BGP-REFLECTION FUNCTORS AND LUSZTIG’S SYMMETRIES OF MODIFIED QUANTIZED ENVELOPING ALGEBRAS

OF MODIFIED QUANTIZED ENVELOPING ALGEBRAS

JIE XIAO AND MINGHUI ZHAO

Abstract. LetU be the quantized enveloping algebra and ˙Uits modified form. Lusztig gives some symmetries onUand ˙U. Since the realization ofU by the reduced Drinfeld double of the Ringel-Hall algebra, one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig’s symmetries onUand their important properties, for instance, the braid rela- tions. In this paper, we define a modified form ˙Hof the Ringel-Hall algebra and realize the Lusztig’s symmetries on ˙U by applying the BGP-reflection functors to ˙H.

1. Introduction

LetUbe the quantized enveloping algebra associated to a symmetrizable general- ized Cartan matrix. Lusztig introduces some symmetriesT_iacting on an integrable U-module and then on the quantized enveloping algebraU([1][2][3]). Let ˙Ube the modified quantized enveloping algebra obtained fromU by modifying the Cartan part U⁰ to ⊕_λ∈PQ(v)1_λ. This algebra has same representations with U. Lusztig also introduces some symmetries T_i acting on the modified quantized enveloping algebra ˙U ([3]).

LetH^∗_q(Λ) be the Ringel-Hall algebra associated to a finite dimensional hered- itary algebra Λ. Then the composition subalgebraC_q^∗(Λ) realizes the positive part U⁺of the quantized enveloping algebra by the Ringel-Green Theorem ([4][5]). One can extend the Ringel-Green theorem to the Drinfeld double version and realize the whole U by the reduced Drinfeld double of the composition algebra ([6]). These work give a connection between the representation theory of finite dimensional hereditary algebras and quantized enveloping algebras.

Via the Ringel-Hall algebra approach, one can apply the BGP-reflection functors to the quantum enveloping algebras U⁺ and U to obtain Lusztig’s symmetries and their properties in a conceptual way ([7][8]). This method gives a precise construction of Lusztig’s symmetries not only in the quantum enveloping algebras, also for the whole Drinfeld doubles of Ringel-Hall algebras ([9][10]).

In this paper, we define a modified form ˙H^∗_q(Λ) of the Ringel-Hall algebraH^∗_q(Λ).

We apply the BGP-reflection functors to obtain Lusztig’s symmetries on ˙H^∗_q(Λ).

Viewing the modified quantized enveloping algebra ˙Uas a subalgebra of ˙H_q^∗(Λ), we get a precise construction of Lusztig’s symmetries on ˙U. From this construction,

Date: February 19, 2017.

2000Mathematics Subject Classification. 16G20, 17B37.

Key words and phrases. BGP-reflection functors, Lusztig’s symmetries, Ringel-Hall algebras.

This work was supported by NSF of China (Grant No. 11131001).

1

we can obtain important properties of Lusztig’s symmetries, for instance, the braid relations.

In Section 2, we first give the basic notation of quantized enveloping algebras and modified quantized enveloping algebras; then we recall the definition of Lusztig’s symmetries on Uand ˙U. In Section 3, we recall the definition of the Ringel-Hall algebra H^∗_q(Λ) and define a modified form ˙H^∗_q(Λ) of it. In Section 4, we recall the BGP-reflection functors and define the corresponding maps from H˙^∗_q(Λ) to H˙^∗_q(σiΛ) induced by them. We prove in Section 6 that these maps induce algebra isomorphisms from ˙Uto itself, which coincide to the Lusztig’s symmetries on ˙Uand satisfy the braid relations. In Section 5, we define Lusztig’s symmetries on ˙H^∗_q(Λ) and find the precise relation between these symmetries and the maps induced by the BGP-reflection functors.

2. Quantized enveloping algebras and their modified forms 2.1. Quantized enveloping algebras. Denote byQthe field of rational numbers andZthe ring of integers. LetIbe a finite index set with|I|=nandA= (a_ij)_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)}

and h = Q⊗_ZP^∨ be the Q-linear space spanned by P^∨. We call P^∨ 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^∗ satisfying αj(hi) =aij and αj(ds) = 0 or 1 for 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 matrix A. Let W be the Weyl group generated by simple reflectionssi for alli∈I. There exists a bilinear form (−,−) onh^∗([11]).

We recall the definition of the quantized enveloping algebras. Assume thatA= (a_ij)_i,j∈I is a symmetrizable generalized Cartan matrix andD = diag(ε_i|i∈I) is its symmetrizing matrix.

Fix an indeterminatev. Forn∈Z, we set [n]_v=vⁿ−v⁻ⁿ

v−v⁻¹ ,

and [0]v! = 1, [n]v! = [n]v[n−1]v· · ·[1]v for n ∈ Z>0. For nonnegative integers m≥n≥0, the analogues of binomial coefficients are given by

hm n i

v

= [m]v! [n]_v![m−n]_v!. Then [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 elementsE_i,F_i(i∈I) andK_µ(µ∈P^∨) subject to the following relations:

(1) K0=1, KµKµ⁰ =Kµ+µ⁰ for allµ, µ⁰∈P^∨;

(2) K_µE_iK_−µ=v^αi(µ)E_i for alli∈I,µ∈P^∨;

(3) K_µF_iK_−µ=v^−αi(µ)E_i for alli∈I,µ∈P^∨;

(4) EiFj−FjEi=δij

K˜_i−K˜_−i

vi−v_i⁻¹ for alli, j∈I;

fori6=j, settingb= 1−aij, (5)

b

X

k=0

(−1)^kE^(k)_i E_jE^(b−k)_i = 0;

fori6=j, settingb= 1−a_ij, (6)

b

X

k=0

(−1)^kF_i^(k)FjF_i^(b−k)= 0.

Here, ˜K_ν = Π_i∈IK_εiνihi forν =P

i∈Iν_ih_i,v_i =v^εi andE_i⁽ⁿ⁾=E_iⁿ/[n]_vi!, F_i⁽ⁿ⁾= F_iⁿ/[n]_vi!.

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

Fi) fori∈I, and letU⁰be the subalgebra of Ugenerated byKµ forµ∈P^∨. We know that the quantized enveloping algebra has the triangular decomposition

U∼=U⁻⊗U⁰⊗U⁺.

Letf be the associative algebra defined by Lusztig in [3], which is generated by θi(i∈I) subject to the following relations

b

X

k=0

(−1)^kθ^(k)_i θ_jθ^(b−k)_i = 0,

wherei6=j,b= 1−a_ij andθ⁽ⁿ⁾_i =θ_iⁿ/[n]_vi!. There exist well-definedQ(v)-algebra monomorphisms f →U(x7→x⁺) andf → U(x7→x⁻) with image U⁺ and U⁻ respectively satisfyingE_i=θ⁺_i andF_i=θ⁻_i .

2.2. Modified quantized enveloping algebras. Let us recall the definition of the modified form ˙UofU in [3].

Ifλ⁰, λ⁰⁰∈P, we set

λ⁰Uλ⁰⁰=U/



 X

µ∈P^∨

(Kµ−v^λ0(µ))U+ X

µ∈P^∨

U(Kµ−v^λ00(µ))



.

Letπ_λ⁰_,λ⁰⁰:U→_λ⁰ U_λ⁰⁰ be the canonical projection and U˙ = M

λ⁰,λ⁰⁰∈P λ⁰U_λ⁰⁰.

Consider the weight space decompositionU=⊕βU(β), whereβruns throughZI andU(β) ={x∈U|KµxK_µ⁻¹=v^β(µ)x for allµ∈P^∨}. The image of summands U(β) under π_λ0,λ⁰⁰ form the weight space decomposition _λ0U_λ00 = ⊕β λ⁰U_λ00(β).

Note that_λ⁰U_λ⁰⁰(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 follows: 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 .

Let1_λ =π_λ,λ(1), where 1is the unit element of U. Then they satisfy1_λ1_λ⁰ = δλ,λ⁰1λ. In general, there is no unit element in the algebra ˙U. However the family (1_λ)_λ∈P can be regarded locally as the unit element in ˙U.

Note that _λ⁰U_λ⁰⁰ = 1_λ⁰U1˙ _λ⁰⁰. We define ˙U1_λ = ⊕λ⁰∈P1_λ⁰U1˙ _λ. Then ˙U =

⊕_λ∈PU1˙ λ.

2.3. Lusztig’s symmetries onU.˙ In [3], Lusztig introduces some symmetries on U, which is now called Lusztig’s symmetries.

Fixi∈I. DefineTi:U→Uon the generators as follows:

T_i(E_i) =−F_iK˜_i, T_i(F_i) =−K˜_−iE_i; Ti(Ej) = X

r+s=−αj(h_i)

(−1)^rv^−r_i E_i^(s)EjE_i^(r)forj6=i;

T_i(F_j) = X

r+s=−α_j(h_i)

(−1)^rv_i^rF_i^(r)F_jF_i^(s)forj 6=i;

Ti(Kµ) =K_µ−αi(µ)h_i.

Lusztig also introduces symmetries T_i : ˙U→U˙ induced by the symmetries on U. We write the following formulas:

Ti(Ei1λ) =−v_i^−λ(hi)Fi1siλ; T_i(F_i1_λ) =−v^−(2−λ(h_i ⁱ⁾⁾E_i1_siλ; T_i(E_j1_λ) = X

r+s=−αj(hi)

(−1)^rv_i^−rE_i^(s)E_jE_i^(r)1_siλ forj 6=i;

Ti(Fj1λ) = X

r+s=−αj(h_i)

(−1)^rv^r_iF_i^(r)FjF_i^(s)1s_iλ forj6=i.

3. Ringel-Hall algebras and their modified form

3.1. Ringel-Hall algebras. In this subsection, we recall the definition of Ringel- Hall algebras, following the notations in [12], [8] and [10].

Let k be a finite field and Λ be a finite dimensional hereditaryk-algebra. Ac- cording to [12], we can identity Λ with the tensor algebra of ak-species. A valued graph (Γ,d) is a finite set Γ together with nonnegative integers d_ij for all i, j∈Γ such thatd_ii= 0 and there exist positive integers{εi}i∈Γ satisfying

dijεj=djiεi fori, j∈Γ.

Given a Cartan datum (A,Π,Π^∨, P, P^∨), there is a valued graph (Γ,d) correspond- ing to it.

An orientation Ω of a valued graph (Γ,d) is given by an order on each edge {i, j}, which is indicated by an arrowi→j. We callQ= (Γ,d,Ω) a valued quiver.

We assume that Q = (Γ,d,Ω) is connected and contains no cycles. Let S = (F_i,_iM_j)_i,j∈Γ be a reducedk-species of type Q, that is, for all i, j∈Γ, _iM_j is an

Fi-Fj-bimodule, where Fi andFj are finite extensions ofk in an algebraic closure and dim(iMj)F_j =dij and dimk(Fi) =εi. Ak-representation (Vi,jϕi) ofS is given by vector spaces (Vi)Fi for anyi∈Γ andFj-linear mappingjϕi:Vi⊗iMj →Vjfor anyi→j. Such a representation is called finite dimensional ifP

i∈Γdim_kV_i<∞.

We denote by rep-S the category of finite dimensional representations of S over k. Let Λ be the tensor algebra of S. Then the category rep-S is equivalent to the module category mod-Λ of finite dimensional modules over Λ.

Given three modulesL, M andN in mod-Λ, denote by g^L_{M N} the number of Λ- submodulesW ofLsuch thatW 'N andL/W 'N in mod-Λ. Letv=p

|k| ∈C, P 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. The Ringel-Hall algebra Hq(Λ) of Λ is by definition the Q(v)-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 that Hq(Λ) is associative Q(v)-algebra with unit u_[0], where 0 denotes the zero module.

For each representation V = (Vi,jϕi) in rep-S, the dimension vector of V is defined to be dimV = (dimF_iVi)_i∈Γ ∈ N^Γ. For V, W ∈rep-S, The Euler form is defined by

hdimV,dimWi=X

i∈Γ

εiaibi−X

i→j

dijεjaibj,

where dimV = (a1, . . . , an) and dimW = (b1, . . . , bn). It is well known that hdimV,dimWi= dimkHomΛ(V, W)−dimkExtΛ(V, W).

Further, the symmetric Euler form is defined as

(dimV,dimW) =hdimV,dimWi+hdimW,dimVi.

Bothh−,−iand (−,−) are well defined on the Grothendieck groupG(Λ) of mod-Λ.

In fact, the Grothendieck group G(Λ) with the symmetric Euler form is a Cartan datum.

Let I ⊂ P be the set of isomorphism classes of (nilpotent) simple Λ-modules, which can be identified with Γ. Then the Euler form and the symmetric Euler form are defined on ZI. We also identify N^Γ with NI and regard dimV as an element in NI for each representation V = (V_i,_jϕ_i) in rep-S. For each α ∈ P, we fix a representationV_α in the isomorphism classα and let M(α) be the corresponding Λ-module. Forα, β∈ P, we set

hα, βi=hdimVα,dimVβi and

(α, β) = (dimVα,dimVβ).

Note that forα, β∈ P, (α, β) = (P

i∈Ia_iα_i,P

i∈Ib_iα_i), where dimV_α=Pa_iiand dimV_β =Pb_ii. Hence, we also use αto express the elementP

i∈Ia_iα_i in P and the elementP

i∈Ia_ih_i inP^∨.

The twisted Ringel-Hall algebraH^∗_q(Λ) is defined as follows. 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].

The composition algebraC_q^∗(Λ) is a subalgebra ofH^∗_q(Λ) generated byu_i =u_[Si], i∈ I, where Si is the (nilpotent) simple module corresponding toi ∈I. For any Λ-moduleM, we denote

hMi=v^{−dimM+dim EndΛ(M)}u_[M].

Note that{hMi|M ∈ P}is a Q(v)-basis ofH^∗_q(Λ).

Then we consider the generic form of Ringel-Hall algebras. LetQ be a valued quiver and Λk the corresponding finite dimensional hereditary algebra of ak-species which is of type Q. Denote by H^∗_q(Λk) the twisted Ringel-Hall algebra of Λk. Let K be a set of finite fields k such that the set {qk = |k||k ∈ K} is infinite and R be an integral domain containing Q and an element vq_k such that v_q²

k = qk for eachk∈ K. For eachk∈ K, we consider the composition algebraC_q^∗(Λk) which is theR-subalgebra ofH^∗_q(Λk) generated by the elements ui(k). Consider the direct product

H^∗(Q) = Y

k∈K

H^∗_q(Λk)

and the elements v= (v_qk)_k∈K, v⁻¹= (v_q⁻¹

k)_k∈K and u_i = (u_i(k))_k∈K. By C^∗(Q)_A we denote the subalgebra of H^∗(Q) generated by v, v⁻¹ and ui over Q, where A = Q[v, v⁻¹]. We may regard it as the A-algebra generated by ui where v is considered as an indeterminate. Finally, denote by C^∗(Q) = Q(v)⊗ C^∗(Q)A the generic twisted composition algebra of typeQ.

Remark 3.1. If Qis a Dynkin quiver, then the generic composition algebra ofQ can be defined directly using Hall polynomials.

Then we have the following well-known result of Green and Ringel ([4][5]).

Theorem 3.2. Let Q be a valued quiver, A be the associated generalized Cartan matrix, andf be the Lusztig’s algebra of typeA. Then the correspondenceui7→θi, i∈I induces an algebra isomorphism fromC^∗(Q)tof.

3.2. Double Ringel-Hall algebras. Let Λ be a finite dimensional hereditary al- gebra. In [6], the reduced Drinfeld doubleD(Λ) of Λ is defined. As an associative algebra,D(Λ) is generated by huα(+)i, huα(−)i(α∈ P) and K_µ(µ∈P^∨) subject

to the following relations ([8]):

K0=hu0(+)i=hu0(−)i=1, KµKµ⁰ =Kµ+µ⁰; (7)

huα(+)ihuβ(+)i=v^−hβ,αiX

λ∈P

g^λ_αβhuλ(+)i;

(8)

huα(−)ihuβ(−)i=v^−hβ,αiX

λ∈P

g^λ_αβhuλ(−)i;

(9)

Kµhuβ(+)i=v^β(µ)huβ(+)iKµ; (10)

Kµhuβ(−)i=v^−β(µ)huβ(−)iKµ; (11)

X

α,α⁰∈P

v^{hα0,αi+(α,α)}aα⁰

aλ⁰

g^λ_α⁰0αK˜_−αhuα⁰(−)ir_α⁰(huλ(+)i)

= X

α,β∈P

vhα,βi+(β,β)aα

aλ

g^λ_αβK˜βhuα(+)irβ(huλ⁰(−)i), (12)

whereα, β, λ, λ⁰ ∈ P,µ, µ⁰ ∈P^∨ and r⁰_α(huλ(+)i) = X

β∈P

vhα,βi+(α,β)g_αβ^λ aαaβ

a_λ huβ(+)i;

r_α(hu_λ(−)i) = X

β∈P

vhα,βi+(α,β)g_αβ^λ a_αa_β aλ

hu_β(−)i.

From the definition ofD(Λ), we have two algebra monomorphisms (+) :H^∗_q(Λ)→ D(Λ) mapping hM(λ)i to uλ(+) and (−) : H^∗_q(Λ) → D(Λ) mapping hM(λ)i to u_λ(−) for allλ∈ P.

Consider the weight space decomposition D(Λ) = ⊕_βD(Λ)(β), where β runs throughZI andD(Λ)(β) ={x∈ D(Λ)|K_µxK_µ⁻¹=v^β(µ)x for allµ∈P^∨}.

LetDc(Λ) be the subalgebra ofD(Λ) generated by hui(±)i(i∈I) and Kµ(µ∈ P^∨). In [6], the Green-Ringel Theorem 3.2 is extended to the Drinfeld double version andDc(Λ) realizes the corresponding quantum enveloping algebra U.

3.3. Another definition of U and a similar form of˙ H^∗(Λ). In [3], Lusztig gives another definition of ˙Uas follows. ˙Ucan be viewed as the algebra generated by the symbolsx⁺1ζx⁰⁻ and x⁻1ζx⁰⁺ with x∈fν, x⁰ ∈ fν⁰ for variousν, ν⁰ ∈NI andζ∈P; these symbols are subject to the following relations (13) to (19):

(13) (θ_i^(a))⁺1ζ(θ_j^(b))⁻= (θ^(b)_j )⁻1ζ+aα_i+bα_j(θ^(a)_i )⁺ifi6=j;

(14)

(θ^(a)_i )⁺1−ζ(θ_i^(b))⁻=X

t≥0

a+b−ζ(hi) t

vi

(θ_i^(b−t))⁻1−ζ+(a+b−t)αi(θ_i^(a−t))⁺;

(15) (θ^(b)_i )⁻1ζ(θ^(a)_i )⁺=X

t≥0

a+b−ζ(hi) t

vi

(θ_i^(a−t))⁺1_{ζ−(a+b−t)αi}(θ_i^(b−t))⁺;

(16) x⁺1_ζ =1_ζ+νx⁺, x⁻1_ζ =1_ζ−νx⁻forx∈f_ν;

(17) (x⁺1_ζ)(1_ζ⁰x⁰⁻) =δ_ζ,ζ⁰x⁺1_ζx⁰⁻,(x⁻1_ζ)(1_ζ⁰x⁰⁺) =δ_ζ,ζ⁰x⁻1_ζx⁰⁺;

(x⁺1ζ)(1ζ⁰x⁰⁺) =δζ,ζ⁰1ζ+ν(xx⁰)⁺,

(x⁻1ζ)(1ζ⁰x⁰⁻) =δζ,ζ⁰1ζ−ν(xx⁰)⁻forx∈fν; (18)

(rx+r⁰x⁰)⁺1ζ =rx⁺1ζ+r⁰x⁰⁺1ζ,(rx+r⁰x⁰)⁻1ζ =rx⁻1ζ+r⁰x⁰⁻1ζ

forx, x⁰∈fν andr, r⁰ ∈Q(v).

(19)

Letkbe a finite field and Λ a finite dimensional hereditaryk-algebra. For each ν∈NI, set

H^∗_q(Λ)ν= span{u_[M]|dimM =ν}.

Similarly, we can define ˙H_q^∗(Λ) as follows. H˙_q^∗(Λ) is the algebra generated by the symbolsx⁺1ζx⁰⁻andx⁻1ζx⁰⁺withx∈ H^∗_q(Λ)ν, x⁰∈ H^∗_q(Λ)ν⁰for variousν, ν⁰∈NI andζ∈P; these symbols are subject to the following relations (20) to (24):

X

α,α⁰∈P

v^hα0,αi+(α,α)+(ζ,−α)aα⁰

a_λ⁰g_α^λ00α(−1)^trα0v^m(α0)hM(α⁰)i⁻1ζ+α⁰(r⁰_α(hM(λ)i))⁺= X

α,β∈P

vhα,βi+(β,β)+(ζ,β)aα

a_λg^λ_αβ(−1)^tr(λ0−β)v^m(λ0−β)hM(α)i⁺1_ζ−α(rβ(hM(λ⁰)i))⁻ for allλ, λ⁰∈ P;

(20)

(21) x⁺1ζ =1ζ+νx⁺, x⁻1ζ =1_ζ−νx⁻forx∈ H^∗_q(Λ)ν; (22) (x⁺1_ζ)(1_ζ⁰x⁰⁻) =δ_ζ,ζ⁰x⁺1_ζx⁰⁻,(x⁻1_ζ)(1_ζ⁰x⁰⁺) =δ_ζ,ζ⁰x⁻1_ζx⁰⁺;

(x⁺1_ζ)(1_ζ⁰x⁰⁺) =δ_ζ,ζ⁰1_ζ+ν(xx⁰)⁺,

(x⁻1_ζ)(1_ζ⁰x⁰⁻) =δ_ζ,ζ⁰1_ζ−ν(xx⁰)⁻forx∈ H^∗_q(Λ)_ν; (23)

(rx+r⁰x⁰)⁺1ζ =rx⁺1ζ+r⁰x⁰⁺1ζ,(rx+r⁰x⁰)⁻1ζ =rx⁻1ζ+r⁰x⁰⁻1ζ

forx, x⁰∈ H^∗_q(Q)ν andr, r⁰∈Q(v).

(24)

Here a_λ is the order of the automorphism group of V_λ for λ∈ P,trα =P

i∈Ia_i, m(α) =P

i∈Ia_iε_iifα=P

i∈Ia_iα_i, and r_α(hM(λ)i) = X

β∈P

vhβ,αi+(β,α)g_βα^λ a_βa_α aλ

hM(β)i;

r⁰_α(hM(λ)i) = X

β∈P

vhα,βi+(β,α)g_αβ^λ aαaβ

a_λ hM(β)i.

Similarly to the case of modified form of quantum group, we have the following direct sums decompositions

H˙^∗_q(Λ) =M

ζ∈P

{x⁺1ζx⁰⁻|x, x⁰∈ H^∗_q(Λ)}

and

H˙^∗_q(Λ) =M

ζ∈P

{x⁻1ζx⁰⁺|x, x⁰∈ H^∗_q(Λ)}.

Let ˙C_q^∗(Λ) be the composition algebra, which is a subalgebra of ˙H_q^∗(Λ) generated byu⁺_i 1_ζu⁻_j andu⁻_i 1_ζu⁺_j for alli, j∈I andζ∈P.

Similarly to the Ringel-Hall algebra case we can consider the generic form H˙^∗(Q) = Y

k∈K

H˙^∗(Λ_k)

and its generic composition subalgebra ˙C^∗(Q) generated byu⁺_i 1_ζu⁻_j andu⁻_i 1_ζu⁺_j for alli, j∈Iandζ∈P, which is isomorphic to the corresponding modified quantum enveloping algebra ˙U. If a formula in ˙C_q^∗(Λ) is independent of the choice of the field, it can be viewed as a formula in ˙C^∗(Q)'U.˙

4. BGP-reflection functors and Lusztig’s symmetries

In this section we apply the BGP-reflection functors to the Ringel-Hall alge- bras and obtain an alternative construction of Lusztig’s symmetries on modified quantum enveloping algebras.

4.1. BGP-reflection functors. LetQ= (Γ,d,Ω) be a valued quiver,S = (F_i,_iM_j)_i,j∈Γ be ak-species of typeQ and pbe a sink or source of (Γ,d,Ω). We define a new orientation σpΩ of (Γ,d) by reversing the direction of arrows along all edges con- taining p and σpQ = (Γ,d, σpΩ). Let σpS be the k-species obtained from S by replacing rMs by its k-dual for r=p or s=p. ThenσpS is a reducedk-species of typeσpQ. Assume Λ is the corresponding finite dimensional hereditary algebra toS. We denote byσiΛ the corresponding finite dimensional hereditary algebra to σiS.

Now, we recall the definition of the Bernstein-Gelfand-Ponomarev (BGP) reflec- tion functorsσ_p^±: rep-S →rep-σpS ([13] [12] [8]).

Letpbe a sink of Ω. For anyV = (Vi,jϕi)∈rep-S, defineσ_p⁺V =W = (Wi,jψi) as follows. Let

Wi=Vi fori6=p, andWp be the kernel of

L

j→pVj⊗jMp

(pϕj)j//Vp ,

that is, we have the following exact sequence of vector spaces 0 //Wp

(jκp)j//L

j→pVj⊗jMp

(pϕj)j//Vp. Let

jψi=jϕi fori6=p, and

jψp=jκ¯p:Wp⊗pMj→Wj, wherej¯κp corresponds tojκp under the natural isomorphism

Hom_Fj(W_p⊗_pM_j, W_j)'Hom_Fp(W_p, W_j⊗_jM_p).

For any morphism f = (f_i) :V →V⁰ in rep-S, defineσ⁺_pf =g = (g_i) as follows.

Let

g_i=h_i fori6=p

andgp:Wp→W_p⁰ be the restriction of⊕j→p(fj⊗1), that is, we have the following commutative diagram

0 //W_p

g_p

(_jκ_p)_j

//L

j→pV_j⊗_jM_p

L

j→p(fj⊗1)

(_pϕ_j)_j

//V_p

f_p

0 //W_p⁰

(_jκ⁰_p)_j

//L

j→pV_j⁰⊗jMp (_pϕ⁰_j)_j

//V⁰_p

Similarly, ifpis a source of Ω, we can defineσ⁻_p from rep-S to rep-σpS.

Fori∈Γ, let rep-Shiibe the full subcategory of rep-S containing all representa- tions which do not haveV_ias a direct summand, whereV_iis the simple representa- tion with dimV_i=i. Ifiis a sink or source, then rep-Shiiis closed under direct sum- mands and extensions. Ifiis a sink (resp. source), thenσ_i⁺: rep-Shii 'rep-σ_iShii (resp. σ⁺_i : rep-Shii 'rep-σiShii) is an equivalence.

4.2. Construction of Lusztig’s symmetries. Assumeiis a sink ofQ. We first define a mapTi from ˙H^∗_q(Λ) to ˙H^∗_q(σiΛ).

For λ∈ P, assume thatVλ =Vλ₀ ⊕tVi and Vλ₀ contains no direct summand isomorphic toVi. Then Hom(Vλ₀, Vi) = 0 and Ext(Vi, Vλ₀) = 0. In this case

hM(λ)i=v^hλ0,tiiu^(t)_i hM(λ₀)i inH^∗_q(Λ). We define a mapTi: ˙H^∗_q(Λ)→H˙^∗_q(σiΛ) given by (25)

Ti(hM(λ)i⁺1ζhM(λ⁰)i⁻) = (−1)^p1v^q1u^−(t)_i hM(σ⁺_i λ0)i⁺1s_iζu^+(t

0)

i hM(σ_i⁺λ⁰₀)i⁻ where p₁ =t+t⁰−λ⁰₀(h_i) and q₁ =−hti, λ0i −t²ε_i+tε_i−(ζ, tα_i) +hλ⁰₀, t⁰ii − (λ⁰₀, i) +t⁰²ε_i−t⁰ε_i+ (ζ, t⁰α_i);

(26)

Ti(hM(λ⁰)i⁻1ζhM(λ)i⁺) = (−1)^p2v^q2u^+(t

0)

i hM(σ_i⁺λ⁰₀)i⁻1s_iζu^−(t)_i hM(σ⁺_i λ0)i⁺ wherep2=t+t⁰−λ⁰₀(hi) andq2=t²εi+tεi+hλ0, tii −(ζ, tαi)− ht⁰i, λ⁰₀i −(λ⁰₀, i)−

t⁰²εi−t⁰εi+ (ζ, t⁰αi).

In fact, the definition ofTi is induced by the following formulas:

T_i(hM(λ)i⁺1_ζ) = hM(σ⁺_i λ)i⁺1_siζ

Ti(hM(λ)i⁻1ζ) = (−1)^λ(hi)v^−(λ,i)hM(σ⁺_i λ)i⁻1siζ

ifV_λ contains no direct summand isomorphic toV_i and Ti(u⁺_i1_ζ) = −v^−(ζ,αi)u⁻_i 1_siζ Ti(u⁻_i 1ζ) = −v^{(ζ,αi)−2εi}u⁺_i 1s_iζ.

Note that, by the relation (24) in the definition of ˙H^∗(Λ), we can defineTion all the generators of ˙H^∗(Λ). If we can prove that Ti keeps the relations (20) to (23), thenTiinduces a map from ˙H^∗(Λ) to ˙H^∗(σiΛ). This is the first main result of this section.

Theorem 4.1. Leti be a sink. The formula (25) and (26) induces aQ(v)-algebra isomorphismTi: ˙H^∗(Λ)'H˙^∗(σ_iΛ)

The proof of Theorem 4.1 will be given in the last section.

Let i be a sink. For j ∈ I, if i = j, we have Ti(u⁺_i 1ζ) ∈ C˙^∗_q(σiΛ) and Ti(u⁻_i 1ζ) ∈ C˙_q^∗(σiΛ) since u⁺_i 1ζ and u⁻_i 1ζ are contained in ˙C_q^∗(σiΛ). If i 6= j, we haveTi(u⁺_j1ζ) =hM(σ_i⁺(j))i⁺1s_iζ. Note thatV_σ+

i(j)is an exceptional object in rep-σ_iS. Hence hM(σ⁺_i (j))i ∈C˙_q^∗(σ_iΛ). Hence T_i(u⁺_j1_ζ)∈C˙_q^∗(σ_iΛ). Similarly we have Ti(u⁻_j1ζ)∈C˙_q^∗(σiΛ). HenceTi induces anQ(v)-algebra homomorphism from C˙_q^∗(Λ) to ˙C_q^∗(σiΛ). Note the formula (25) and (26) are independent of the choice of the field. We can consider them as formulas in ˙C^∗(Q) and ˙C^∗(σiQ). Since both C˙^∗(Q) and ˙C^∗(σiQ) are isomorphic to ˙U, Ti induces a endomorphism on ˙U, if we identify ˙C^∗(Q) and ˙C^∗(σiQ) with ˙U.

Assume iis a source. Forλ∈ P, assume thatVλ=Vλ₀⊕tVi and Vλ₀ contains no direct summand isomorphic toVi. Then Hom(Vi, Vλ₀) = 0 and Ext(Vλ₀, Vi) = 0.

In this case

hM(λ)i=v^hti,λ0ihM(λ₀)iu^(t)_i inH^∗_q(Λ). We define a mapT_i⁰ : ˙H^∗_q(Λ)→H˙^∗_q(σ_iΛ) given by

T_i⁰(hM(λ)i⁺1ζhM(λ⁰)i⁻) = (−1)^p1v^q1hM(σ⁺_i λ0)i⁺u^−(t)_i 1s_iζhM(σ⁺_i λ⁰₀)i⁻u^+(t

0) i

where p1=t−t⁰−λ⁰₀(hi) andq1 =hti, λi+tεi+ (ζ, tαi)−(λ⁰₀, i)−t⁰εi−t⁰²εi− (ζ, t⁰αi)− hλ⁰₀, t⁰ii;

T_i⁰(hM(λ⁰)i⁻1ζhM(λ)i⁺) = (−1)^p2v^q2hM(σ⁺_i λ⁰₀)i⁻u^+(t

0)

i 1s_iζhM(σ⁺_i λ0)i⁺u^−(t)_i wherep2=t−t⁰−λ⁰₀(hi) andq2=−t²εi+tεi+ (ζ, tαi)− hλ0, tii −(λ⁰₀, i)−t⁰εi− (ζ, t⁰αi) +ht⁰i, λ⁰i.

By a similar way, we can prove thatT_i⁰ induces a Q(v)-algebra homomorphism from ˙U to ˙U.

Now assume i is a sink ofQ. Theni is a source of σiQ. We can easily check that TiT_i⁰ = 1 and T_i⁰Ti = 1. Hence Ti is a Q(v)-algebra isomorphism with T_i⁰ as its inverse.

Hence, we have the following theorem.

Theorem 4.2. Leti be a sink. The formula (25) and (26) induces aQ(v)-algebra isomorphismTi: ˙U'U.˙

Then we will prove thatTi coincides withTi. Proposition 4.3 ([8]). Let i6=j∈I and n=aij.

(1) Ifiis a sink, then in H_q^∗(Λ) we have hM(λ)i=

n

X

t=0

(−1)^tv^−t_i u^(t)_i uju^(n−t)_i

whereλ∈ P is the unique isomorphism class of indecomposable representation with the dimension vector j+ni.

(2) Ifiis a source, then in H^∗_q(Λ)we have hM(λ)i=

n

X

t=0

(−1)^tv^−t_i u^(n−t)_i uju^(t)_i

whereλ∈ P is the unique isomorphism class of indecomposable representation with the dimension vector j+ni.

Sinceiis a sink inQ,iis a source inσiQ, andV_σ+

i(j)is a unique indecomposable module in rep-σiS with dimension vector j +ni where n = aij. Thus by the Proposition 4.3,

hM(σ_i⁺(j))i⁺1s_iζ =

n

X

t=0

(−1)^tv_i^−tu^+(n−t)_i u⁺_ju^+(t)_i 1s_iζ. Hence

Ti(u⁺_j1ζ) =

n

X

t=0

(−1)^tv^−t_i u^+(n−t)_i u⁺_ju^+(t)_i 1s_iζ =Ti(u⁺_j1ζ).

Similarly we can checkTi=Ti on other generators.

Hence, we have the following theorem.

Theorem 4.4. Ifi is a sink, then the isomorphismT_i: ˙U→U˙ coincides with T_i. 4.3. Braid group relations. Let A= (aij)i,j∈I be a symmetrizable generalized Cartan matrix. Ifd(i, j) =a_ija_ji≤3, then the orderm(i, j) ofs_is_j is finite ([11]).

In fact, we have

m(i, j) =















2 ifd(i, j) = 0;

3 ifd(i, j) = 1;

4 ifd(i, j) = 2;

6 ifd(i, j) = 3;

∞ ifd(i, j)≥4.

The braid group of typeAis defined by the generators{κi}i∈I and relations κiκj· · ·=κjκi. . .

fori6=j withm(i, j)≤+∞factors on both sides, wherem(i, j) is the order ofs_is_j inW, that is,

κ_iκ_j=κ_jκ_i ifm(i, j) = 2;

κiκjκi =κjκiκj ifm(i, j) = 3;

κ_iκ_jκ_iκ_j =κ_jκ_iκ_jκ_i ifm(i, j) = 4;

κiκjκiκjκiκjκi =κjκiκjκiκjκiκj ifm(i, j) = 6.

(27)

Let Λ be a finite dimensional hereditary algebra, andAbe the corresponding gen- eralized Cartan matrix. In [8], the Lusztig’s symmetries on Dc(Λ) are constructed as follows.

Theorem 4.5. Letibe a sink. For allλ∈ P andµ∈P^∨, we writeVλ'Vλ₀⊕tVi

whereVλ₀ contain no direct summand isomorphic toVi. Then the mapT˜i is defined as follows:

T˜i(huλ(+)i) =v^hλ,tiiK˜tihui(−)i^(t)hu_σ+ iλ0(+)i;

(28)

T˜i(huλ(−)i) =v^hλ,tiiK˜_−tihui(+)i^(t)hu_σ+ iλ0(−)i;

(29)

T˜i(Kµ) =K_si(µ), (30)

induces aQ(v)-algebra isomorphism: Dc(Λ)' Dc(σiΛ).

In [8], the following theorem is proved.

Theorem 4.6. For anyi6=j ∈I such that m=m(i, j)≤+∞,T˜i andT˜j satisfy braid group relations (27) of typeA as maps onDc(Λ).

Let Λ be a finite dimensional hereditary algebra. Similarly to the the relation between ˙U and U, We consider the relation between ˙H^∗_q(Λ) and D(Λ). For any ζ∈P, we have a surjective linear mapping

πζ :D(Λ) → H˙_q^∗(Λ)1ζ

huα(+)ihuβ(−)iKµ 7→ (−1)^tr(β)v^m(β)+ζ(µ)hM(α)i⁺hM(β)⁻i1ζ

whereβ =P

i∈Ibiαi,tr(β) =P

i∈Ibi andm(β) =P

i∈Ibiεi. The kernel ofπζ is X

µ∈P^∨

D(Λ)(Kµ−v^ζ(µ)).

For anyζ, ζ⁰∈P,β ∈ZIand any x∈ D(Λ), y∈ D(Λ)(β), πζ(x)πζ⁰(y) =

πζ⁰(xy) if ζ=ζ⁰+β 0 otherwise . Our main result in this subsection is the following.

Theorem 4.7. LetΛ be a finite dimensional hereditary algebra, andAbe the cor- responding generalized Cartan matrix. For any i6=j∈I such that m=m(i, j)≤ +∞,Ti andTj satisfy braid group relations (27) of typeAas maps on C˙^∗_q(Λ).

Proof. For all λ∈ P andµ ∈P^∨, we write V_λ 'V_λ0⊕tV_i where V_λ0 contain no direct summand isomorphic toVi. We need to check that for any ζ∈P

π_siζ( ˜Ti(huλ(+)i)) =Ti(π_ζ(huλ(+)i));

(31)

π_siζ( ˜Ti(huλ(−)i)) =Ti(π_ζ(huλ(−)i));

(32)

π_siζ( ˜Ti(K_µ)) =Ti(π_ζ(K_µ)).

(33) First

π_siζ( ˜Ti(huλ(+)i)) = π_siζ(v^hλ,tiiK˜_tihui(−)i^(t)hu_σ⁺

iλ₀(+)i)

= v^{hλ,tii+(σi+λ0−ti,ti)}πsiζ(hui(−)i^(t)hu_σ+

iλ₀(+)iK˜ti)

= v^{hλ,tii+(σi+λ0−ti,ti)+(siζ,tαi)}(−1)^tv^m(ti)u^−(t)_i hM(σ⁺_i λ0)i⁺1s_iζ

= (−1)^tv^{−hti,λ0i−t2εi+tεi−(ζ,tαi)}u^−(t)_i hM(σ⁺_i λ₀)i⁺1_siζ

= Ti(hM(σ_i⁺λ0)i⁺1ζ)

= Ti(πζ(huλ(+)i)).

Hence we have formula (31). Similarly, we can get formula (32) and (33). Then

Theorem 4.6 implies this theorem.

5. Lusztig’s symmetries on the modified form of Ringel-Hall algebras

5.1. The structure of Ringel-Hall algebras. First we recall the structure of the Ringel-Hall algebra considered in [14] and [9].

We consider a bilinear formψ:H^∗_q(Λ)× H^∗_q(Λ) as ψ(hM(β)i,hM(β⁰)i) =|Vβ|

a_β δββ⁰

forβ, β⁰∈ P.