
Moreover, the restriction ϕ: +(Λ) −(Λ) Q(v) is non-degenerate (see[HX]).
C ×C −→
In addition, D(Λ) also admits an involution ω defined by
ω(u+) = u−,ω(u−)= u+, for all λ ;
λ λ λ λ ∈ P
ω(K ) = K , for all µ N[I].
µ −µ
∈
Then ϕ(x,y) = (ω(x),ω(y)). Obviously, ω induces an involution of (Λ).
C
2. Singularity Ringel-Hall algebras
2.1 Singularity Ringel-Hall algebras s(Λ). Let , , be all nonhomogeneous tubes
1 l
H {T ··· T }
inmod-Λ(in fact,l 6 3), r = r( )theperiodof .Itiswellknownthat l (r 1) = I 2.
i Ti Ti i=1 i− | |−
Wedefine s(Λ)tobethesubalgebraof ∗(Λ)generatedby ui,u[M] : i PI,M j,1 6 j 6 l .
H H { ∈ ∈ T }
It is called Singularity Ringel-Hall algebra of Λ. From the definition, it is clear that s(Λ)
H
depends only on the type of the quiver Q, and does not depend on the finite field F . We now
q
set Ds(Λ) to be the subalgebra of D(Λ) generated by u±,u± : i I,M ,1 6 j 6 l .
{ i [M] ∈ ∈ Tj }
According to the AR quiver of tame quiver (see [CB]), it is easy to see that:
−
Lemma 2.1.1 Let M , for some i,1 6 i 6 l, and
i
∈ T
0 M M M 0
2 1
−→ −→ −→ −→
ashortexactsequence. ThenM = I N ,M = P N ,whereP ispreprojective,N ,N ,
1 ∼ 1⊕ 1 ∼ 2⊕ 2 2 1 2 ∈Ti
and I is preinjective. (cid:3)
1
Lemma 2.1.2 Ds(Λ) is a Hopf algebra depends on the type of Q.
Proof. By Lemma 2.1.1, it is easy to see that Ds(Λ) is closed under comultiplication. So it is
sufficient to prove that Ds(Λ) is closed under antipode S.
Since comultiplication is an algebra homomorphism , it is sufficient to prove that S(u+ )
[M] ∈
Ds(Λ), for some M ,1 6 i6 l.
i
∈ T
We may assume that S(u+ ) Ds(Λ) for any dimM′ < dimM. Because of µ(S 1) = ηǫ,
[M′] ∈ ⊗ △
we have
a a
S(u+ )+S(K )u+ + vhdimM1,dimM2i M1 M1gM S(u+ K )u+ . (2.1.1)
[M] dimM [M] M1X,M26=0 aM M1M2 [M1] dimM2 [M2]
Suppose M = I N ,M = P N . Using induction on dimM, we have, by Lemma 2.1.1,
1 ∼ 1⊕ 1 ∼ 2⊕ 2
S(u+ ) Ds(Λ).
[N1] ∈
Since u+ u+ = vhdimN1,dimI1iu+ , and S(u+ ) Ds(Λ) by Lemma 6.1 and 6.2 in [LXZ],
[N1]∗ [I1] M1 [I1] ∈
we have S(u+ ) Ds(Λ). Therefore (2.1.1) implies S(u+ ) Ds(Λ). (cid:3)
[M1] ∈ [M] ∈
6 GUANGLIANZHANG
2.2 Decomposition of s(Λ). In the following, we follow an idea of Sevenhant and Van den
H
Bergh to obtain subalgebras of s(Λ) and Ds(Λ). (see also [HX].)
H s,+ s,−
According to the definition of ϕ, it is easy to see that the restriction of ϕ on , for
α α
H ×H
all α N[I], is also non-degenerate.
∈
For α,β N[I], we use α β to means that β α N[I]. Clearly, (Λ) = s(Λ) if β < δ.
β β
∈ ≤ − ∈ C H
We now define
+ = x+ s,+(Λ) ϕ(x+, −(Λ)) = 0 .
Lδ { ∈ H | C }
Dually, let
− = x− s,−(Λ) ϕ(x−, +(Λ)) = 0 .
Lδ { ∈ H | C }
The non-degeneracy of ϕ implies
s,+(Λ) = +(Λ) +,
H δ C δ ⊕Lδ
s,−(Λ) = −(Λ) −.
H δ C δ ⊕Lδ
Let Ds(1) be the subalgebra of Ds generated by ±(Λ) and ±. Then we also have the
C Lδ
triangular decomposition
Ds(1) = Ds(1)− Ds(1)+.
⊗T ⊗
Suppose ± has been defined, we inductively define ± as follows:
L(m−1)δ Lmδ
+ = x+ s,+ ϕ(x+,Ds(m 1)−)= 0 ,
Lmδ { ∈ Hmδ| − }
− = x− s,− ϕ(x−,Ds(m 1)+) = 0 .
Lmδ { ∈Hmδ| − }
Let Ds(m) be the subalgebra of Ds generated by Ds(m 1)± and L± . Therefore
− mδ
Ds(m) = Ds(m)− Ds(m)+.
⊗T ⊗
Lemma 2.2.1 Let η = dim + = dim − . Then η = l.
nδ Q(v)Lnδ Q(v)Lnδ nδ
Proof. By proposition 7.5 in[LXZ], weknow that Ψ(u )for α Φ+ ; Ψ(u )for α ,
[M(α)] ∈ Prep α,i ∈Ti
therealroots: i =1, ,l;Ψ(u u ),m 1,1 j r 1,i= 1, ,l;Ψ(E˜ ),n
j,mδ,i j+1,mδ,i i nδ
··· − ≥ ≤ ≤ − ··· ≥
1 and Ψ(u ) for β Φ+ form a Z-basis of n+, that is, the basis of +(Λ)/(v 1).
[M(β)] ∈ Prei C −
According to the definition of s, Ψ(u ) for α Φ+ ; Ψ(u ) for α the real root:
H [M(α)] ∈ Prep α,i ∈ Ti
i = 1, ,l; Ψ(u ), m 1, 1 j r , i = 1, ,l; Ψ(E˜ ),n 1 and Ψ(u ) for
j,mδ,i i nδ [M(β)]
··· ≥ ≤ ≤ ··· ≥
β Φ+ form a Z-basis of s/(v 1).
∈ Prei H −
Since l (r 1) = I 2, we have η = l by the construction of + for all n. (cid:3)
i=1 i− | |− nδ Lnδ
For eachPnδ, there exist bases x1, ,xl of + and a basis y1, ,yl of − such that
{ n ··· n} Lnδ { n ··· n} Lnδ
1
ϕ(xp,yq)= δ .
n n v v−1 pq
−
Thus, we have
K K
xpyq yq xq = nδ − −nδδ δ ,
n m− m n v v−1 pq mn
−
for all m,n,1 6 p,q 6 l (see [HX]).
We now set x = u+,y = v−1u− for each i I, and let J = (nδ,p) : 16 p 6 l . Define
i i i − i ∈ { }
θ = i if i I,θ = nδ if i J, and x = xp if i = (nδ,p),y = v−1yp if i= (nδ,p).
i ∈ i ∈ i n i − n
Moreover, by a theorem of Sevenhaut and Van den Bergh, we have that Ds is generated by
x ,y i I J K :µ N[I]
i i µ
{ | ∈ ∪ }∪{ ∈ }
CANONICAL BASES 7
with the defining relations
K = 1,K K = K for all µ,ν N[I] (2.2.1)
0 µ ν µ+ν
∈
K x = v(µ,θi)x K , and K y = v−(µ,θi)y K for all i I J,µ N[I] (2.2.2)
µ i i µ µ i i µ
∈ ∪ ∈
x y y x = Kθi−K−θiδ for all i,j I J (2.2.3)
i j − j i v−v−1 ij ∈ ∪
( 1)px(p)x x(p′) = 0 and ( 1)py(p)y y(p′) = 0 (2.2.4)
p+p′=1−aij − i j i p+p′=1−aij − i j i
xPx = x x and y y = y y for all i,jP I J with (θ ,θ ) = 0. (2.2.5)
i j j i i j j i i i
∈ ∪
Applying the relations above, the next statement is clear.
Proposition 2.2.2
(a) xi n N,1 6 i6 l are central in s,+. Dually , yi n N,16 i 6 l are central in s,−.
{ n| ∈ } H { n| ∈ } H
(b) s,+ = + Q(v)[xi n N,1 6 i6 l].
H ∼ C ⊗ n| ∈
(c) xj commutes with s,−,yj commutes with s,+.
n n
H H
(cid:3)
Inparticular,wehavel = 1incaseQ = A ,n 3,thatis s,+ = U (sl )+ Q(v)[x , ,x , ,].
n ∼ q n 1 n
≥ H ⊗ ··· ···
e e
3. PBW bases of Singularity Ringel-Hall algebras
−
3.1 It is known that there exists full subcategory C(P,L) in modΛ (see [LXZ] 7.1). Moreover
C(P,L) is equivalent to the module category of the Kronecker quiver over F . Thus it induces an
q
exactembeddingF : modK ֒ modΛ,whereK isthepathalgebraoftheKroneckerquiverover
→
F . We note here that the embedding functor F is essentially independent of the field F . This
q q
gives rise to an injective homomorphism of algebras, still denoted by F : ∗(K) ֒ ∗(Λ). In
H → H
∗(K)wehavedefinedtheelementE form 1.SetE = F(E ).SinceE ∗(K),
H mδK ≥ mδ mδK mδK ∈ C
and L , P ∗(Λ), so E is in ∗(Λ) and even in ∗(Λ) . Let be the subalgebra of ∗(Λ)
mδ Z
h i h i ∈ C C C K C
generated by E for m N, it is a polynomial ring on infinitely many variables E m 1 ,
mδ mδ
∈ { | ≥ }
and its integral form is the polynomial ring on variables E m 1 over .
mδ
{ | ≥ } Z
We denote by C (resp. C ) the full subcategory of C(P,L) consisting of the Λ-modules which
0 1
belong to homogeneous (resp. non-homogeneous) tubes of modΛ.
We now decompose E as follows
nδ
E = E +E +E ,
nδ nδ,1 nδ,2 nδ,3
where
Enδ,1 = v−ndimS1−ndimS2 [M],M∈C1,dimM=nδu[M] (3.1.1)
E = v−ndimS1−ndimS2P u (3.1.2)
nδ,2 [M],dimM=nδ [M]
PM=M1⊕M2,06=M1∈C1,06=M2∈C0
Enδ,3 = v−ndimS1−ndimS2 [M],M∈C0,dimM=nδu[M]. (3.1.3)
P
Note that dimS = dim S ,i = 1,2, but the values are independent of the choice of finite
i Fq i
field F . Let w = (w , ,w ) be a partition of n, we then define
q 1 t
···
Ewδ,3 = Ew1δ,3∗···∗Ewtδ,3.
Let P(n) be the set of all partitions of n, and N = v−dimN+dimEnd(N)u . Set
[N]
h i
B = {hPi∗hMi∗Ewδ,3∗hIi| P ∈ Pprep,M ∈ ⊕li=1Ti,I ∈ Pprei,w ∈ P(n),n ∈N}.
Then we have the following:
8 GUANGLIANZHANG
Theorem 3.1.1 The set B is Q(v) bases of s,+.
− H
Proof. Let Πa be the set of aperiodic r tuples of partitions, for all 16 i 6 l. Set
i i−
Bc = {hPi∗Eπ1 ∗···∗Eπl ∗Ewδ ∗hIi : P ∈ Pprep,I ∈ Pprei,πi ∈ Πai,w ∈ P(n),1 6 i6 l,n ∈ N},
B′ = x f(xj) :x Bc,f(xj) Q(v)[xj :1 6 j 6 l,n N] .
n n n
{ ∗ ∈ ∈ ∈ }
By Proposition 7.2 in [LXZ], we know that Bc is a Q(v)-basis of ∗(Λ). Following Proposi-
C
tion 2.2.2, we conclude that B′ is Q(v) basis of s,+.
− H
Using induction on n, it follows from (3.1.1), (3.1.2) and (3.1.3) that E s,+. Therefore
nδ,3
∈ H
B s,+.
⊆ H
Because B is linear independent over Q(v), we will see that each
hPi∗Eπ1 ∗···∗Eπl ∗Ewδ ∗hIi∗f(xjn)
in B′ may be represented by B.
If M is a homogeneous module in some tubes, N a regular module in other tubes, then
M N = N M . Thus, (3.1.1), (3.1.2) and (3.1.3) imply that Ewδ is represented by
hB. iFu∗rhtheirmorhe, iB∗chmaiy also be represented by B. So it is sufficient to prove that xj satisfy
n
the property. By the definition of s,+, we know that s,+ is generated by Bc and M :
H H {h i
M l . Note that M or M = 0 if 0 P P M M 0 is a short exact
∈ ⊕i=1Ti} 1 ∈ Ti 1 −→ −→ 1 ⊕ 1−→ −→
sequence, where P,P ,M for some i. Similarly, we have M or M = 0 if
1 prep i 1 i 1
∈ P ∈ T ∈ T
0 M I M I 0 is a short exact sequence, where I,I ,M . It implies
1 1 1 prei i
−→ −→ ⊕ −→ −→ ∈ P ∈ T
that every element in s,+ , in particular xj, my be represented by B.
n
H
4. Purity Properties of Perverse sheaves of closure of semi-simple objects in
i
T
4.1 We denote by M(x),x E , the Λ module of dimension vector α corresponding to x. For
α
∈ −
subsets E and E , we define the extension set ⋆ of by to be
α β
A⊂ B ⊂ A B A B
⋆ = z E there exists an exact sequence
α+β
A B { ∈ |
0 M(x) M(z) M(y) 0 with x , y .
→ → → → ∈ B ∈ A}
It follows from the definition that ⋆ = p p (p−1( ). Because p is a locally trivial
A B 3 2 1 A×B 1
fibration (see Lemma 2.3 in [LXZ]), we then have ⋆ ⋆ . In particular, ⋆ =
M N
A B ⊆ A B O O
if Ext(M,N) = 0, i.e., is open and dense in ⋆ .
M⊕N M⊕N M N
O O O O
Set codim = dimE dim . We will need the following:
α
A − A
Lemma 4.1.1 [Re] Given any α,β NI, if E and E are irreducible algebraic
α β
∈ A ⊂ B ⊂
varieties and are stable under the action of G and G respectively, then ⋆ is irreducible
α β
A B
and stable under the action of G , too. Moreover,
α+β
codim ⋆ = codim +codim β,α +r,
A B A B−h i
where 0 r min dim Hom(M(y),M(x))y ,x . (cid:3)
k
≤ ≤ { | ∈ B ∈ A}
Let M(x),N(y) be modules corresponding to x,y E respectively. We denote by ( or
α x
∈ O
) the G orbits of x. We now introduce two orders in Λ mod as follows:
M(x) α
O − −
N M if .
deg N M
• ≤ O ⊆ O
N M if there exists M ,U ,V and short exact sequence
ext i i i
• ≤
0 U M V 0
i i i
−→ −→ −→ −→
such that M = M ,M = U V ,1 6 i 6p, and N = M for some natural number p.
1 i+1 i i p+1
⊕
CANONICAL BASES 9
It follows from G.Zwara [Z] that:
Proposition 4.1.2 The orders , are equivalent in Λ mod. (cid:3)
deg ext
≤ ≤ −
We denote by the category of perverse sheaves on algebraic variety X. Let f be a locally
X
P
closed embedding from X to Y. One has the intermediate extension functor
f : ,P Im p 0(fP) p 0(f P) .
!∗ X Y ! ∗
P −→ P 7−→ { H −→ H }
Inparticular,suppose bealocalsystemonanonsingularZariskidenseopensubsetj :U Y
L −→
of the irreducible n dimensional Y. Then IC ( ) := j [n] .
Y !∗ Y
− L L ∈ P
Definition 4.1.3 Let be a Weil complex. Then is said to have the purity property if on
K K
all stalks of the semisimplication H v( )ss(v/2) of cohomology sheaves the Frobenius F acts
K x x
trivially.
Lemma 4.1.4 Let p : X Y be a smooth morphism of relative dimension d,dimY = m, a
−→
nonsingular Zariski dense open subset U of the irreducible n dimensional Y and the diagram
−
as follows is a cartesian square
U = p−1(U ) j X
0
−→
p p
|U0 ↓ ↓
U j0 Y.
−→
If j Q [m] has the purity property, then j Q [d+m] has the purity property.
0!∗ l !∗ l
Proof. By the definition of j and j, we have a natural morphism
∗ !
ϕ :p 0(j Q [m]) p 0(j Q [m]).
Y 0! l 0∗ l
H −→ H
It induces an intermediate extension functor
j :
0!∗ U Y
P −→ P
such that j Q [m] = Im p 0(j Q [m]) p 0(j Q [m]) . Furthermore,
0!∗ l 0! l 0∗ l
{ H −→ H }
p∗[d] ϕ : p∗[d](p 0(j Q [m]) p∗[d](p 0(j Q [m]).
Y 0! l 0∗ l
◦ H −→ H
Since p : X Y is a smooth of relative dimension d, p∗[d] = p![ d] are t exact, we have
−→ − −
ϕ :p 0(p∗j Q [m+d]) p 0(p∗j Q [m+d]),
X 0! l 0∗ l
H −→ H
ϕ :p 0(jQ [m+d]) p 0(j Q [m+d]).
X ! l ∗ l
H −→ H
So, p∗[d](j Q [m]) = j Q [d+m]. By the same argument, we get p∗[d] F = F p∗[d]. Since
0!∗ l !∗ l
◦ ◦
j Q [m] has the purity property, the statement of Lemma is true. (cid:3)
0!∗ l
Let and are, respectively, the union of orbits of regular modules of C(P,L) and
Nwi Nwi,3
C0(P,L) with dimension vector wiδ. Set Nw = Nw1 ⋆···⋆Nwt and Nw,3 = Nw1,3⋆···⋆Nwt,3.
For any P ,M l ,I ,π Πa,1 6 i 6 l,w P(n),n N, we define the
∈ Pprep ∈ ⊕i=1Ti ∈ Pprei i ∈ i ∈ ∈
varieties
OP,π1,···,πl,w,I = OP ⋆Oπ1···⋆Oπl ⋆Nw ⋆OI, OP,M,w,I = OP ⋆OM ⋆Nw,3⋆OI.
According to [L4] and [L5], we know that IC (Q ) have purity property. In order
OP,π1,···,πl,w,I l
to construct the canonical basis of s(Λ), we need to study the purity property of P,M,w,I.
H O
Theorem 4.1.5 Let X = OP,M,w,I, for all P ∈ Pprep,M ∈ ⊕li=1Ti,I ∈ Pprei,w ∈ P(n),n ∈ N.
Then IC (Q ) has the purity property.
X l
10 GUANGLIANZHANG
InordertoproveTheorem4.1.5,weneedtoproveanumberofLemmasaboutpurityproperties
of perverse sheaves of closure of semi-simple objects in .
i
T
4.2 For the tame quivers A ,D ,E ,E and E . The orientations are given ( see blow)
n n 6 7 8
2 3
~e e e e e
A
n 1 k
n+1 n
Fig.4.2.1
2 n+1
~
D
3 4 n-2 n-1
n
1 n
Fig.4.2.2
7
6
~
E
6
1 2 3 4 5
Fig.4.2.3
8
~
E
7
1 2 3 4 5 6 7
Fig.4.2.4
9
~
E
8
1 2 3 4 5 6 7 8.
Fig.4.2.5
We then have ( or following [DR])
(4.2.1) the real regular simples of A have period n, and have dimension vectors
n
(1,0,0, ,0,1),(0,1,0, ,0,0), ,(0,0,0, ,1,0).
··· ··· ·e·· ···
(4.2.2) the real regular simples of D with dimension vectors
n
(1,0,1,1 ,1,1,0),(0,1,1,1, ,1,e0,1)(with period 2)
··· ···
(1,0,1,1 ,1,0,1),(0,1,1,1, ,1,1,0)(with period 2)
··· ···
(1,1,1,0 ,0,0,0),(0,0,1,1, ,1,1,1),(0,0,0,1,0, ,,0,0,0), ,(0,0,0,0, ,0,1,0,0)
··· ··· ··· ··· ···
(with period n-2)