This gives the algebraic realization of the positive/negative part of a (Kac-Moody type) quantum group

GROUPS I: RESTRICTION FUNCTOR AND GREEN’S FORMULA

JIE XIAO, FAN XU AND MINGHUI ZHAO

Abstract. In this paper, we generalize the categorifical construction of a quantum group and its canonical basis by Lusztig ([16, 17]) to the generic form of whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula in [7] and the restriction functor in [17]. By a geometric way to prove Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig’s framework.

1. Introduction

Based on the classical works of Hall ([8]) and Steineiz ([31]), the Ringel-Hall algebra H(A) of a (small) abelian category A was introduced by Ringel in [24], as a model to realize the quantum group. When A is the category Rep_F

qQ of finite dimensional representations for a simply-laced Dynkin quiverQover a finite field Fq, the Ringel-Hall algebra H(A) is isomorphic to the positive part of the corresponding quantum group ([24]). For any acyclic quiverQandA= Rep_FqQ, the composition subalgebra ofH(A) generated by the elements corresponding to simple representations is isomorphic to the positive part of the quantum group of typeQ.

This gives the algebraic realization of the positive/negative part of a (Kac-Moody type) quantum group. This realization was achieved by Green ([7]), through solving a natural question whether there is a comultiplication on H(A) compatible with the corresponding multiplication so that the above isomorphism is an isomorphism between bialgebras. Now it is well-known that Green’s comultiplication depends on a remarkable homological formula in [7] (called the Green formula in the following).

In the seminal papers [16] and [17], Lusztig gave the geometric realization of the positive/negative part of a quantum group and then constructed the canonical basis for it. LetQ= (Q₀, Q₁) be a quiver and

Ed:= M

α∈Q1

Hom_K(K^ds(α),K^dt(α)) be the variety with the natural action of the algebraic group

Gd:= Y

i∈Q0

GL(di,K)

Date: September 13, 2016.

Key words and phrases. Hall algebra, induction functor, restriction functor, simple perverse sheaves.

Jie Xiao was supported by NSF of China (No. 11131001), Fan Xu was supported by NSF of China (No. 11471177) and Minghui Zhao was supported by NSF of China (No. 11526037).

1

for a given dimension vector d∈NQ0. Lusztig ([17]) defined the flagFi,a and the subvariety

Eei,a⊆Ed× Fi,a.

Fix any type (i,a), consider the canonical proper morphism π_i,a: Eei,a → Ed. By the decomposition theorem of Beilinson, Bernstein and Deligne ([1]), the complex π_i,a!1 is semisimple, where 1 is the constant perverse sheaf on Eei,a. Let Q_d be the category of complexes isomorphic to sums of shifts of simple perverse sheaves appearing in πi,a!1, Kd the Grothendieck group of Qd and K = L

dKd. Lusztig ([17]) already endowed K with the multiplication and comultiplication structures by introducing his induction and restriction functors. He proved that the comul- tiplication is compatible with the multiplication inK and K is isomorphic to the positive part of the corresponding quantum group as bialgebras up to a twist.

By this isomorphism, the isomorphism classes of simple perverse sheaves inQd

provide a basis of the corresponding quantum group, which is called the canonical basis. The canonical basis theory of quantum algebras is crucially important in Lie theory. This basis has many remarkable properties such as such as integrality and positivity of structure constants, compatibility with all highest weight integrable representations, etc. Lusztigs approach essentially motivates the categorification of quantum groups (for example, see [12],[28] and [32]) or quantum cluster algebras (see [9],[21],[13], etc.), i.e., a quantum group/quantum cluster algebra can be viewed as the Grothendieck ring of a monoidal category and some simple objects provides a basis (see also [33]).

For a long time we have been asked what the explicit relation exists between the Green’s comultiplication and Lusztig’s restriction functor. As one of the main result in the present paper, the following Theorem 4.8 and the definition of the comultiplication operator ∆ provide us this strong and clear link. Thanks to an embedding property as in [14] and [27], we can lift the Green formula from finite fields to the level of sheaves. This is finally suitable to apply the Lusztig’s restriction functor to the larger categories of the so-called Weil complexes, whose Grothendieck groups realize the weight spaces of a generic Ringel-Hall algebra.

The second aim of this paper is to give the categorification of Ringel-Hall algebras via Lusztig’s geometric method. In Section 2, we recall the theory of Ringel-Hall algebras, focusing on the Hopf structure of Ringel-Hall algebras. In Section 3, we recall Lusztig’s construction of Hall algebras via functions invariant under the Frobenius map. Then we obtain an algebra CF^F(Q). The comultiplication over CF^F(Q) is just a twist of Green’s comultiplication. However, the proof of the compatibility of Lusztig’s comultiplication and multiplication for the whole Ringel- Hall algebra essentially depends on the proof of the Green formula. In the end of this section, we show that the twist ofCF^F(Q) is isomorphic toH^tw(A) as a Hopf algebra. Hence, Lusztig’s comultiplication can be applied to Ringel-Hall algebras.

In Section 4, we extend the geometric realization of a quantum group to the whole Ringel-Hall algebra under Lusztig’s framework. We obtain the generic Ringel-Hall algebra as the direct sum of Grothendieck groups of the derived categories of a class of Weil complexes. The simple perverse sheaves provide the caonical basis.

We show that the compatibility of the induction and restriction functor holds for these perverse sheaves. Therefore the generic Ringel-Hall algebra has a structure of Hopf algebra. In Section 5, we construct the Drinfeld double of the generic Ringel-Hall algebra.

Acknowledgments. The authors are very grateful to Hiraku Nakajima for telling us that Lusztig’s restriction is a hyperbolic localization and Reference [3]. The sec- ond named author thanks Sheng-Hao Sun for explaining the contents in Reference [14] and many helpful comments.

2. A revisit of Ringel-Hall algebras as Hopf algebras

We recall the definition of the Ringel-Hall algebra H(A) for the hereditary abelian category A= modkQ= Rep_kQ, wherek=Fq is a finite field withq=p^e elements for some prime numberpandQis a finite quiver without oriented cycles.

For M ∈ A, we denote by dimM the dimension vector in NQ0 and define the Euler-Ringel form onNQ₀ as follows:

hM, Ni= dimkHom_A(M, N)−dimkExt¹_A(M, N).

ForM, N andL∈modkQ, we denote byF_{M N}^L the set{X ⊂L|X ∈modkQ, X∼= N, L/X ∼=M}and Ext¹_A(M, N)L the subset of Ext¹_A(M, N) with the middle term isomorphic to L. Write F_{M N}^L =|F_{M N}^L |and h^{M N}_L = ^|Ext_|Hom^1A(M,N)L|

A(M,N)|. ForX ∈ A, set aX=|AutAX|.

The ordinary Ringel-Hall algebra H(A) is a C-space with isomorphism classes [X] of allkQ-modulesX as a basis and the multiplication is defined by

[M]∗[N] =X

[L]

F_{M N}^L [L]

forM, N and L∈modkQ. We can endow H=H(A) a comultiplicationδ:H → H ⊗_CHby setting

δ([L]) = X

[M],[N]

h^{M N}_L [M]⊗[N].

The comultiplication is compatible with the multiplication via Green’s theorem.

Theorem 2.1. [7] The map δ is an algebra homomorphism with respect to the twisted multiplication onH ⊗ H as follows:

([M1]⊗[N1])◦([M2]⊗[N2]) =q^hM1,N2i([M1]∗[M2])⊗([N1]∗[N2]).

The theorem is equivalent to the following Green formula holds:

a_M1a_M2a_N1a_N2X

[L]

F_M^L

1N₁F_M^L

2N₂a⁻¹_L

= X

[X],[Y1],[Y2],[Z]

|Ext¹_A(X, Z)|

|HomA(X, Z)|F_XY^M1

1F_XY^M2

2F_Y^N1

2ZF_Y^N2

1Za_Xa_Y1a_Y2a_Z.

Define a symmetric bilinear form on H by setting ([M],[N]) = ^δM,N_a

M . We call the form Green’s Hopf pairing. It is clear that the Green Hopf pairing is a non degenerated bilinear form overH. The following proposition shows that the comul- tiplication is dual to the multiplication, i.e., the comultiplication can be viewed as the multiplication overH^∗= Hom_C(H,C).

Proposition 2.2. The comultiplication is left adjoint to the multiplication with respect to Green’s Hopf pairing, i.e., fora, b, c∈ H, (a, bc) = (δ(a), b⊗c).

The proposition is equivalent to the Riedtmann-Peng formula F_{M N}^L aMaN =h^{M N}_L aL

holds.

It is easy to generalize the multiplication and comultiplication to the r-fold versions forr≥2.ForM₁,· · ·, M_r, M ∈ A, setF_M^M

1,···,M_r to be the set

{0 =X₀⊆X₁⊆ · · · ⊆X_r=M |X_i∈ A, Xi+1/X_i∼=M_r−i, i= 0,1,· · · , r−1}

andF_M^M

1···M_r =|F_M^M

1,···,M_r|.Then

[M1]∗[M2]∗ · · · ∗[Mr] =X

[M]

F_M^M_1···Mr[M].

The r-fold comultiplication δ^r can be defined inductively. For r = 1,δ¹ =δ and δ^r+1= (1⊗ · · · ⊗1⊗δ)◦δ^r forr≥1.Set

δ^r−1([M]) = X

[M₁],···,[M_r]

h^M_M^1···Mr[M1]⊗ · · · ⊗[Mr]

forr≥2. It is clear that the Riedtmann-Peng formula can be reformulated as h^M_M^1M2···Mr =F_M^M

1···M_raM₁· · ·aM_ra⁻¹_M forr≥2.

Letσ:H → Hbe a map such that σ([M]) =δM,0+X

r≥1

(−1)^r· X

[N],[M1],···,[Mr]6=0

h^M_M^1···MrF_M^N_1···Mr[N].

forM ∈ A. We callσthe antipode ofH.

One can also define the twisted version of the multiplication overH(A) by setting [M]·[N] =v^hM,Ni[M]∗[N],

where v = √

q. Similarly, the comultiplication, the antipode and Green’s Hopf pairing can be twisted (see [34] for more details). We denote byH^tw(A) the twisted version ofH(A).

Theorem 2.3. [24, 7, 34] The algebra H^tw(A) is a Hopf algebra with the above twisted multiplication, comultiplication and antipode. Let gQ be the Kac-Moody algebra associated to the quiverQ andU_v⁺(gQ)be the positive part of the quantum group Uν(gQ) specialized at ν = v with v = q¹². Let C^tw(A) be the subalgebra of the twisted Ringel-Hall algebraH^tw(A)generated by isomorphism classes of simple kQ-modules. Denoted by C^tw

Z[v,v⁻¹](A) the integral form of C^tw(A). Then there is an isomorphism of Hopf algebras

Ψ :U_v⁺(gQ)→ C_Z^tw_[v,v−1](A) O

Z[v,v⁻¹]

Q(v) sendingEi to[Si]fori∈Q0.

3. Lusztig’s construction of Hall algebras via functions In this section, we recall Lusztig’s construction of Hall algebras via functions in [19] and compare it with Ringel-Hall algebras. Let k=Fq as above and K=Fq. Given a dimension vectord= (d_i)_i∈Q0,define the variety

Ed :=Ed(Q) = M

α∈Q1

Hom_K(K^ds(α),K^dt(α)).

Any elementx= (xα)_α∈Q1 inEd(Q) defines a representationM(x) = (K^d, x) ofQ withK^d=L

i∈Q0K^di. The algebraic group Gd:=Gd(Q) = Y

i∈Q0

GL(di,K) acts onEdby (x_α)^g_α∈Q

1 = (g_t(α)x_αg_s(α)⁻¹ )_α∈Q1forg= (g_i)_i∈Q0 ∈G_dand (x_α)_α∈Q1 ∈ Ed.The isomorphism class of aKQ-moduleX is just the orbit ofX. The quotient stack [Ed/Gd] parametrizes the isomorphism classes of KQ-modules of dimension vectord.

LetF be the Frobenius automorphism ofK, i.e.,F(x) =x^q. TheF-fixed subfield is justFq. This induces an isomorphismEd→Ed sending (((xα)ij)d_s(α)×d_t(α))α∈Q1

to (((xα)^q_ij)d_s(α)×dt(α))_α∈Q1. We will denote all induced map by F if it does not cause any confusion.

For anyKQ-moduleM(x) = (K^d, x), set M(x)^[q] =F(M(x)). The representa- tionM(x)∈Ed is F-fixed if M(x)∼=M(x)^[q]. The last condition is equivalent to say that M(x) is defined over Fq, i.e., there exists a kQ-moduleM₀(x) such that M(x)∼=M0(x)⊗_FqK([10]). We denoted byE^Fd and G^F_d theF-fixed subset of Ed

andG_d respectively. For akQ-moduleM ∈E^Fd, letOM denote the orbit of M in Ed andO^F_M theF-fixed subset ofO_M.

Let l 6= p be a prime number and Ql be the algebraic closure of the field of l-adic numbers. Fix a square root v =q¹² ∈ Ql. Define CF^F_d to be the Ql-space generated by G^F_d-invariant functions: E^Fd → Ql. We will endow the vector space CF^F(Q) =L

dCF^F_d with a multiplication and comultiplication.

First we recall two functors: the pushforward functors and the inverse image functors in [19]. Given two finite setsX, Y and a map φ:X →Y. LetCF(X) be the vector space of all functionsX →QloverX.Define the pushforward ofφto be

φ!:CF(X)→ CF(Y), φ!(f)(y) = X

x∈φ⁻¹(y)

f(x) and the inverse image ofφ

φ^∗:CF(Y)→ CF(X), φ^∗(g)(x) =g(φ(x)).

Let E

00 be the variety of all pairs (x, W) where x ∈ Eα+β and (W, x|W) is a KQ-submodule of (K^α+β, x) with dimension vector β. LetE⁰ be the variety of all quadruples (x, W, ρ₁, ρ₂) where (x, W)∈E

00 andρ₁:K^α+β/W∼=K^α, ρ₂:W ∼=K^β are linear isomorphisms. Consider the following diagram

Eα×Eβ E⁰

p₂ //

p₁

oo E⁰⁰

p₃ //Eα+β

wherep₂, p₃are natural projections and p₁(x, W, ρ₁, ρ₂) = (x⁰, x⁰⁰) such that x⁰_h(ρ₁)_s(h)= (ρ₁)_t(h)x_h and x⁰⁰_h(ρ₂)_s(h)= (ρ₂)_t(h)x_h

for anyh∈Q1.

The groups Gα×Gβ and Gα+β naturally act on E⁰. The map p1 is Gα+β× Gα×Gβ-equivariant under the trivial action ofGα+β onEα×Eβ.The map p2 is a principalG_α×G_β-bundle.

Applying the Frobenius map F, we can define the above diagram over Fq as follows:

E^Fα×E^Fβ E^0F

p₂ //

p₁

oo E^00F

p₃ //E^Fα+β. There is a linear map (called the induction)

m:CFG_α×Gβ(E^Fα×E^Fβ)→ CFG_α+β(E^Fα+β) =CF^F_α+β

sendinggto|G^F_α×G^F_β|⁻¹(p₃)_!(p₂)_!p^∗₁(g).Iteratively, one can define ther-fold version m^rofmforr≥1 by settingm¹=id,m²=mandm^r+1=m◦(1⊗m^r) forr≥2.

Now we can define the multiplication over CF^F(Q). For fα∈ CF^F_α, fβ ∈ CF^F_β and (x1, x2)∈Eα×Eβ, setg(x1, x2) =fα(x1)fβ(x2). Theng∈ CF^F_α+β and define the multiplcation

f_α∗f_β =m(g).

Lemma 3.1. [15] Given threekQ-modulesM, N andL, let1_OM,1_ON and1_OL be the characteristic functions over orbits, respectively. Then1_OF

M∗1_OF

N(L) =F_{M N}^L . We now turn to define the comultiplication overCF^F(Q).Fix a subspace W of K^α+β with dimW =β and linear isomorphismsρ1:K^α+β/W ∼=K^α, ρ2:W ∼=K^β. LetFα,β be the closed subset ofEα+βconsisting of allx∈Eα+βsuch that (W, x|W) is aKQ-submodule of (K^α+β, x) with dimension vectorβ. Consider the diagram

Eα×Eβ Fα,β

oo κ ⁱ //Eα+β ,

where the mapiis the inclusion andκ(x) =p1(x, W, ρ1, ρ2).For (x1, x2)∈Eα×Eβ, the fibreκ⁻¹(x₁, x₂)∼=L

h∈Q1Hom_K(K^αs(h),K^βt(h)) and thenκis a vector bundle of dimensionP

h∈Q1α_s(h)β_t(h).

Applying the Frobenius map F, we can define the above diagram over Fq as follows:

E^Fα×E^Fβ oo ^κ F_α,β^{F i} //E^Fα+β. There is also the restriction map

δα,β :CFG_α+β(E^Fα+β)→ CFG_α×Gβ(E^Fα×E^Fβ)

sendingf ∈ CFG_α+β(E^Fα+β) toκ!i^∗(f).It defines the comultiplicationδoverCF^F(Q), i.e., for f ∈ CF^F_γ and α+β =γ, δ(f) = P

α,β;α+β=γδα,β(f). Iteratively, we can defineδ^r forr≥1 by settingδ¹=δandδ^r+1= (1⊗ · · · ⊗δ)◦δ^rforr≥1.

ForM, N andL inA= Rep_kQ, we set D^{M N}_L =δ(1_OF

L)(M, N) =κ!i^∗(1_OF

L)(M, N).

In order to compare this with the comultiplication of Ringel-Hall algebras, we define the twistδ_α,β^tw =q^{−Pi∈Q0αiβi}δα,β andδ^tw in the same way.

Lemma 3.2. With the notations in Lemma 3.1 and dimM =α,dimN = β, we haveδ_α,β^tw(1_OF

L)(M, N) =h^{M N}_L .

Proof. SupposeM = (K^α, x1) andN = (K^β, x2). The linear isomorphisms ρ1, ρ2

induce the module structures ofK^α+β/W andW, denoted by (K^α+β/W, y_Kα+β/W) and (W, yW) respectively. Consider the set

S={x∈E^α+β|(W, x|W) = (W, y_W),

(K^α+β/W, x|_Kα+β/W) = (K^α+β/W, y_Kα+β/W),(K^α+β, x)∼=L}.

Fix a decomposition of the vector spaceK^α+β=W⊕K^α+β/W.Then S={x=

(yW)h d(h) 0 (y_Kα+β/W)h

h∈Q1

|d(h)∈Hom_K(K^αs(h),K^βt(h)),(K^α+β, x)∼=L}.

SetD(α, β) =L

h∈Q1Hom_K(K^αs(h),K^βt(h)). Apply the Frobenius mapF, we have the following long exact sequence (see [4])

0 //HomkQ(M, N) //⊕i∈Q₀Homk(k^αi, k^βi) //

//D^F(α, β) ^π //Ext¹_kQ(M, N) //0.

We denote by D^F(α, β)L the inverse image of Ext¹_kQ(M, N)L under the map π.

Then D_L^{M N} =|D^F(α, β)L| =q^{Pi∈Q0αiβi}h^{M N}_L . By definition,δα,β(1_OF

L)(M, N) =

|D^F(α, β)_L|=D_L^{M N}.This completes the proof.

In order to compare these with Lusztig’s construction, we consider the subalgebra of CF^F(Q) generated by 1Si = 1_OF

Si for all i ∈ Q0, denoted by F^F(Q). Then F^F(Q) =L

dF_d^F.

Lemma 3.3. Given a sequence i = (i1, i2,· · ·, im) in Q0 such that ij 6= ik for j 6= k ∈ {1,2,· · · , m} and let f = 1S_i1 ∗1S_i2 ∗ · · · ∗1S_im ∈ CF^F(Q). Then δ(f) =δ^tw(f).

The Riedtmann-Peng formula can be reformulated to the following form, which generalizes [19, Lemma 1.13] fromF^F(Q) toCF^F(Q).

Proposition 3.4. Letfi∈ CF^F_d

i fori= 1,2 andg∈ CF^F_d ford=d₁+d₂.Then

|G^F_d|X

x,y

f1(x)f2(y)δ_d^tw

1,d₂(g)(x, y) =|G^F_d

1×G^F_d

2|X

z

f1∗f2(z)g(z) wherex∈E^Fd₁, y∈E^Fd₂ andz∈E^Fd₁.

Proof. Given a dimension vectord, takef ∈E^Fd, then f =Pl

i=1a_i1_OF

Mi for some ai∈Ql,l∈ZandkQ-modulesM1,· · ·, Ml.With loss of generality, we may assume thatf1= 1_OF

M,f2= 1_OF

Nandg= 1_OF

L for somekQ-modulesM, NandL.Following Lemma 3.1 and 3.2, the left side of the equation is equal to

|G^F_d| · |O^F_M| · |O_N^F| ·h^L_{M N} and the right side of the equation is equal to

|G^F_d

1| · |G^F_d

2| · |O_L^F| ·F_{M N}^L

UsingaL=|G^F_d|/|O_L^F|and the Riedtmann-Peng formula, we prove the proposition.

By definition, dimkG^F_d =P

i∈Q0d²_i and then we obtain the following lemma ([30, Section 1.2]).

Lemma 3.5. With the above notation, we have dimkG^F_d −dimkG^F_d

1−dimkG^F_d

2= X

i∈Q0

(d₁)i(d₂)i. Hence, the equation in the above proposition can also be written as

|G^F_d|

|g^F_d| ·X

x,y

f1(x)f2(y)δd₁,d₂(g)(x, y) =

|G^F_d

1|

|g^F_d

1|

|G^F_d

2|

|g^F_d

2| X

z

f1∗f2(z)g(z) by substitutingδ for δ^tw, where g^F_d, g^F_d

1 and g^F_d

2 are the Lie algebras of G^F_d, G^F_d

1

andG^F_d

2, respectively.

Fix dimension vectorsα, β, α⁰, β⁰ with α+β =α⁰+β⁰ =d. LetN be the set of quadruplesλ= (α1, α2, β1, β2) of dimension vectors such that α=α1+α2, β= β1+β2, α⁰=α1+β1 andβ⁰=α2+β2. Consider the following diagram

E^Fα×E^Fβ E^0Fα,β p₁

oo ^p2 //E

00F α,β

p₃ //E^Fd

`

λ∈NF_λ^F

i⁰

OO

κ⁰

F_α^F0,β⁰

κ

i

OO

`

λ∈NE^F(λ) `

λ∈NE^0Fλ p⁰₁

oo ^p02 //`

λ∈NE

00F(λ) ^p

0

3 //E^Fα⁰×E^Fβ⁰

where E^F(λ) =E^Fα1×E^Fα2×E^Fβ₁ ×E^Fβ₂, E^0F(λ) =E^0Fα₁,β₁ ×E^0Fα₂,β₂ and E^00F(λ) = E^00Fα1,β1 ×E^00Fα2,β2 for λ = (α₁, α₂, β₁, β₂). This induces the maps between F-fixed subsets and then the maps between vector spaces of functions as follows:

CF^F_α× CF^F_β ^mα,β //

δ^tw

CF^F_d

δ_α^tw0,β0

CF^F(Q

λ∈NEλ) ^m //CF^F_α0× CF^F_β0

The following theorem can be viewed as the geometric analog of Green’s theorem.

Theorem 3.6. With the above notation, the above diagram is commutative, i.e., δ^tw_α0,β⁰m_α,β =mδ^tw.

In order to prove the theorem, we introduce some notations. Set C_α,β,α^0F 0,β⁰ ={(x, W, ρ₁, ρ₂)∈E^0Fα,β|x∈F_α^F0,β⁰} and

C_α,β,α^00F 0,β⁰ ={(x, W)∈E

00F

α,β|x∈F_α^F0,β⁰}.

The sets can be illustrated by the following diagram W⁰

W //(K^d, x) //

(K^d, x)/W

(K^d, x)/W⁰

where (x, W⁰)∈E^00Fα⁰,β⁰.Consider the following diagram E^Fα×E^Fβ C_α,β,α^0F 0,β⁰

oo p ^q //C_α,β,α^00F 0,β⁰

r //F_α^F0,β⁰

κ //E^Fα⁰×E^Fβ⁰ . Then by definition, we have

δ_α^tw0,β⁰m_α,β=|G^F_α×G^F_β|⁻¹q⁻

P

i∈Q0α0 iβ0

i(κ)!(r)!(q)!p^∗. ForM ∈E^Fα, N ∈E^Fβ, M⁰ ∈E^Fα⁰, N⁰∈E^Fβ⁰, one can check

δ_α^tw0,β⁰m_α,β(1_OF M,1_OF

N)(M⁰, N⁰) = X

[L]∈E^Fd/G^F_d

F_{M N}^L h^M_L^0N0.

Similarly, we can define the set

S_λ^0F ={(xα, xβ, xα⁰, xβ⁰, W1, W2, ρ11, ρ12, ρ21, ρ22)|(xα⁰, W1, ρ11, ρ12)∈E

0F α2,β2, (xβ⁰, W2, ρ21, ρ22)∈E

0F

α1,β1,(xβ, W2)∈E^00Fβ1,β2,(K^β, xβ)/W2∼= (W1, x_α⁰_|W

1),

∃W3,(x_α, W₃)∼= (K^β

0, x_β⁰)/W₂,(K^α, x_α)/W₃∼= (K^α

0, x_α⁰)/W₁}

where λ = (α1, α2, β1, β2) and W1, W2 and W3 are graded vector spaces of di- mensions β2, β1 and α1, respectively. The set can be illustrated by the following diagram:

W₂ //

(K^β

0, x_β⁰) //W₃

(K^β, xβ)

(K^α, xα)

W1 //(K^α

0, x_α⁰) //(K^α, x_α)/W₃∼= (K^α

0, x_α⁰)/W₁. Set

S_λ^00F ={(xα, xβ, xα⁰, xβ⁰, W1, W2)| ∃ρ11, ρ12, ρ21, ρ22, (x_α, x_β, x_α⁰, x_β⁰, W₁, W₂, ρ₁₁, ρ₁₂, ρ₂₁, ρ₂₂)∈S_λ^0F}.

Then there is a projectionS_λ^0F →S_λ^00F which is a principalGα₁×Gα₂×Gβ₁×Gβ₂- bundle. We also have the following diagram

E^Fα×E^Fβ

`

λ∈NF_λ^F

i⁰

oo `

λ∈NS_λ^0F

p⁰

oo ^q0 //`

λ∈NS_λ^{00F r}

0 //E^Fα⁰×E^Fβ⁰ . Then we have

mδ^tw =|G^F_α1×G^F_α2×G^F_β1×G^F_β2|⁻¹q^{−hα2,β1i−Pi∈Q0[(β1)i(β2)i+(α1)i(α2)i]}(r⁰)_!(q⁰)_!(p⁰)^∗(i⁰)^∗.

ForM ∈E^Fα, N ∈E^Fβ, M⁰ ∈E^Fα⁰, N⁰∈E^Fβ⁰, one can check mδ^tw(1_OF

M,1_OF

N)(M⁰, N⁰) = X

[X],[Y₁],[Y₂],[Z]

F_XY^M0

2F_Y^N0

1Zh^XY_M ¹h^Y_N^2Z.

where [X]∈E^Fα₂/G^F_α

2,[Y₁]∈E^Fα₁/G^F_α

1,[Y₂]∈E^Fβ2/G^F_β

2,[Z]∈E^Fβ1/G^F_β

1. We obtain the following diagram

E^Fα×E^Fβ C^0F oo p

κrq

`

λ∈NS^0F_λ

p⁰i⁰

OO

r⁰q⁰ //E^Fα⁰×E^Fβ⁰

and the proof of Theorem 3.6 is deduced to prove the identity X

[L]∈E^F_d/G^F_d

F_{M N}^L h^M_L^0N0 = X

[X],[Y₁],[Y₂],[Z]

q^−hX,ZiF_XY^M0₂F_Y^N₁⁰_Zh^XY_M ¹h^Y_N^2Z.

Applying the Riedtmann-Peng formula, it is equivalent to the Green formula in Section 2. The following we refer the reformulation of the proof of Green’s theorem in [29].

The left side of the identity is the number counting the set of crossings with the group action. More precisely, fixM, N, M⁰, N⁰ and consider the diagram

0

N⁰

a⁰

0 //N ^a //L ^b //

b⁰

M //0

M⁰

0

Set

Q={(a, b, a⁰, b⁰)|a, b, a⁰, b⁰ as in the above crossing}.

By calculation,|Q| =P

[L]∈E^F_d/G^F_d F_{M N}^L h^M_L^0N0|G^F_d||G^F_α||G^F_β|.Consider the natural action of G^F_d on Q, and the orbit space is denoted by Q.e The fibre of the map Q→Qe has cardinality ^|G

F d|

|Hom(Cokerb⁰a,Kerb⁰a)|.

The right side of the identity is the number counting the squares with the group action. Consider the diagram

0

0

0 //Z ^e1 //

u⁰

N^{0 e2} //Y1 //

x

0

N

v⁰

M

y

0 //Y2

e₃ //M^{0 e4} //X //

0

0 0

Set

O={(e₁, e₂, e₃, e₄, u⁰, v⁰, x, y)| all morphisms occur in the above diagram}.

The groupG^F_α

1×G^F_α

2×G^F_β

1×G^F_β2 freely acts onOwith orbit spaceO. Note thate

|O|e = X

[X],[Y1],[Y2],[Z]

F_XY^M0₂F_Y^N₁⁰_Zh^XY_M ¹h^Y_N^2Z|G^F_α||G^F_β|.

There is a canonical mapfe:Qe→O, which the cardinality of fibre ise |Ext¹(X, Z)|.

As Ringel-Hall algebras, we can define the analogue σ:CF^F(Q)→ CF^F(Q) of the antipode by setting

σ(f) = X

r≥1∈Z

(−1)^r X

α1,···,αr6=0

m^r_α

1,···,α_r◦δ^tw,r_α

1,···,α_r(f) forf 6= 0∈ CF^F(Q).

In order to compare Lusztig’s Hall algebras with twisted Ringel-Hall algebras.

We twistCF^F(Q) by settingm^t_α,β =v^hα,βim_α,β,δ_α,β^t =v^hα,βiδ_α,β^tw and σ^t(f) = X

r≥1∈Z

(−1)^r X

α₁,···,α_r6=0

m^t,r_α

1,···,α_r◦δ_α^t,r_1,···,αr(f).

We denote the twist version by CF^F,tw(Q).By applying Lemma 3.1, 3.2 and The- orem 3.6, we obtain the following result.

Theorem 3.7. The algebra CF^F,tw(Q) is a Hopf algebra with the multiplication m^t, comultiplication δ^t and antipode σ^t. Fix an isomorphism τ :Ql→C, there is an isomorphism of Hopf algebra Φ : CF^F,tw(Q) → H^tw(A)where A = Rep_kQ by settingΦ(1_OF

M) = [M]forM ∈ A.

4. The categorification of Ringel-Hall algebras

LetFqbe a finite field withqelements. In the following,Kis an algebraic closure ofFq.

Let X be a scheme of finite type over K. We say thatX has an Fq-structure if there exists a variety X₀ over Fq such that X = X₀ ×spec(Fq) Spec(k). Let F_X0 :X₀→X₀ be the Frobenius morphism. It can be extended to the morphism F_X :X →X.LetX^F be the set of closed points of X fixed byF, i.e., the set of F^q-rational points. For anyn∈N, let X^Fn be the set of closed points of X fixed byFⁿ. Note thatX^F1=X^F.

Denote byD^b(X) =D^b(X,Ql) the bounded derived category ofQl-constructible complexes onX.

The morphismF_X :X →X naturally induces a functorF_X^∗ :D^b(X)→ D^b(X).

A Weil complex is a pair (F, j) such that F ∈ D^b(X) and j : F_X^∗(F) → F is an isomorphism. Let D_w^b(X) be the triangulated subcategory of D^b(X) of Weil complexes andKw(X) be the Grothendieck group ofD^b_w(X).

Given a Weil complexF= (F, j)¹in D_w^b(X), let x∈X^F be a closed point. We get automorphisms

Fi,x:Hⁱ(F)_|x→ Hⁱ(F)_|x.

One can define aF-invariant function χ^F_F:X^F →Ql via defining χ^F_F(x) =X

i

(−1)ⁱtr(F_i,x,Hⁱ(F)_|x) =X

i

(−1)ⁱtr(F_i,x).

Similarly, one can defineχ^F_Fⁿ:X^Fn→Ql via defining χ^F_Fⁿ(x) =X

i

(−1)ⁱtr(F_i,xⁿ ,Hⁱ(F)_|x) =X

i

(−1)ⁱtr(F_i,xⁿ ).

In particular,χ^F1 =χ^F.

Theorem 4.1. [14, Theorem 12.1] LetX be as above. Then we have

(1) Let K //L //M //K[1] be a distinguished triangle inD^b_w(X).

Thenχ^F_K+χ^F_M=χ^F_L.

(2) Let g:X →Y be a morphism. Then forK ∈ D^b_w(X)andL ∈ D_w^b(Y), we haveχ^F_Rg

!K=g!(χ^F_K)andχ^F_g∗L =g^∗(χ^F_L).

(3) ForK ∈ D_w^b(X), we have χ^F_K[d]= (−1)^dχ^F_K andχ^F_K(n)=q⁻ⁿχ^F_K.

By Theorem 4.1(1), the functionχ^F_Fⁿ only depends on the isomorphism class of F inD_w^b(X). LetCF(X^Fn) be the vector space of all functionsX^Fn→Ql.Hence, we obtain a mapχ^Fn:Kw(X)→ CF(X^Fn).

Let G be an algebraic group over K and X be a scheme of finite type over K together with an G-action. Assume that X and G have Fq-structures and X = X0×spec(Fq)Spec(k), G = G0×spec(Fq)Spec(k). Let FG0 : G0 → G0 be the Frobenius morphism. It can be extended to the morphism F_G : G → G.

Denote byD_G^b(X) =D^b_G(X,Ql) theG-equivariant bounded derived category ofQl- constructible complexes onX andD^b_G,w(X) the subcategory ofD_G^b(X) consisting of Weil complexes. LetKG,w(X) be the Grothendieck group ofD_G,w^b (X).

1All Weil complexes considered here are induced by the complexes inD^b(X0,Ql).

Assume that we have the following commutative diagram G×X

F_G×FX

//X

F_X

G×X //X

Then, the morphismFX :X →X naturally induces a functor F_X^∗ :D_G,w^b (X)→ D^b_G,w(X).

Letx∈X^F be a closed point. For any (F, j) in D_G,w^b (X), we get automorphisms Fi,x:Hⁱ_G(F)_|x→ Hⁱ_G(F)_|x.

In the same way, one can define the G-equivariant version of χ^Fn for n ∈ N as follows:

χ^F_F(x) =X

i

(−1)ⁱtr(F_i,x,H_Gⁱ(F)_|x) and

χ^F_Fⁿ(x) =X

i

(−1)ⁱtr(F_i,xⁿ ,Hⁱ_G(F)_|x).

In particular,χ^F1 =χ^F.

Lemma 4.2. For any (F, j)∈ D^b_G,w(X),χ^F_Fⁿ is aG-equivariant function.

Proof. For anyx andy in the same G-orbit of X, there exists an element g ∈G such thatg.x=y. Consider the following commutative diagram

Hⁱ_G(F)_|x

g^∗

Fx //Hⁱ_G(F)_|x

g^∗

H_Gⁱ(g^∗F)_|y ^Fy //Hⁱ_G(g^∗F)_|y Sinceg^∗F ' F, we have

Hⁱ_G(F)_|x

g^∗

Fx //Hⁱ_G(F)_|x

g^∗

Hⁱ_G(F)_|y ^Fy //Hⁱ_G(F)_|y

By the definition ofχ^Fn,χ^F_Fⁿ(x) =χ^F_Fⁿ(y). That isχ^F_Fⁿis aG-equivariant function.

In the G-equivariant case, we also have Theorem 4.1. Hence the functionχ^F_Fⁿ also only depends on the isomorphism class of F in D^b_G,w(X). Hence, we obtain a mapχ^Fn:K_G,w(X)→ CFG(X^Fn).

LetQbe a finite quiver without loops. Given a dimension vector d= (d_i)_i∈Q0, the variety Ed and the algebraic group G_d are defined in Section 3. Both of them have naturalFq-structures. Consider the following diagram

Eα×Eβ E⁰

p₂ //

p₁

oo E

00 p₃ //Eα+β .