Introduction Let U+ be the positive part of the quantized enveloping algebra U associated with a Cartan datum

AFFINE MODIFIED QUANTIZED ENVELOPING ALGEBRAS

JIE XIAO AND MINGHUI ZHAO

Abstract. For a symmetrizable Kac-Moody Lie algebrag, Lusztig introduced the corresponding modified quantized enveloping algebra ˙Uand its canonical basis ˙Bin [13]. In this paper, in casegis a symmetric Kac-Moody Lie algebra of finite or affine type, we define a set M˜ which depends only on the root categoryRand prove that there is a bijection betweenM˜ and ˙B, whereR is the T²-orbit category of the bounded derived category of corresponding Dynkin or tame quiver. Our method is based on a result of Lin, Xiao and Zhang in [10], which gives a PBW-type basis ofU⁺.

1. Introduction

Let U⁺ be the positive part of the quantized enveloping algebra U associated with a Cartan datum. In the case of finite type, Lusztig gave two approachs to con- struct the canonical basisBofU⁺ ([11]). The first one is an elementary algebraic construction. By using Ringel-Hall algebra realization of U⁺, the isomorphism classes of representations of the corresponding Dynkin quiver form a PBW-type basis ofU⁺ and there is an order on this basis. Under this order, the transition matrix between this basis and a monomial basis is a unipotent lower triangular ma- trix. By a standard linear algebra method one can get a bar-invariant basis, which is the canonical basis B. The second one is a geometric construction. Lusztig constructed the canonical basisBby using perverse sheaves and intersection coho- mology. The geometric construction ofB was generalized to the cases of all types in [12]. In the case of affine type, Lin, Xiao and Zhang in [10] provided a process to construct a PBW-type basis ofU⁺ and the canonical basisBby using Ringel-Hall algebra approach ([10]).

Let ˙U be the modified quantized enveloping algebra obtained fromUby mod- ifying the Cartan partU⁰ to L

λ∈PQ(v)1_λ, whereP is the weight lattice. ˙U can be considered as the limit of tensor products of highest weight modules and lowest weight modules. Lusztig introduced the canonical bases of the tensor products and then the canonical basis ˙Bof ˙U([13, 14]). Kashiwara also studied the algebra ˙U and its canonical basis ˙B([9]).

Happel studied the bounded derived category D^b(Λ) of a finite dimensional al- gebra Λ in [6, 7]. In case Λ is hereditary and representation-finite, he proved that there is a bijection between the isomorphism classes of indecomposable objects in

Date: February 19, 2017.

2000Mathematics Subject Classification. 16G20, 17B37.

Key words and phrases. Ringel-Hall algebras, Root categories, Modified quantized enveloping algebras, Canonical bases .

Project supported by the Fundamental Research Funds for the Central Universities (No.

BLX2013014) and the National Natural Science Foundation of China (No. 11131001).

1

R=D^b(Λ)/T² and all the roots of the corresponding Lie algebra, where T is the translation functor in the triangulated category D^b(Λ). HenceR is called a root category. It was proved in [15] thatRis still a triangulated category. In [15, 16], Peng and Xiao gave a realization of all symmetrizable Kac-Moody Lie algebras via the root categories of finite dimensional hereditary algebras.

Note that the construction of canonical basis ˙B is abstract and depends on the construction of canonical basisBofU⁺. Inspired by the method of Peng and Xiao, we want to study the relations between the canonical basis ˙Band the corresponding root categoryR. In this paper, first we associate a set ˜MtoR. In [10], Lin, Xiao, and Zhang associated a setMto a hereditary category and the definition of ˜Mis based on that ofM. However, ˜Mis independent of the embedding of the hereditary category toR. Fixing an embedding of the hereditary category toR, we can get a bijection between ˜M and the canonical basis ˙B_λ of ˙U1_λ for everyλ∈P. Hence we say that the set ˜Mprovides a parameterization of the canonical basis ˙B.

Since [21], it has been an open problem: how to realize the whole quantized en- veloping algebras by using Hall algebras from derived categories or root categories.

A lot of efforts have been paid on the progress ([3, 8, 20, 22]) and the most recent progress is given by Bridgeland in [1]. We hope that the main result in the present paper can provide a strong evidence for the connection between canonical bases and root categories.

In Section 2, we first give some notations of quantized enveloping algebras and modified quantized enveloping algebras. Then we review the definitions of Ringel- Hall algebras and root categories. In Section 3, we study the case of finite type, which is simpler and can reflect the idea clearly. In Section 4, we study the case of affine type. We first review the construction of the PBW-type basis ofU⁺ in [10].

Then we define a set M˜ depending on the corresponding root category Rand a PBW-type basis of ˙U1λwith ˜Mas an index. By a standard linear algebra method, we get a bar-invariant basis and prove that each element in it is the leading term of an element in ˙Bλ. At last, we prove that there is a bijection between ˜Mand ˙Bλ.

2. Preliminaries

2.1. Quantized enveloping algebras. LetQbe the field of rational numbers and Zbe the ring of integers. LetIbe a finite index set with|I|=nandA= (aij)_i,j∈I be a generalized Cartan matrix. Denote by r(A) the rank of A. LetP^∨ be a free abelian group of rank 2n−r(A) with aZ-basis{hi |i∈I} ∪ {ds |s= 1, . . . , n− r(A)}andh=Q⊗_ZP^∨be theQ-linear space spanned byP^∨. We callP^∨the dual weight lattice andhthe Cartan subalgebra. We also define the weight lattice to be P ={λ∈h^∗ |λ(P^∨)⊂Z}.

Set Π^∨ ={hi |i ∈I} and choose a linearly independent subset Π = {αi |i ∈ I} ⊂h^∗ satisfyingα_j(h_i) =a_ij andα_j(d_s) = 0 or 1 for all i, j∈I,s= 1, . . . , n− rankA. The elements of Π are called simple roots, and the elements of Π^∨ are called simple coroots. The quintuple (A,Π,Π^∨, P, P^∨) is called a Cartan datum associated with the generalized Cartan matrixA.

We shall review the definition of quantized enveloping algebras ([14]). From now on, assume that the generalized Cartan matrixA= (aij)_i,j∈I is symmetric.

Fix an indeterminatev. For anyn∈Z, set [n]v=vⁿ−v⁻ⁿ

v−v⁻¹ .

Set [0]v! = 1 and [n]v! = [n]v[n−1]v· · ·[1]v for anyn∈Z>0. For any two nonneg- ative integersm≥n, the analogue of binomial coefficient is given by

hm n i

v

= [m]v! [n]_v![m−n]_v!. Note that [n]_v andm

n

v are elements of the fieldQ(v).

The quantized enveloping algebraUassociated with a Cartan datum (A,Π,Π^∨, P, P^∨) is an associative algebra overQ(v) with1generated by the elementsEi,Fi(i∈I) andKµ(µ∈P^∨) subject to the following relations:

K₀=1, K_µK_µ0 =K_µ+µ0 for allµ, µ⁰∈P^∨; KµEiK−µ=v^αi(µ)Ei for alli∈I,µ∈P^∨; KµFiK−µ=v^−αi(µ)Fi for alli∈I,µ∈P^∨;

EiFj−FjEi=δij

Ki−K_−i

v−v⁻¹ for alli, j∈I;

for alli6=j, settingb= 1−aij,

b

X

k=0

(−1)^kE^(k)_i E_jE^(b−k)_i = 0;

b

X

k=0

(−1)^kF_i^(k)F_jF_i^(b−k)= 0.

Here,Ki=Khi andE_i⁽ⁿ⁾=Eⁿ_i/[n]v!, F_i⁽ⁿ⁾=F_iⁿ/[n]v!.

LetU⁺ (resp. U⁻) be the subalgebra ofUgenerated by the elementsE_i(resp.

F_i) for alli∈I, andU⁰ be the subalgebra ofU generated byK_µ for all µ∈P^∨. The quantized enveloping algebraUhas the following triangular decomposition

U∼=U⁻⊗U⁰⊗U⁺.

Denote by ¯() the uniqueQ-algebra automorphism of Ugiven by E¯i=Ei,F¯i=Fi,K¯µ=K−µ for alli∈I, µ∈P^∨,

f x= ¯fx,¯ for allf ∈Q(v), x∈U, where ¯f(v) =f(v⁻¹).

Let f be the associative algebra defined by Lusztig in [14]. f is generated by θi(i∈I) subject to the following relations:

b

X

k=0

(−1)^kθ_i^(k)θ_jθ^(b−k)_i = 0 for alli6=j, whereb= 1−aij andθ⁽ⁿ⁾_i =θⁿ_i/[n]v!.

There are two well-definedQ(v)-algebra homomorphisms⁺:f →Uand⁻ :f → U satisfyingEi =θ⁺_i andFi =θ_i⁻ for all i∈I. The images of ⁺ and⁻ are U⁺ andU⁻ respectively.

Denote by ¯() the uniqueQ-algebra automorphism of f given by θ¯_i=θ_i for alli∈I,

f x= ¯f¯x for allf ∈Q(v),x∈f. Note that, for allx∈f,x^±= ¯x^±.

LetA=Q[v, v⁻¹] andZ=Z[v, v⁻¹]. Denote by U^±_Z theZ-subalgebras of U^± generated byE_i^(s) and F_i^(s) for all i ∈I and s∈N respectively. Also, denote by U_Z theZ-subalgebra of U generated by E_i^(s), F_i^(s) and K_µ for all i ∈ I, s ∈ N

and µ ∈ P^∨. Let U^±_A = U^±_Z ⊗Z A and UA = UZ ⊗Z A. Similarly, let fZ be the Z-subalgebra of f generated by θ^(s)_i for all i ∈ I and s ∈ N. At last, let f_A=f_Z⊗_ZA.

In [11, 12, 14], Lusztig defined the canonical basisBoff.

2.2. Modified quantized enveloping algebras. Let us review the definition of the modified form ˙UofU ([13, 14]).

For anyλ⁰, λ⁰⁰∈P, set

λ⁰Uλ⁰⁰=U/



 X

µ∈P^∨

(Kµ−q^λ0(µ))U+ X

µ∈P^∨

U(Kµ−q^λ00(µ))



. Letπ_λ⁰_,λ⁰⁰:U→_λ⁰U_λ⁰⁰ be the canonical projection and

U˙ = M

λ⁰,λ⁰⁰∈P λ⁰Uλ⁰⁰.

Consider the weight space decompositionU=L

β∈ZIU(β), where U(β) ={x∈U| KµxK_µ⁻¹=v^β(µ)x for allµ∈P^∨}.

Here, the setI is viewed as a subset ofP andiis identified withαifor each i∈I.

The images of summandsU(β) underπλ⁰,λ⁰⁰ form the weight space decomposition

λ⁰U_λ⁰⁰= M

β∈ZI

λ⁰U_λ⁰⁰(β).

Note thatλ⁰Uλ⁰⁰(β) = 0 unlessλ⁰−λ⁰⁰=β.

There is a natural associative Q(v)-algebra structure on ˙U inherited from that of U. It is defined as follow: for any λ⁰₁, λ⁰⁰₁, λ⁰₂, λ⁰⁰₂ ∈ P, β1, β2 ∈ ZI such that λ⁰₁−λ⁰⁰₁ =β1, λ⁰₂−λ⁰⁰₂ =β2 and anyx∈U(β1), y∈U(β2),

π_λ⁰

1,λ⁰⁰₁(x)π_λ⁰

2,λ⁰⁰₂(y) =

πλ⁰₁,λ⁰⁰₂(xy) if λ⁰⁰₁ =λ⁰₂ 0 otherwise .

For anyλ∈P, let1_λ=π_λ,λ(1), where1is the unit element ofU. They satisfy the following relations:

1λ1λ⁰ =δλ,λ⁰1λ.

In general, there is no unit element in the algebra ˙U. However the family (1λ)λ∈P

can be regarded locally as the unit element in ˙U.

Note that λ⁰Uλ⁰⁰ = 1λ⁰U1˙ λ⁰⁰. Define ˙U1λ = L

λ⁰∈P1λ⁰U1˙ λ. Then ˙U = L

λ∈PU1˙ _λ.

TheQ-algebra automorphism ¯() :U→Uinduces a linear isomorphism_λ⁰U_λ⁰⁰→

λ⁰U_λ⁰⁰ for anyλ⁰, λ⁰⁰∈P. Taking direct sums, we obtain an algebra automorphism () : ˙¯ U→U. Note that˙ 1λ=1λ for anyλ∈P.

The elementsb⁺b⁰⁻1λfor allb, b⁰∈Bform a basis of theQ(v)-vector space ˙U1λ

([14], 23.2.1). Denote by ˙U_Zthe subalgebra of ˙Ugenerated by the elementsE_i⁽ⁿ⁾1_λ andF_i⁽ⁿ⁾1λoverZ for alli∈I,n∈Nandλ∈P. The set{b⁺b⁰⁻1λ |b, b⁰∈B, λ∈ P}is a Z-basis of ˙U_Z.

As the notations in [14], the canonical basis of ˙Uis denoted by B˙ ={b♦λb⁰ | b, b⁰∈B, λ∈P}.

Note that{b♦λb⁰ | b, b⁰∈B}is also a Z-basis of ˙UZ1λ. According to the proof of Theorem 25.2.1 in [14]

(1) b♦_λb⁰ ≡b⁺b⁰⁻1_λ modP(tr|b| −1,tr|b⁰| −1).

HereP(tr|b| −1,tr|b⁰| −1) is theQ(v)-submodule of ˙Uspanned by the set

{b⁺₁b⁻₂1λ |b1, b2∈Bsuch that tr|b1| ≤tr|b| −1, tr|b2| ≤tr|b⁰| −1 and|b1| − |b2|=|b| − |b⁰|}, where|b|is the weight ofband trµ=Pa_i ifµ=Pa_iα_i.

2.3. Ringel-Hall algebras. In this subsection, we shall review the definition of Ringel-Hall algebras ([5, 10, 18]).

A quiverQ= (I, H, s, t) consists of a vertex setI, an arrow setH, and two maps s, t:H→I such that an arrowρ∈H starts ats(ρ) and terminates att(ρ).

Let k be a field and Λ = kQ be the path algebra of Q over k. Denote by mod-Λ the category of finite dimensional left Λ-modules and rep-Q the category of finite dimensional representations of Q overk. It is well-known that mod-Λ is equivalent to rep-Q. We shall identify Λ-modules with representations of Qunder this equivalence.

LetP be the set of isomorphism classes of finite dimensional nilpotent Λ-modules and ind(P) be the set of isomorphism classes of indecomposable finite dimensional nilpotent Λ-modules. For any α ∈ P, fix a Λ-module M(α) in the isomorphism classα.

The set of isomorphism classes of nilpotent simple Λ-modules is indexed by the setIand the Grothendieck groupG(Λ) of mod-Λ is the free abelian groupZI. For any Λ-moduleM, the dimension vector dimM ofM is an element inG(Λ) =ZI.

The Euler formh−,−i onG(Λ) =ZI is defined by hα, βi=X

i∈I

a_ib_i−X

ρ∈H

a_s(ρ)b_t(ρ)

whereα=P

i∈Iaii, β=P

i∈Ibii∈ZI. For any Λ-modulesM andN, one has hdimM,dimNi= dimkHomΛ(M, N)−dimkExtΛ(M, N).

The symmetric Euler form is defined by (α, β) =hα, βi+hβ, αifor all α, β∈ZI.

This gives rise to a symmetric generalized Cartan matrix A = (a_ij)_i,j∈I where aij = (i, j). The generalized Cartan matrix A depends only on the underlying graph of quiverQ.

From now on, letkbe a finite field withqelements. Given three modulesL, M and N in mod-Λ, let g^L_{M N} be the number of Λ-submodules W of L such that W 'N andL/W 'M in mod-Λ. Letv=√

q∈C. By definition, the Ringel-Hall algebra Hq(Λ) of Λ is the Q(v)-vector space with basis {u_[M] |[M] ∈ P} whose multiplication is given by

u_[M]u_[N] = X

[L]∈P

g_{M N}^L u_[L].

It is easily seen thatHq(Λ) is an associative Q(v)-algebra with unit u_[0], where 0 denotes the zero module. Note that, the Ringel-Hall algebraHq(Λ) is aNI-graded algebra by dimension vectors of modules.

The twisted Ringel-Hall algebraH^∗_q(Λ) is defined as follow. SetH^∗_q(Λ) =Hq(Λ) asQ(v)-vector space and define the multiplication by

u[M]∗u[N] =v^hdimM,dimNi X

[L]∈P

g^L_{M N}u[L].

LetSi be the nilpotent simple module corresponding to i ∈I and defineui = u[S_i]. The composition algebraC_q^∗(Λ) is a subalgebra ofH^∗_q(Λ) generated byui for alli∈I. For any Λ-moduleM, denote

hMi=v^{−dimkM+dimkEndΛ(M)}u[M]. Note that{hM(α)i |α∈ P}is a Q(v)-basis ofH^∗_q(Λ).

LetQbe a finite quiver. Then consider the generic Ringel-Hall algebra associated withQ. Letkbe a finite field and Λk=kQ. Denote byH^∗_q(Λk) the corresponding twisted Ringel-Hall algebra. Let K be a set of some finite fields k such that the set{qk =|k| |k∈ K}is an infinite set. Let Rbe an integral domain containingQ andvq_k, where vq_k =√

qk for anyk∈ K. For each k∈ K, the composition algebra C_q^∗(Λk) is theR-subalgebra ofH^∗_q(Λk) generated by the elementsui(k) for alli∈I.

Consider the direct product

H^∗(Q) = Y

k∈K

H^∗_q(Λ_k)

and the elements v= (vq_k)_k∈K, v⁻¹= (v_q⁻¹_k)_k∈K and ui = (ui(k))_k∈K. By C^∗(Q)_A we denote the subalgebra ofH^∗(Q) generated byv, v⁻¹ and u_i over Q. We may regard it as anA-algebra generated byu_i, wherev is viewed as an indeterminate.

Finally, define C^∗(Q) = Q(v)⊗ C^∗(Q)_A, which is called the generic composition algebra ofQ.

Then we have the following well-known result of Green and Ringel ([5, 18]).

Theorem 2.1. Let Q be a connected quiver, A be the corresponding generalized Cartan matrix, andf be the Lusztig’s algebra of type A. Then there is an isomor- phism of algebras:

C^∗(Q) ∼= f u_i 7→ θ_i. Hence, we always identifyC^∗(Q) withf.

2.4. Root categories. A triangulated category (C, T) is called 2-periodic if the translation functorT satisfiesT²'id.

Let k be a field. Given a finite dimensional hereditary k-algebra Λ, denote by D^b(Λ) the bounded derived category of the abelian category mod-Λ and T the translation functor in this triangulated category. Consider the orbit category R(Λ) = D^b(Λ)/T² of D^b(Λ) under the equivalent functor T². Let F : D^b(Λ) → R(Λ) be the canonical functor. The translation functor T of D^b(Λ) induces an equivalent functor inR(Λ) of order 2, which is still denoted byT. By [15], (R(Λ), T) is also a triangulated category and the functor F : D^b(Λ) → R(Λ) sends each triangle inD^b(Λ) to a triangle inR(Λ). It is clear that the root categoryR=R(Λ) is a 2-periodic triangulated category.

LetQbe a connected quiver andR(Q) =D^b(kQ)/T². Denote by ˜P the set of isomorphism classes of objects inR(Q) and ind( ˜P) the set of isomorphism classes of

indecomposable objects inR(Q). Note that mod-kQcan be embeded intoR(Q) as a full subcategory and ind( ˜P) = ind(P) ˙∪ind(T(P)), where ˙∪means disjoint union.

3. Finite type

3.1. PBW-type basis of U⁺. In this section, letQbe a connected Dynkin quiver, kbe a finite field and Λ =kQ. Denote by Φ⁺ (resp. Φ⁻) the set of positive (resp.

negative) roots of the Dynkin quiver Q. Note that Φ⁺ and Φ⁻ can be viewed as subsets of ZI. By Gabriel’s Theorem, the map dim induces a bijection between ind(P) and Φ⁺. Given a positive rootα, the corresponding isomorphism class is also denoted byα.

SinceQis representation-directed, we can define a total order on the set Φ⁺={α1, α2,· · ·, αn}

such that the corresponding indecomposable Λ-modules {M(α1), M(α2),· · · , M(αn)}

satisfy the following conditions

Hom(M(αi), M(αj))6= 0⇒i≤j.

Define

N^ind(P)={a: Φ⁺→N}.

For anya∈N^ind(P), we can define a representation M(a) = M

α∈Φ⁺

a(α)M(α) and any representation can be written in this form.

Since the Hall polynomials exist in this case, we can consider the generic com- position algebra C^∗(Q) directly. Note that the set N^ind(P) is independent of the choice of the finite field andhM(a)ican be viewed as an element in C^∗(Q) for any a∈N^ind(P).

By [19], we have

Proposition 3.1. The set{hM(a)i |a∈N^ind(P)} is anA-basis of C_A^∗(Q).

3.2. PBW-type basis of U1˙ λ. Let R(Q) be the root category corresponding to a connected Dynkin quiver Qover some finite field k. Remember that ind( ˜P) is the set of isomorphism classes of indecomposable objects in R(Q). Let Φ = {dim(M)|M ∈ind( ˜P)}. Then Φ is the root system of the corresponding Lie alge- bra and there is a bijection between ind( ˜P) and Φ by Gabriel’s Theorem. Note that Φ = Φ⁺∪Φ˙ ⁻. For any element α∈Φ, we also denote byM(α) the corresponding object inR(Q).

Define

N^{ind( ˜P)}={˜a: Φ→N}.

For any ˜a∈N^{ind( ˜P)}, we can define an object M(˜a) =M

α∈Φ

˜

a(α)M(α) and any object inR(Q) can be written in this form.

Note that the categoryR(Q), so the setN^{ind( ˜P)}, depends only on the underlying graph of Q. If Q⁰ is another quiver such that D^b(kQ) ' D^b(kQ⁰), they give the same setN^{ind( ˜P)}.

Given any symmetric generalized Cartan matrix A = (aij)_n×n of finite type, consider a quiverQ, the quantum enveloping algebraUand the modified quantized enveloping algebra ˙Ucorresponding toA.

Remember that mod-kQcan be embedded intoR(Q) as a full subcategory and ind( ˜P) = ind(P) ˙∪ind(T(P)).

For any ˜a ∈ N^{ind( ˜P)}, let a1 = ˜a|_ind(P) and a2 = ˜a|_ind(T(P)), which is denoted by

˜

a= (a1,a2). Since we always identifyC^∗(Q) withf, the elements in the following set

{hM(˜a)iλ=hM(a1)i⁺· hM(a2)i⁻1λ |a˜∈N^{ind( ˜P)}} can be regarded as elements in ˙U1λ.

We have the following proposition.

Proposition 3.2. The set{hM(˜a)i_λ |a˜∈N^{ind( ˜P)}} is aQ(v)-basis of U1˙ _λ. Proof The modified quantized enveloping algebra ˙U is a freef⊗f^opp-module with basis (1λ)_λ∈P ([14], 23.2.1). So the set

{hM(a₁)i⁺· hM(a₂)i⁻1_λ |a₁∈ind(P),a₂∈ind(T(P))}

is a basis of ˙U1λ. By ˜a= (a1,a2), the set{hM(˜a)iλ |˜a∈N^{ind( ˜P)}} is aQ(v)-basis of ˙U1λ.

Denote by BQ( ˙U1λ) the PBW-type basis {hM(˜a)iλ |˜a ∈N^{ind( ˜P)}}. Note that this PBW-type basis depends on the embedding of mod-kQintoR(Q).

3.3. A bar-invariant basis of U1˙ λ. As before, let Q be a connected Dynkin quiver and R(Q) be the corresponding root category. Remember that the set of positive roots Φ⁺ = {α1, . . . , αn}. For any a,b : Φ⁺ → N, define b≺ a if and only if there exists some 1 ≤ j ≤ n such that b(α_i) = a(α_i) for all i < j and b(αj) > a(αj). For any ˜a,b˜ : Φ → N, define ˜a ≺ b˜ if and only if a1 b1 and a₂b₂but ˜a6= ˜b, where ˜a= (a₁,a₂) and ˜b= (b₁,b₂).

For anyc: Φ⁺→N, there exists a monomialw_∗(c) on Chevalley generatorsu_i satisfying

w_∗(c) =hM(c)i+X

c⁰≺c

a_cc⁰hM(c⁰)i,

wherea_cc⁰ ∈ A([19]). Note that the transition matrixa= (a_cc⁰) from{hM(c)i |c: Φ⁺→N}to{w_∗(c)|c: Φ⁺→N}satisfies thata_cc= 1 anda_cc⁰ = 0 unlessc⁰≺c.

That is,ais a unipotent lower triangular matrix.

Let ¯a= (acc⁰). Sincew_∗(c) =w_∗(c), we have w_∗(c) =w_∗(c) =X

c⁰

¯

acc⁰hM(c⁰)i, thus

hM(c)i=X

c⁰

¯

a⁻¹_cc0w_∗(c⁰) =X

c⁰

X

c⁰⁰

¯

a⁻¹_cc0a_c⁰_c⁰⁰hM(c⁰⁰)i.

Let h = ¯a⁻¹a. The matrix h is again a unipotent lower triangular matrix and

¯h=h⁻¹. There exists a unique unipotent lower triangular matrixd= (d_cc⁰) with

off-diagonal entries inv⁻¹Q[v⁻¹], such thatd= ¯dh. Then the canonical basis off is

E^c=hM(c)i+X

c⁰≺c

dcc⁰hM(c⁰)i, withdcc⁰ ∈v⁻¹Q[v⁻¹] ([19]).

Similarly, we can get a bar-invariant basis of ˙U1λ from

BQ( ˙U1λ) ={hM(c1)i⁺· hM(c2)i⁻1λ |˜c: Φ→N,˜c= (c1,c2)}

and

{w∗(c1)⁺·w∗(c2)⁻1λ |˜c: Φ→N,˜c= (c1,c2)}

under the order≺onN^{ind( ˜P)}defined above.

First definew_∗(˜c)_λ=w_∗(c₁)⁺·w_∗(c₂)⁻1_λ, where ˜c= (c₁,c₂). Since w∗(c) =hM(c)i+X

c⁰≺c

acc⁰hM(c⁰)i, we have

w∗(c1)⁺=hM(c1)i⁺+ X

c⁰₁≺c1

ac₁c⁰₁hM(c⁰₁)i⁺ and

w_∗(c₂)⁻=hM(c₂)i⁻+ X

c⁰₂≺c2

a_c2c⁰

2hM(c⁰₂)i⁻ inU^± respectively. Hence, we have

w_∗(˜c)λ = w_∗(c1)⁺·w_∗(c2)⁻1λ

= (hM(c₁)i+ X

c⁰₁≺c1

a_c1c⁰

1hM(c⁰₁)i)⁺·(hM(c₂)i+ X

c⁰₂≺c2

a_c2c⁰

2hM(c⁰₂)i)⁻1_λ

= hM(c1)i⁺· hM(c2)i⁻1λ+hM(c1)i⁺· X

c⁰₂≺c2

a_c2c⁰

2hM(c⁰₂)i⁻1λ+ X

c⁰₁≺c1

a_c1c⁰

1hM(c⁰₁)i⁺· hM(c2)i⁻1λ+ X

c⁰₁≺c1

a_c1c⁰

1hM(c⁰₁)i⁺· X

c⁰₂≺c2

a_c2c⁰

2hM(c⁰₂)i⁻1λ

= hM(˜c)iλ+X

˜ c⁰≺˜c

˜

a˜c˜c⁰hM(˜c⁰)iλ, where ˜c= (c₁,c₂), ˜c⁰= (c⁰₁,c⁰₂) and ˜a˜c˜c⁰ =a_c1c⁰

1a_c2c⁰

2 ∈ A.

As before, the transition matrix ˜a = (˜a˜c˜c⁰) from {hM(˜c)iλ |˜c : Φ → N} to {w_∗(˜c)λ |˜c: Φ→N} satisfies that ˜a˜c˜c= 1 and a

ec˜c⁰ = 0 unless ˜c⁰ ≺˜c. That is, ˜a is also a unipotent lower triangular matrix with off-diagonal entries inA.

Let ¯˜a= (˜a˜c˜c⁰). Sincew_∗(˜c)_λ=w_∗(˜c)_λ, we have w_∗(˜c)λ=w_∗(˜c)λ=X

˜ c⁰

¯

a˜c˜c⁰hM(˜c⁰)iλ, thus

hM(˜c)iλ=X

˜ c⁰

¯˜

a⁻¹_˜c˜c0w_∗(˜c⁰)λ=X

˜ c⁰

X

˜ c⁰⁰

¯˜

a⁻¹_˜c˜c0˜a˜c⁰˜c⁰⁰hM(˜c⁰⁰)iλ.

Let ˜h= ¯a˜⁻¹˜a. The matrix ˜his again a unipotent lower triangular matrix and

¯˜

h= ˜h⁻¹. There exists a unique unipotent lower triangular matrix ˜d= ( ˜d_˜c˜c⁰) with

off-diagonal entries inv⁻¹Q[v⁻¹], such that ˜d=d¯˜˜h. Then we get a bar-invariant basis of ˙U1λ

E_λ^˜c=hM(˜c)iλ+X

˜ c⁰≺˜c

d˜˜c˜c⁰hM(˜c⁰)iλ, with ˜d_˜c⁰_˜c∈v⁻¹Q[v⁻¹]. We denote this basis by BQ( ˙U1_λ).

Theorem 3.3. BQ( ˙U1_λ) ={E_λ^˜c |˜c: Φ→N}={b⁺b⁰⁻1_λ| b, b⁰∈B}.

We omit the proof of this theorem. The proof of Theorem 3.3 is simpler than Theorem 4.9 of affine case, which will be proved in next section.

3.4. A parameterization of the canonical basis ofU1˙ _λ. Let ˙U=L

λ∈PU1˙ _λ be the modified quantized enveloping algebra corresponding to the quiver Q and B˙λ be the canonical basis of ˙U1λ.

Theorem 3.4. We have a bijection

Ψ_Q :N^{ind( ˜P)}→B˙_λ given by

˜

c7→ E^c1♦λE^c2, which is the composition of the following two bijections

N^{ind( ˜P)} → B_Q( ˙U1_λ)

˜

c 7→ E_λ^˜c, and

BQ( ˙U1λ) → B˙λ

b⁺b⁰⁻1λ 7→ b♦λb⁰.

Proof The first bijection fromN^{ind( ˜P)}toBQ( ˙U1λ) comes from our construction of E_λ^˜c and the second bijection fromBQ( ˙U1λ) to ˙Bλ comes from formula (1). By Theorem 3.3,E_λ^˜c=E^c1+E^c2−1_λ. Hence, Ψ_Q:N^{ind( ˜P)}→B˙_λ is a bijection.

4. Affine type

4.1. PBW-type basis of U⁺. We first review the construction of the PBW-type basis in [2, 4, 10, 23].

4.1.1. The integral basis arising from the Kronecker quiver. LetQbe the Kronecker quiver withI={1,2} andH={ρ₁, ρ₂}:

1

ρ1

,,

ρ₂

222

Letkbe a finite field withq elements,v=√

qand Λ =kQbe the path algebra ofQ.

The set of dimension vectors of indecomposable Λ-modules is

Φ⁺={(l+ 1, l),(m, m),(n, n+ 1)|l, m, n∈Z, l≥0, m≥1, n≥0}.

The dimension vectors (l+ 1, l) and (n, n+ 1) correspond to preprojective and preinjective indecomposable Λ-modules respectively.

Remember that P is the set of isomorphism classes of finite dimensional Λ- modules. Denote by Hq (resp. H^∗_q) the Ringel-Hall (resp. twisted Ringel-Hall) algebra of Λ.

Define

E_(n+1,n)=hM(n+ 1, n)iandE_(n,n+1)=hM(n, n+ 1)i,

whereM(n+ 1, n) (resp. M(n, n+ 1)) is the corresponding Λ-module of dimension vector (n+ 1, n) (resp. (n, n+ 1)) for anyn∈N. Let δ= (1,1). For any n≥1, define

E˜nδ=E_(n−1,n)∗E(1,0)−v⁻²E(1,0)∗E_(n−1,n). Then, define inductively

E0δ = 1, Ekδ= 1 [k]

k

X

s=1

v^s−kE˜sδ∗E_(k−s)δ for allk≥1.

Consider the generic composition algebraC^∗. SinceEkδ,E(m+1,m)andE(n,n+1)

are defined in each H^∗_q, they can be regarded as elements in Q

qH_q^∗. Note that, these elements also belong toC_A^∗.

Denote byP(m) the set of all partitions ofm. For any partition w= (w1≥w2≥. . .≥wt)∈P(m),

define

Ewδ =Ew₁δ∗Ew₂δ∗ · · · ∗Ew_tδ. Proposition 4.1. (cf. [2, 10, 23]) The set

{hPi ∗E_wδ∗ hIi}

is anA-basis ofC_A^∗, whereP∈ P is preprojective,w∈P(m),I∈ Pis preinjective andm∈N.

4.1.2. The integral basis arising from a tube. Let ∆ = ∆(n) be the cyclic quiver whose vertex set is ∆₀=Z/nZ={1,2,. . . ,n}and arrow set is ∆₁={i→i+ 1 |i∈

∆₀}:

2 //3 //4

1

??

5

n

__

7oo 6 Letkbe a finite field withqelements,v=√

qandT =T(n) be the category of finite dimensional nilpotent representations of ∆(n) overk. For anyi∈∆0, denote bySi the corresponding simple object in T. For anyi∈∆0 and l∈N, denote by Si[l] the indecomposable object in T with top Si and length l. Note that Si[l] is independent of the choice of finite fields. LetP be the set of isomorphism classes of objects in T. Denote by H(resp. H^∗) the corresponding Ringel-Hall algebra (resp. twisted Ringel-Hall algebra). Since the Hall polynomials always exist in this case, they are regarded as generic forms. Denote by C^∗ the twisted composition subalgebra ofH^∗.

Let Π be the set ofn-tuples of partitionsπ= (π⁽¹⁾, π⁽²⁾, . . . , π⁽ⁿ⁾), whereπ⁽ⁱ⁾= (π₁⁽ⁱ⁾ ≥ π₂⁽ⁱ⁾ ≥ · · ·) is a partition of some non-negative integer. For each π ∈ Π, define an object inT

M(π) = M

i∈∆₀,j≥1

Si[π⁽ⁱ⁾_j ].

In this way, we obtain a bijection between Π andP.

Ann-tupleπ= (π⁽¹⁾, π⁽²⁾, . . . , π⁽ⁿ⁾) of partitions in Π is called aperiodic, if for eachl≥1 there exists some i=i(l)∈∆0 such that π_j⁽ⁱ⁾6=l for allj ≥1. Denote by Π^a the set of aperiodic n-tuples of partitions. An object M in T is called aperiodic if M ' M(π) for some π ∈ Π^a. For any dimension vector α ∈ N∆0, define Πα={λ∈Π| dimM(λ) =α}and Π^a_α= Π^a∩Πα.

For any objects M and N in T, there exists a unique (up to isomorphism) extension L of M byN with minimal dimEnd(L). The extensionL is called the generic extension ofM byN, which is denoted byL=MN.

Let Ω be the set of all words on the alphabet ∆₀. For eachw=i₁i₂· · ·i_m∈Ω, set M(w) =S_i1S_i2 · · · S_im. Then there is a unique p(w) = π∈Π such that M(π) 'M(w). It has been proved in [17] that π = p(w) ∈Π^a and p induces a surjectionp: ΩΠ^a.

For each objectM inT and s≥1, denote bysM the direct sum ofs copies of M. For anyw∈Ω, writewin tight formw=j^e₁¹j₂^e2· · ·j^e_t^t ∈Ω withj_r−16=jr for allr. Letµrbe the element in Π such thatM(µr) =erSj_r. For anyλ∈Π^Pt

r=1erjr, writeg^λ_wfor the Hall polynomial g_M(µ^M(λ)

1),...,M(µ_t). A wordw is called distinguished if the Hall Polynomialgw^p(w)= 1. For anyπ∈Π^a, there exists a distinguished word wπ=j₁^e1j₂^e2· · ·j_t^et ∈p⁻¹(π) in tight form by [4]. From now on, fix a distinguished wordwπ ∈p⁻¹(π) for anyπ∈Π^a. Thus we have a sectionD={wπ |π∈Π^a} of pover Π^a. Dis called a section of distinguished words in [4].

For eachw=j^e₁¹j₂^e2· · ·j_t^et ∈Ω in tight form, define inC^∗ a monomial m^(w)=E_j^∗e1

i ∗ · · · ∗E_j^∗et

t .

Then defineEπ for allπ∈Π^a inductively by the following relations Eπ=m^(wπ)ifπ∈Π^a_α is minimal,

and

Eπ=m^(wπ)− X

λ≺π,λ∈Π^a_α

v^dg^λ_wπ(v²)Eλ

whereα=Pt

i=1e_rj_r,d=−dimM(π)+dim EndM(π)+dimM(λ)−dim EndM(λ) andλ≺µ⇔dim Hom(M, M(λ))≤dim Hom(M, M(µ)) for all objectsM in T.

For anyπ∈Π^a,Eπ is contained inC^∗. We have the following proposition.

Proposition 4.2. (cf. [4, 10]) LetD={wπ |π∈Π^a}be a section of distinguished words. Then both {m^(wπ)|π ∈Π^a} and {Eπ |π∈ Π^a} are A-bases of C_A^∗. And the transition matrix between these two bases is a unipotent lower triangular matrix with off-diagonal entries inA.

It has been proved in [4] that the basis {Eπ | π ∈ Π^a} is independent of the choice of the sections of distinguished words.

4.1.3. The integral bases arising from preprojective and preinjective components.

Letkbe a finite field withqelements andv=√

q. LetQbe a connected tame quiver without oriented cycles and Λ =kQbe the corresponding path algebra. Denote by P repandP reithe sets of isomorphism classes of indecomposable preprojective and preinjective Λ-modules respectively, which are independent of the choice of finite fields. LetH_q (resp. H^∗_q) be the Ringel-Hall (resp. twisted Ringel-Hall) algebra of Λ.

SinceP reiis representation-directed, we can define a total order on the set Φ⁺_{P rei}={· · ·, β₃, β₂, β₁}

of all positive real roots appearing inP reisuch that the corresponding Λ-modules {· · ·, M(β₃), M(β₂), M(β₁)}

satisfy the following conditions

Hom(M(β_i), M(β_j))6= 0⇒i≥j.

Similarly, since P rep is representation-directed, we can define a total order on the set

Φ⁺_{P rep}={α1, α2, α3,· · · }

of all positive real roots appearing inP repsuch that the corresponding Λ-modules {M(α₁), M(α₂), M(α₃),· · · }

satisfy the following conditions

Hom(M(α_i), M(α_j))6= 0⇒i≤j.

Define

N^{P rei}f ={b: Φ⁺_{P rei}→N|bis support-finite}.

For anyb∈N^{P rei}f , we can define a preinjective representation

M(b) = M

βi∈Φ⁺_{P rei}

b(β_i)M(β_i)

and any preinjective representation can be written in this form.

Define

N^{P rep}f ={a: Φ⁺_{P rep}→N|a is support-finite}.

For anya∈N^{P rep}f , we can define a preprojective representation

M(a) = M

α_i∈Φ⁺_{P rep}

a(α_i)M(α_i)

and any preprojective representation can be written in this form.

For any three elements b,b₁,b₂ ∈ N^{P rei}f (resp. a,a₁,a₂ ∈ N^{P rep}f ), the Hall polynomialg_M^M(b)_(b

1)M(b₂)(resp. g^M_M(a^(a)

1)M(a₂)) always exists.

Consider the generic composition algebra C^∗ of Q. Note that hM(b)i ∈ C_A^∗ (resp. hM(a)i ∈ C_A^∗) for any b ∈ N^{P rei}f (resp. a ∈ N^{P rep}f ). Denote by C_{P rei}^∗ (resp. C_{P rep}^∗ ) the A-submodule of C_A^∗ generated by {hM(b)i |b ∈ N^{P rei}f } (resp.

{hM(a)i |a∈N^{P rep}f }).

Proposition 4.3. (cf. [10]) The A-submoduleC_{P rei}^∗ (resp. C_{P rep}^∗ ) is a subalgebra of C^∗_A and {hM(b)i |b ∈ N^{P rei}f } (resp. {hM(a)i |a ∈ N^{P rep}f }) is an A-basis of C_{P rei}^∗ (resp. C^∗_{P rep}).

4.1.4. The integral basis for the generic composition algebra. Letkalso be a finite field with q elements and v = √

q. Let Q be a connected tame quiver without oriented cycles and Λ =kQbe the corresponding path algebra.

First consider the embedding of the category of representations of Kronecker quiver into that ofQ.

Letebe a extending vertex ofQandP=P(e) be the projective cover of simple moduleS_e. Letp= dimP(e) andδbe the minimal imaginary root vector. Note that hp,pi= 1 =hp, δiand there exists a unique indecomposable preprojective module L such that dimL = p+δ. Moreover, HomΛ(L, P) = 0 and ExtΛ(L, P) = 0.

Let C(P, L) be the smallest full subcategory of mod-Λ which contains P and L and is closed under taking extensions, kernels of epimorphisms and cokernels of monomorphisms. The category C(P, L) is equivalent to the module category of Kronecker quiver K over k. Thus we have an exact embedding F : mod-kK ,→ mod-Λ. Note that the embedding F is independent of the choice of finite fields.

Hence, this gives rise to a monomorphism of algebras F : H^∗(K) → H^∗(Q). In H^∗(K), we have defined Emδ_K for any m ∈ N. Define Emδ = F(Emδ_K). Since EmδK ∈ C^∗(K),Emδ∈ C^∗(Q).

List all non-homogeneous tubes T1,T2, . . . ,Ts in mod-Λ (in fact s ≤ 3). For eachTi, letr_i be the period ofTi,C^∗(Ti) be the corresponding generic composition algebra and C^∗(T_i)_A be its integral form as we did in Section 4.1.2. For each T_i, denote by Π^a_i the set of aperiodic r_i-tuples of partitions. We have constructed in Section 4.1.2 the elementsE_πi for any π_i ∈Π^a_i and the set {E_πi |π_i ∈Π^a_i} is an A-basis ofC^∗(Ti)_A.

LetMbe the set of

c= (ac,bc, πc, wc)

where ac ∈N^{P rep}f ,bc ∈N^{P rei}f ,πc = (π1c, π2c, . . . , πsc)∈Π^a₁×Π^a₂× · · · ×Π^a_s and wc= (w1≥w2≥ · · · ≥wt) is a partition ofm∈N.

For eachc∈ M, define

E^c=hM(ac)i ∗Eπ_1c∗Eπ_2c∗ · · · ∗Eπ_sc∗Ew_cδ∗ hM(bc)i,

wherehM(a_c)iandhM(b_c)iare defined in Section 4.1.3,E_πic is defined in Section 4.1.2 andE_wcδ is defined in Section 4.1.1.

Note that the set{E^c| c∈ M} is contained inC^∗(Q). We have the following proposition.

Proposition 4.4. (cf. [10]) The set{E^c |c∈ M}is an A-basis of C^∗(Q)_A. From this basis we can get a bar-invariant basis. But it is not the one considered by Lusztig. Hence in [10], another PBW-type basis is constructed. Let us review its definition.

There is a bilinear form (−,−) onH^∗_q(Λ) defined in [5]. It is also well-defined on C^∗(Q) which coincides with the one defined by Lusztig in [14]. Consider theQ(v)- basis{E^c|c∈ M}. Let R(C^∗(Q)) be the Q(v)-subspace ofC^∗(Q) with the basis {Eπ1c∗E_π2c∗ · · · ∗E_πsc∗E_wcδ}, whereπ_c= (π_1c, π_2c, . . . , π_sc)∈Π^a₁×Π^a₂× · · · ×Π^a_s, andw_c= (w₁≥w₂≥ · · · ≥w_t) is a partition. R(C^∗(Q)) is a subalgebra ofC^∗(Q).

Let R^a(C^∗(Q)) be the subalgebra of R(C^∗(Q)) with the basis {Eπ_1c ∗Eπ_2c ∗

· · · ∗Eπ_sc |πc = (π1c, π2c, . . . , πsc) ∈ Π^a₁ ×Π^a₂× · · · ×Π^a_s}. For any α, β ∈ NI, define α ≤ β if β −α ∈ NI. If β < δ, R(C^∗(Q))β = R^a(C^∗(Q))β. Define Fδ = {x|(x, R^a(C^∗(Q))_δ) = 0}.

In [10], it is proved that

R(C^∗(Q))_δ =R^a(C^∗(Q))_δ⊕ F_δ

and dimF_δ = 1. By the method of Schmidt orthogonalization, we may set E⁰_δ=E_δ− X

M(πic),dimM(πic)=δ,1≤i≤s

a_πicE_πic

satisfyingFδ =Q(v)E_δ⁰.

Now let R(C^∗(Q))(1) be the subalgebra of R(C^∗(Q)) generated by R^a(C^∗(Q)) andFδ. Ifβ <2δ,R(C^∗(Q))(1)_β =R(C^∗(Q))_β. Define

F2δ ={x|(x, R(C^∗(Q))(1)_2δ) = 0}.

Then dimF_2δ = 1 andR(C^∗(Q))_2δ =R(C^∗(Q))(1)_2δ⊕ F_2δ. In general, define

Fnδ={x|(x, R(C^∗(Q))(n−1)nδ) = 0}.

LetR(C^∗(Q))(n) be the subalgebra ofR(C^∗(Q)) generated byR(C^∗(Q))(n−1) and Fnδ. Then dimFnδ = 1 andR(C^∗(Q))nδ =R(C^∗(Q))(n−1)nδ⊕ Fnδ. Similarly, choose E_nδ⁰ such thatEnδ−E_nδ⁰ ∈R(C^∗(Q))(n−1)nδ andFnδ =Q(v)E⁰_nδ for all n≥1.

Let Pnδ = nE_nδ⁰ . For a partition w = (1^r12^r2· · ·t^rt) of m ∈ N, let Pwδ = P_1δ^∗r1∗ · · · ∗P_tδ^∗rt. For anyc∈ M, letSw_cδ be the Schur function corresponding to Pw_cδ and

F^c=hM(ac)i ∗Eπ1c∗Eπ2c∗ · · · ∗Eπsc∗Swcδ∗ hM(bc)i.

Proposition 4.5. (cf. [10]) The set {F^c| c ∈ M} is an almost orthonormal Q(v)-basis of C^∗(Q)'f.

4.2. PBW-type basis of U1˙ λ. LetQ be a connected tame quiver without ori- ented cycles andR(Q) be the corresponding root category over some finite fieldk.

Remember that ˜P is the set of isomorphism classes of objects inR(Q) and ind( ˜P) is the set of isomorphism classes of indecomposable objects inR(Q). The set ind( ˜P) can be divided into four parts as follow

ind( ˜P) =P ∪˙ T∪˙ T(P) ˙∪T(T).

Fix an embedding of mod-kQintoR(Q). ThenP=P rep(Q) ˙∪T(P rei(Q)) andTis the set of isomorphism classes of all indecomposable regular representations ofQ. T consists of isomorphism classes of indecomposable representations in homogeneous tubes and non-homogeneous tubesT1,T2,· · · ,Ts appearing in mod-kQ.

Let ˜Mbe the set of

˜c= (d˜c, π˜c, w˜c,d⁰_˜c, π⁰_˜c, w⁰_˜c) where

d˜c∈N^Pf, d⁰_˜c∈N^T_f^(P),

π˜c= (π1˜c, π2˜c, . . . , πs˜c)∈Π^a₁×Π^a₂× · · · ×Π^a_s, π⁰_˜c= (π⁰_1˜c, π⁰_2˜c, . . . , π_s˜⁰_c)∈Π^a₁×Π^a₂× · · · ×Π^a_s,