Cluster realization of positive representations of split real quantum Borel subalgebra

arXiv:1712.00013v1 [math.QA] 30 Nov 2017

Cluster realization of positive representations of split real quantum Borel subalgebra

Ivan Chi-Ho Ip

October 24, 2018

Dedicated to the memory of Ludvig D. Faddeev

Abstract

In our previous work [13], we studied the positive representations of split real quantum groups Uq˜q(g_R) restricted to its Borel part, and showed that they are closed under taking tensor products. However, the tensor product decomposition was only constructed abstractly using the GNS-representation of a C^∗-algebraic version of the Drinfeld-Jimbo quantum groups. In this paper, using the recently discovered cluster realization of quantum groups [14], we write down the decomposition explicitly by realizing it as a sequence of cluster mutations in the corresponding quiver diagram representing the tensor product.

2010 Mathematics Subject Classification. Primary 81R50, Secondary 22D25 Keywords. positive representations, split real quantum groups, modular double, quantum cluster algebra, tensor category

1 Introduction

1.1 Positive representations of split real quantum groups

To any finite dimensional complex simple Lie algebra g, Drinfeld [3] and Jimbo [15] de- fined a remarkable Hopf algebraU^q(g) known as the quantum group. The notion ofpositive representations was introduced in [8] as a new research program devoted to the represen- tation theory of its split real formUqeq(g_R) which uses the concept of Faddeev’s modular double [4, 5], and generalizes the case ofUqqe(sl(2,R)) studied extensively by Teschneret al. [2, 18, 19] from the physics point of view.

Explicit construction of the positive representationsPλofUqqe(g_R), parametrized by the R₊-span of positive weightsλ∈P_R⁺, was constructed in [11, 12] for simple Lie algebragof all types, where the generators of the quantum groups are realized by positive essentially self-adjoint operators acting on certain Hilbert space, and compatible with the modular double structure (see Section 2.2 for a review). Although the representations involve

∗Center for the Promotion of Interdisciplinary Education and Research/

Department of Mathematics, Graduate School of Science, Kyoto University, Japan Email: ivan.ip@math.kyoto-u.ac.jp

unbounded operators, the algebraic relations are well-defined and are unitary equivalent to theintegrable representations of the quantum plane in the sense of Schm¨udgen [10, 24].

A long standing conjecture in the theory of positive representations is the closure under taking tensor product:

P^α⊗ Pβ≃ Z

γ∈P⁺ R

P^γ⊗ M^γαβdµ(γ) (1.1)

for some Plancherel measure dµ(γ) and multiplicity moduleM^γαβ. The simplest case of Uqeq(sl(2,R)) has been proved previously in [19], which is related to the fusion relations of quantum Liouville theory. The case in typeAn is recently solved algebraically in [23]

using a cluster realization [14, 22] of the positive representations discussed below, and is related to certain quantum open Toda systems. For other types however, the question is still open, but as a first step, we showed in [13] that the restriction ofP^λto the Borel part Uqqe(b_R) is indeed closed under taking tensor product:

Theorem 1.1. Let Pλ^b be the positive representations Pλ restricted to the Borel part U^qqe(bR). ThenPλ^b≃ P^b does not depend on λ, and we have the unitary equivalence

P^b⊗ P^b≃ P^b⊗ M, (1.2)

whereMis a multiplicity module in whichU^qqe(bR)acts trivially.

As applications, this gives a new candidate for quantum higher Teichm¨uller theory, where the above unitary equivalence gives thequantum mutation operator satisfying the pentagon relation (see [9, 13] for a review). It also provides a major step towards proving the tensor product decomposition of Pλ in general. Together with the braiding by the universalRoperator, the positive representations will carry a (continuous) braided tensor category structure, which may give rise to new class of TQFT’s in the sense of Reshetikhin- Turaev [20, 21].

The construction of the unitary equivalence in [13] utilizes the language ofmultiplier Hopf algebra and the GNS representations of C^∗-algebra, which leads to the existence of a unitary operator W called themultiplicative unitary giving the desired intertwiner (1.2). However, the construction ofW on theC^∗-algebraic level is quite abstract and it is very difficult to write down explicitly the intertwiner as a product of quantum dilogarithm functions. We presented several examples in [13] for typeAn and hint at a relationship to the Heisenberg double, but we were not able to generalize it to other types nor write down the formula in general.

1.2 Cluster realization of positive representations

In this paper, we reprove Theorem 1.1 using a new technique which comes with the discovery of the cluster realization of quantum groups in [22] for type An and [14] for general types, where we found an embedding of the Drinfeld’s double of the Borel part D(U^q(b))֒→ X^D2,1 into a quantum cluster algebra associated to a quiver QD_2,1 on the triangulation of a once punctured disk with two marked points. The quiver itself has a geometric meaning representing the Poisson structure of the moduli space of framed local systems for general groups [7, 17]. A polarization of the quantum cluster variables, i.e. a choice of representations by the canonical variables on some Hilbert spaceL²(R^N) through the exponential functions, recovers the positive representationsPλofUqqe(g_R), which is the quotient ofD(U^q(b)) by identifying the Cartan partKiKi^′= 1.

In particular, the generators ofU^q(g) can be represented combinatorially using certain paths on QD_2,1 representing a telescoping sums of quantum cluster variables, while the coproduct is given by the quiverQD_2,2 on twice-punctured disk with two marked points, which is simply concatenating two copies of QD_2,1 along one edge. Furthermore, cluster mutations correspond to unitary equivalence of the representationsP^λ. Finally, to each triangulation, we constructed explicitly in [14] using the universalR matrix the sequence of cluster mutationsµrealizing quiver mutations associated to the flip of triangulation as in Figure 1.

Q Q^′

Q Q^′ µ

Figure 1: Flip of triangulation, withQ, Q^′ the associated quivers.

In this new language, it turns out that the restrictionPλ^bcan be easily describe by the so-calledFi-paths as shown schematically in red below in Figure 2. A flip of triangulation in this case creates a self-folded triangle, but since for the study of Pλ^b, the mutation sequence does not involve the gluing edgeE in the quiver, we can relax by un-gluing the edgesE and consider only a normal flip of triangulation as in Figure 6 of Section 6, where we prove the main result:

Theorem 1.2. The sequence of cluster mutations realizing the flip of triangulation pre- serves theFi-paths representing the restriction of positive representations toU^qqe(b_R).

P^b

λ

F_in F_out

Q Q

E

≃Φ₁

λ

Pf^b

F_in F_out

Figure 2: Flipping to self-folded triangle, where theFi-paths are preserved.

In particular, this shows thatPλ^b is unitary equivalent to a representationPf^b repre- senting the generators on the basic quiver QD_3,0 associated only to a single triangle. In [13] we call this thestandard form, which is obtained by omitting half of the operators in the explicit representation ofPλ^b, and coincides with the so-calledFeigin’s homomorphism.

Finally, the tensor product realized by concatenating two copies of QD_2,1, can be decomposed using three flips of triangles as shown schematically in Figure 3 (which is the same as the first three steps in the proof of tensor product decomposition of P^λ for the whole quantum group in typeAnconstructed in [23]). Since we found in [14] an explicit formula for the cluster mutations using a product of quantum dilogarithms, wecompletely solve the combinatorial problemposed in [13] for explicitly writing down the tensor product decomposition of positive representations restricted toU^qqe(bR) ofall types, which cannot be done previously without the cluster realization.

P_λ^b⊗ P_µ^b

F_in F_out ≃Φ₁⊗Φ₁

Pf^b⊗Pf^b

F_in F_out ≃Φ₃

Pf^b⊗ M F_in F_out

Figure 3: Tensor product decompositionP_λ^b⊗ P_λ^b ≃Pf^b⊗ M.

1.3 Outline of the paper

The paper is organized as follows. In Section 2, we recall the positive representationsP^λ and some identities of the quantum dilogarithm function gb. In Section 3, we recall the definitions and results concerning quantum torus algebraXⁱ. In Section 4, we reconstruct the basic quiverQⁱin [14] in the notation of the current paper, and in Section 5 we recall the cluster realization of positive representations. Finally in Section 6, we prove the main result and in Section 7 we give several examples to demonstrate the algorithms of the construction of the tensor product decomposition of positive representations restricted to the Borel partUqqe(b_R).

Acknowledgments

I would like to thank Gus Schrader, Alexander Shapiro and Ian Le for valuable discussions and hospitality at University of Toronto and Perimeter Institute at which this work is inspired. This work is partly based on the talks given at CQIS-2017 held in Dubna, Russia, and the 2017 Autumn Meeting of the Mathematics Society of Japan. This work is supported by Top Global University Project, MEXT, Japan at Kyoto University, and JSPS KAKENHI Grant Numbers JP16K17571.

2 Preliminaries

In this section, let us recall several notations and definitions that will be used throughout the paper. We will follow mostly the convention used in [13] and [14].

2.1 Definition of the modular double U

(g

)

Let g be a simple Lie algebra over C, I = {1,2, ..., n} denotes the set of nodes of the Dynkin diagram ofgwheren=rank(g). Let{αi}i∈I be the set of positive simple roots, and letw0 ∈W be the longest element of the Weyl group ofg, whereN :=l(w0) is the length of the longest word. We call a sequence

i= (i1, ..., iN)∈ I^N

a reduced word ofw0 ifw0 =si1...si_N is a reduced expression, wheresi_k are the simple reflections of the root space. We will denote the reversed word by

i:= (iN, ..., i1).

Definition 2.1. Letqbe a formal parameter. Let(−,−)be theW-invariant inner product of the root lattice, and we define

aij:=2(αi, αj) (αi, αi) , such that A:= (aij)is the Cartan matrix.

We normalize(−,−) as follows: fori∈ I, we define the multipliersto be

di:=1

2(αi, αi) :=





1 iis long root or in the simply-laced case,

1

2 iis short root in typeB, C, F,

1

3 iis short root in typeG2,

(2.1)

where(αi, αj) =−1when i, j are adjacent in the Dynkin diagram, such that diaij=djaji.

We then defineqi:=q^di, which we will also write as

ql:=q, (2.2)

qs:=

( q¹² gis of typeBn, Cn, F4,

q¹³ gis of typeG2, (2.3)

for theq parameters corresponding to long and short roots respectively.

Definition 2.2. [3, 15] The Drinfeld-Jimbo quantum group Uq(g_R) is the Hopf algebra generated by{Ei, Fi, K_i^±1}ⁱ∈I overCsubjected to the relations fori, j ∈ I:

KiEj=q_i^aijEjKi, KiFj=q_i^−aijFjKi, [Ei, Fj] =δij

Ki−K_i⁻¹

qi−q⁻¹_i , (2.4) together with the Serre relations fori6=j:

1−a_ij

X

k=0

(−1)^k [1−aij]q_i!

[1−aij−k]q_i![k]q_i!Xi^kXjX_i^1−aij−k= 0, X =E, F, (2.5) where[k]q:= ^q_q−q^k−q−1^−k.

The Hopf algebra structure ofU^q(g)is given by

∆(Ei) =1⊗Ei+Ei⊗Ki, (2.6)

∆(Fi) =Fi⊗1 +K_i⁻¹⊗Fi, (2.7)

∆(Ki) =Ki⊗Ki. (2.8)

We will not need the counit and antipode in this paper.

In the split real case, it is required that|q|= 1. Throughout the paper, we let

q:=e^π√−1b2 (2.9)

with 0< b²<1 andb²∈R\Q. We also write qi:=e^π√−1b2i such that

bi:=

bl:=b αiis long root orgis simply-laced, bs:=√

dib αiis short root. (2.10)

We defineU^q(gR) to be the real form ofU^q(g) induced by the star structure

E^∗i =Ei, Fi^∗=Fi, K^∗i =Ki. (2.11) Finally, from the results of [11, 12], letqe:=e^{π√−1b−s2} and we define the modular double to be

Uqeq(g_R) :=U^q(g_R)⊗ Uqe(g_R) gis simply-laced, (2.12) U^qqe(gR) :=U^q(gR)⊗ Ueq(^LgR) otherwise, (2.13) where^Lg_Ris the Langlands dual obtained by interchanging the long and short roots ofg_R.

2.2 Positive representations of U

(g

)

In [8, 11, 12], a special class of representations forUqeq(g_R), called the positive representa- tions, is defined. The generators ofU^qqe(g_R) are realized by positive essentially self-adjoint operators on certain Hilbert space, and satisfy the transcendental relations (2.15). In particular the quantum group and its modular double counterpart are represented on the same Hilbert space, generalizing the situation of Uqqe(sl(2,R)) introduced in [4, 5] and studied in [19]. More precisely,

Theorem 2.3. [8, 11, 12] Define the rescaled generators to be

ei:= 2 sin(πb²i)Ei, fi:= 2 sin(πb²i)Fi. (2.14) There exists a family of representations PλofUqqe(g_R)parametrized by theR+-span of the cone of positive weightsλ∈P_R⁺, or equivalently by λ∈R^rank(g)₊ , such that

• The generatorsei,fi, Kiare represented by positive essentially self-adjoint operators acting onL²(R^N)whereN=l(w0).

• Define the transcendental generators:

e ei:=e

1 b2 i

i , fei:=f

1 b2 i

i , Kfi:=K

1 b2 i

i . (2.15)

– if gis simply-laced, the generators eei,fei,Kfi are obtained by replacingb with b⁻¹ in the representations of the generators ei,fi, Ki.

– Ifgis non-simply-laced, then the generatorsfEi,Fei,Kfiwitheei:= 2 sin(πb⁻²_i )fEi

andfei:= 2 sin(πb⁻²_i )Fei generateUqe(^Lg_R).

• The generatorsei,fi, Ki andeei,fei,Kficommute weakly up to a sign.

Let P^λ ≃ L²(R^N, du1...duN), and letuk, pk := _2π√¹

−1

∂

∂u_k be the standard position and momentum operator respectively acting on L²(R^N) as self-adjoint operators. Then the representations can be constructed explicitly as follows:

Theorem 2.4. [11, 12] For a fixed reduced wordi= (i1, ..., iN), the positive representation Pλⁱ parametrized by λ = (λ1, ..., λn) ∈ Rⁿ_≥0 is given by positive essentially self-adjoint operators

Ki= exp −2πbiλi−π XN k=1

ai,i_kbi_kuk

!

, (2.16)

fi=f_i⁻+f_i⁺ (2.17)

:= X

k:ik=i

f^k,−+ X

k:ik=i

f^k,+,

acting onL²(R^N), where f^k,±:= exp ±

k−1

X

j=1

πbi_jai_j,i_kuj+πbi_kuk+ 2πbi_kλi_k

!

+ 2πbi_kpk

!

. (2.18)

TheEigenerators corresponding to the right most root iN ofiis given explicitly by

ei_N =e⁻_iN+e⁺_iN (2.19)

:=e^{πbiN(uN−2pN)}+e^{πbiN(−uN−2pN)}, while for the other generators we have in general

ei=e⁻_i +e⁺_i, (2.20)

wheree^±_i can be obtained from conjugation by quantum dilogarithms ofe^±_i

N above, and can be realized explicitly as theEi-paths polynomials on the cluster realization [14] (cf. Section 5).

Theorem 2.5. For any reduced wordsiandi^′, we have unitary equivalence

Pλ≃ Pλⁱ ≃ Pλ^i′. (2.21)

In particular, for any root indexi∈ I, we can chooseiwithiN=iin order to observe that

Corollary 2.6. For anyi= 1, ..., n, we have

f_i⁻f_i⁺=q⁻²_i f_i⁺f_i⁻, e⁻_ie⁺_i =q_i⁻²e⁺_ie⁻_i, (2.22) [e^±_i,f_j^∓]

qi−q_i⁻¹ =∓δijK_i^±1, (2.23)

f^k,±f^l,±=q^±a_ik^klf^l,±f^k,±, 1≤l < k≤N. (2.24) Remark 2.7. We see that, for example, the triple{e⁻_i ,f_i⁺, K_i⁻¹}forms the commutation relation of the Heisenberg double [16], which will be needed to prove the tensor product decomposition of the positive representations restricted to the Borel part in Section 6, thus confirming the intuition discussed in [13].

2.3 Quantum dilogarithm identities

We recall the quantum dilogarithm identities needed in this paper. More details can be found in [6, 14]. The non-compact quantum dilogarithm is a meromorphic function that can be represented as an integral expression:

gb(x) := exp 1 4 Z

R+i0

x^ibt

sinh(πbt) sinh(πb⁻¹t) dt

t

!

, (2.25)

such that by functional calculus, it is unitary whenxis positive self-adjoint.

Remark 2.8. During formal algebraic manipulation, one may consider its compact ver- sion instead, which in terms of formal power series are related by

gb(x)∼Ψ^q(x)⁻¹, (2.26)

where the Faddeev-Kashaev’s quantum dilogarithm is

Ψ^q(x) :=

Y∞ r=0

(1 +q^2r+1x)⁻¹=Exp_q−2

u q−q⁻¹

, (2.27)

with

Expq(x) :=X

k≥0

x^k

(k)q!, (k)q :=1−q^k

1−q . (2.28)

We will only need the following identities in this paper.

Lemma 2.9(Quantum dilogarithm identities). Letu, vbe positive self-adjoint variables.

Ifuv=q²vu, then we have the quantum exponential relation:

gb(u+v) =gb(u)gb(v), (2.29) and the conjugation

gb(v)ugb(v)^∗=qvu+u, (2.30)

gb(u)^∗vgb(u) =v+qvu.

We also have in the doubly-laced case (√ 2bs=b)

gb_s(u+v) =gb_s(u)gb(q⁻¹uv)gb_s(v). (2.31) Let againu, v be positive self-adjoint and define

c:= [u, v]

q−q⁻¹,

such that uc=q²cuandcv=q²vc. Then we have the generalized equation:

gb(v)ugb^∗(v) =c+u, (2.32)

gb(u)^∗vgb(u) =v+c, in which (2.30)is a special case.

3 Quantum cluster algebra

In this section, let us recall the notation of the quantum torus algebra used in this paper.

We will follow mostly the notations used in [14] and [22] but with some modifications.

3.1 Quantum torus algebra and quiver

Definition 3.1 (Cluster seed). A cluster seed is a datum i= (I, I0, B, D) where I is a finite set, I0⊂I is a subset called the frozen subset,B= (bij)i,j∈I a skew-symmetrizable

1

2Z-valued matrix called the exchange matrix, andD=diag(di)_i∈Iis a diagonalQ-matrix called the multipliersuch that DB=−B^TDis skew-symmetric.

Notation 3.2. In the rest of the paper, we will write i^′ = (I^′, I₀^′, B^′, D^′) and i = (I, I0, B, D) and the respective elements asi^′∈I^′, i∈I etc. for convenience.

Definition 3.3 (Quantum torus algebra). Let q be a formal parameter. We define the quantum torus algebra Xⁱassociated to a cluster seedi to be an associative algebra over C[q^d], whered= mini∈I(di), generated by{Xi}ⁱ∈I subject to the relations

XiXj=q^−2wijXjXi, i, j∈I, (3.1) where

wij=dibij=−wij. (3.2)

The generators Xi∈ Xⁱ are called the quantum cluster variables, and they are frozen if i∈I0. We denote byTⁱthe non-commutative field of fraction ofXⁱ.

Alternatively, given B and D as above, let Λi be a lattice with basis {ei}ⁱ∈I, and define a skew symmetric dZ-valued form on Λⁱ by (ei, ej) :=wij. Then Xⁱis generated by{Xλ}λ∈Λ_i withX0:= 1 subject to the relations

q^(λ,µ)XλXµ=Xλ+µ. (3.3)

Notation 3.4. Under this realization, we shall writeXi=Xe_i, and define the notation

Xi₁,...,i_k:=Xe_i1+...+e_ik, (3.4)

or more generally forn1, ..., nk∈Z, X_iⁿ1

1 ,...,i^nk_k :=Xn₁e_i1+...+n_ke_ik. (3.5) Letcij∈ ¹2Zfori, j∈I be defined bycij=

bij ifdi=dj, wij otherwise.

Definition 3.5 (Quiver associated to i). We associate to each seed i = (I, I0, B, D) a quiverQⁱwith vertices labeled by Iand adjacency matrix(cij)_i,j∈I. We calli∈Ia short (resp. long) node ifq^di =qs (resp. q^di =q). An arrowi−→j represents the algebraic relation

XiXj=q_∗⁻²XjXi (3.6)

whereq∗=qs if bothi, j are short nodes, orq∗=q otherwise.

We will use squares to denote frozen nodesi∈I0 and circles otherwise. We will also use dashed arrow if cij= ¹₂, which only occurs between frozen nodes. We will represent the algebraic relations (3.1) by thick or thin arrows (see Figure 4) for display convenience (thickness isnotpart of the data of the quiver). Thin arrows only occur in the non-simply- laced case between two short nodes.

i j XiXj =q⁻²XjXi

i j X_iX_j =q⁻¹X_jX_i

i j XiXj =q⁻²_s XjXi

Figure 4: Arrows between nodes and their algebraic meaning.

Definition 3.6. A positive representationof the quantum torus algebra Xⁱon a Hilbert spaceH=L²(R^M) is an assignment

Xi=e^2πbLi, i∈I, (3.7)

whereLi:=Li(uk, pk, λk)is a linear combination of the position and momentum operators {uk, pk}^Mk=1and complex parameters λk with

[Li, Lj] = wij

2π√

−1, (3.8)

such that Xi acts as a positive self-adjoint operator onH.

3.2 Quantum cluster mutation

Next we define the cluster mutations of a seed and its quiver, and the quantum cluster mutations for the algebra. Here we will use the notion that keeps the indexingI of the seeds, which ensures the consistency of the relationµ²k= Id.

Definition 3.7 (Cluster mutation). Given a pair of seeds i,i^′ with I =I^′, I0 =I₀^′, and an elementk ∈I\I0, a cluster mutation in directionk is an isomorphism µk :i −→i^′ such that µk(i) =ifor alli∈I,d^′i=di, and

b^′ij=

( −bij ifi=korj=k,

bij+^bik|bkj|+|₂^bik|bkj otherwise. (3.9) Then the quiver mutation Qⁱ −→ Q^i′ corresponding to the mutation µk can be per- formed by the well-known rule (with obvious generalization for dashed arrows)

(1) reverse all the arrows incident to the vertexk;

(2) for each pair of arrowsi−→ k and k −→ j, add n^k_ij arrows from i−→j, where n^kij=d⁻¹_k ifk is a short node and both i, j are long nodes, orn^kij= 1 otherwise;

(3) delete any 2-cycles.

Definition 3.8(Quantum cluster mutation). The cluster mutationµk:i−→i^′, induces an isomorphismµ^q_k:T^i′−→Tⁱcalled thequantum cluster mutation, which can be written as a composition of two homomorphisms

µ^q_k=µ^#_k ◦µ^′k, (3.10)

whereµ^′k:T^i′−→Tⁱ is a monomial transformation defined by µ^′k(Xbi) :=





X_k⁻¹ ifi=k,

Xi ifi6=kandbki≤0, q_i^bikbkiXiX_k^bki ifi6=kandbki≥0,

(3.11)

andµ^#_k :Tⁱ−→Tⁱis a conjugation by the quantum dilogarithm function µ^#_k :=Adg^∗

bk(Xk), (3.12)

wherebk=√ dkb.

It is also useful to recall the following lemma from [22, Lemma 1.1]:

Lemma 3.9. Letµi₁, ..., µi_k be a sequence of mutation, and denote the intermediate seeds byij:=µi_j· · ·µi1(i). Then the induced quantum cluster mutationµ^q_i1· · ·µ^q_ik:T^ik−→Tⁱ can be written as

µ^q_i1· · ·µ^q_ik= Φk◦Mk, (3.13) whereMk:T^ik−→TⁱandΦk:Tⁱ−→Tⁱare given by

Mk:=µ^′i1µ^′i2· · ·µ^′i_k, (3.14) Φk:=Adg^∗

bi1

(X_i1)Ad_g∗

bi2(M1(X⁽¹⁾_i

2 ))· · ·Ad_g∗ bik

(M_k−1(X_ik^(k−1)), (3.15) andX_i^(j)∈ X^ij denotes the corresponding quantum cluster variables of the algebra X^ij.

3.3 Amalgamation

Finally we briefly recall the procedure ofamalgamationof two quantum torus algebra [7]:

Definition 3.10. Leti,i^′be two cluster seeds and letXⁱ,X^i′be the corresponding quantum torus algebra. Let J ⊂ I0 and J^′ ⊂ I₀^′ be subsets of the frozen nodes with a bijection φ:J−→J^′ such that

d^′_φ(i)=di, i∈J.

Then the amalgamation ofXⁱandX^i′alongφis identified with the subalgebraX ⊂ Xe ⁱ⊗X^i′ generated by the variables {Xei}i∈I∪I^′ where

Xei:=

Xi⊗1 ifi∈I\J,

1⊗X_i^′ ifi∈I^′\J^′, (3.16) Xei=Xe_φ(i):=Xi⊗X_φ(i)^′ i∈J.

Equivalently, the amalgamation of the corresponding quiversQ, Q^′ is a new quiverQe constructed by gluing the vertices along φ, defreezing those vertices that are glued, and removing any resulting 2-cycles.

4 The basic quivers

In [14], we constructed the basic quiverQassociated to a triangle such that the quantum group can be embedded into an amalgamation ofQand its mirror imageQ (See Figure 2). Let us recall its construction relevant to this paper, with a slightly different indexing.

First of all, following [1], we define the notationk⁺ andk⁻as follows

Notation 4.1. Given a reduced wordi= (i1, ..., iN), fork∈ {1, ..., N}, we denote byk⁺ the smallest indexl > k such thatik=ilif it exists, orN+ik otherwise.

Similarly, for k ∈ {1, ..., N+n}, we denote by k⁻ the largest indexl < k such that l⁺=k if it exists, or0otherwise. Finally we defineiN+j:=j∈ Iforj= 1, ..., n.

We define the set of extremal indices by

Fⁱⁿ:={k:k⁻= 0}, F^out:={N+ 1, ..., N+n}, (4.1) and letk^∗:= (N+k)⁻be the largest index which is not extremal and such thatik^∗ =k.

By abuse of notation we will also useito denote the cluster seed corresponding to the quiverQⁱ, called thebasic quiver, described as follows.

Definition 4.2. Thebasic quiverQⁱhas nodesI={1, ..., N+2n}. It contains a subquiver QⁱF withN+nnodes{1,2, ..., N+n}where the frozen nodes areFⁱⁿ∪ F^out, such that for k= 1, ..., N:

• there is a single arrowk−→k⁺,

• there is a single arrowl−→k ifl⁻< k⁻< l < k, and for the frozen nodes,

• there is a half arrowk l ifk, l∈ Fⁱⁿ andk > l,

• there is a half arrowk l ifk, l∈ F^outandk⁻> l⁻.

The multiplierD= (di)of the cluster seed iis defined to bedk:=di_k.

We attach the frozen nodes to two sides of the triangle, and we will usually display the arrowsk−→k⁺ inQⁱ_F in horizontal rows. The full quiverQⁱis constructed by addingn more frozen nodes

E :={N+n+ 1, ..., N+ 2n}

attached to the third side with multiplier dN+n+k := dk, and with additional arrows between them and QⁱF, but in this paper we do not need them, so we will not review the construction. Alternatively, the subquiverQⁱ_F can also be constructed from building blocks called theelementary quivers, see e.g. [14, 17].

Example 4.3. Our running example will be the quiver for type A3 (Figure 5a) and type B3(Figure 5b). For completeness we will show the full quiverQⁱ, but grayed out the part of the quiver outsideQⁱ_F which is not needed in the construction of this paper. We highlighted theFi-paths (cf. Definition 4.6) in red.

Definition 4.4. Given two basic quivers Qⁱ, Q^i′ associated to two reduced wordsi,i^′, we denote byQ^ii′ the quiver obtained by amalgamation along k7→φ(k) wherek∈ F^out⊂Qⁱ andφ(k)∈ Fⁱⁿ⊂Q^i′ such thati^′_φ(k)=ik. We denote by

X^ii′⊂ Xⁱ⊗ X^i′

the corresponding quantum torus algebra. Similarly, we let Qⁱⁱ_F^′ be the subquiver obtained by amalgamatingQⁱF andQⁱF^′ along the same mapφ, and let

XF^ii′⊂ X^ii′

be the corresponding subalgebra.

Definition 4.5. Given a pathP= (i1, ..., im)on the quiverQⁱwithik−→ik+1for every k, we write

X_P:=Xi₁,...,i_m (4.2)

for the product of all cluster variables along the path, and we define the path polynomial (note that we ignore the last index) by

X(P) :=X(i1, ..., im−1) :=

mX−1 k=1

Xi₁,...,i_k∈ Xⁱ. (4.3) Given two pathsP,P^′withim=i^′1, we simply denoted by

PP^′= (i1, ..., im=i^′1, ..., i^′m^′) the path inQ^ii′ obtained by concatenating the two paths.

Definition 4.6. An Fi-pathis the path on the quiverQⁱwhich starts fromFⁱⁿ and ends atF^outalong the horizontal path on the quiver with nodes{k:ik=i}(see e.g. Figure 5).

1

2 3

4 5

6 7

8

9

12 11 10

F_in F_out

E

(a) TypeA3 withi= (1,2,1,3,2,1)

1

2 3

4

5 6

7

8

9

10

11

12

13 14 15

F_in F_out

E

(b) TypeB3withi= (1,2,1,2,3,2,1,2,3)

Figure 5: The quiverQⁱ_F ⊂Qⁱattached to a triangle, and theF_i-paths highlighted in red.

5 Cluster realization of quantum groups

Taking Q :=Qⁱ as the basic quiver, we have an embedding ofU^q(g) into the quantum torus algebraXⁱⁱ (where the subquiverQ:=Qⁱis a mirror image ofQⁱ). In particular, the restriction to the Borel part Uqeq(b_R) generated by {Fi, K_i⁻¹}ⁱ∈I only requires the subalgebraXFⁱⁱ. Following [14], we will use

Ki^′:=K_i⁻¹ (5.1)

instead for typesetting purpose, which originally refers to the opposite Cartan generators of the Drinfeld’s double.

Leti= (i1, ..., iN) andi = (i1, ..., iN) := (iN, ..., i1). LetI ≃I ≃ {1, ..., N+ 2n}be the corresponding vertices ofQⁱ, Qⁱ, and by abuse of notation we write

ik:=i_k.

Let us define a bijectionσ:I−→I, k7→lsuch thatik=iland

|{j:ij=il, j < l}|={j:ij=ik, j < k}, (5.2) i.e. the indices appear in the same horizontal position in the quiverQⁱ, Qⁱfrom the left.

Recall the expression of the positive representationP^λ given in Theorem 2.4. If we redefine

f^k,−:=f^σ(k),− and rewrite the sum as

fi=f_i⁻+f_i⁺ (5.3)

= X

k:i_k=i

f^k,−+ X

k:ik=i

f^k,+,

then we observe that each monomial willq_i⁻² commute with all the terms to the right.

Define the consecutive ratios of thef^k,±monomials as X_k=





f^k,− k∈ Fⁱⁿ, q_i⁻¹f^k,−(f^k

−,−

)⁻¹ k /∈ Fⁱⁿ∪ F^out,

(5.4)

Xk=







q_i⁻¹f^k,+(f^k

∗,−

)⁻¹ k∈ Fⁱⁿ, q_i⁻¹f^k,+(f^k−,+)⁻¹ k /∈ Fⁱⁿ∪ F^out, q_i⁻¹K_i⁻¹

k (f^k∗,+)⁻¹ k∈ F^out.

(5.5)

LetPλ^b be the positive representationsP^λrestricted to the Borel partU^qqe(bR). Then from the explicit expression (2.17) of the operators and the commutation relations from Definition 4.2, we have

Theorem 5.1. [14] The assignment (5.4)-(5.5) gives a positive representation of XFⁱⁱ

generated by{X_k, Xk}^N+nk=1 in the sense of Definition 3.6.

Equivalently, letP^Fi,P^Fi be theFipaths ofQⁱandQⁱrespectively. Let

fi:=X(P^FiP^Fi)∈ XFⁱⁱ, (5.6) Ki^′:=X_P

FiP_Fi ∈ XFⁱⁱ

as in Definition 4.5. Then the assignment (5.4)-(5.5)coincides with the positive represen- tationPλ^b of the Borel part. We also have

f_i⁻=X(PF_i).

For the tensor productPλ^b⊗Pµ^b, the coproduct∆(fi),∆(Ki^′)are represented onXⁱⁱ⊗Xⁱⁱ in a similar way by concatenation of the Fi-paths of two copies of the quiver Qⁱⁱ, Q^′ii amalgamated alongF^out=Fin^′ .

On the other hand, we observe in [13] that the Borel part can alternatively be repre- sented using just half of the sum given by

fi= X

k:ik=i

f^k,+, (5.7)

which coincides with the so-calledFeigin’s homomorphism. Let us call this representation Pfλ^b (which we called it thestandard formin [13]).

Lemma 5.2. Pfλ^b≃Pf^b is independent of the parameter λ.

Proof. The proof is similar but easier than the one in [13]. From the explicit expression for Ki, since the Cartan matrix A = (aij) is invertible, we can always make a shift of variables

uk7→uk−ck

for some constantck, wherek∈ Fⁱⁿ, such that the parameterλivanishes in the expression of Ki, while each monomial term f^k,+ picks up some new parameters c^′k. Now since each f^k,+ has a unique momentum operator 2pk, by the unitary transformation given by multiplication bye^πiukc′k for some constantc^′k, which acts as

2pk7→2pk−c^′k, we can make the resulting parametersc^′k vanish.

Modifying (5.5) slightly with

Xk=







f^k,+ k∈ Fⁱⁿ,

q⁻¹_i f^k,+(f^k−,+)⁻¹ k /∈ Fⁱⁿ∪ F^out, q⁻¹_i K_i⁻¹(f^k∗,+)⁻¹ k∈ F^out,

(5.8)

we conclude that

Theorem 5.3. The assignment (5.8)gives a positive representation of the quantum torus algebraXFⁱ ⊂ Xⁱ associated toQⁱF. Equivalently, let

fi:=X(P^Fi)∈ XFⁱ, (5.9)

Ki^′:=X_PFi ∈ XFⁱ,

then the assignment (5.8)gives the representation Pf^b ofU^qqe(bR).

We also have the notion of anEi-path. Consider the quiverQⁱⁱD_2,1 associated to a once punctured disk with two marked points, obtained by gluingQⁱ, Qⁱalong F^out=Fⁱⁿ as well asE =E (see Figure 2). Recall that due to Theorem 2.4, the generators ei can be represented as a sum of monomials

ei=e⁻_i +e⁺_i.

In particular, ifiis well-chosen,ei can be represented again by a path polynomial (with slight modification in typeCn, E8, F4 andG2, see [14]) of the form

ei=X(P^EiP^Ei)∈ XDⁱⁱ2,1, (5.10) whereP^Ei andP^Eiare respectively a path on theQⁱ, Qⁱquivers, called theEi-paths. The pathP^Ei starts atF^outand ends atE, whileP^Ei starts atE and ends atFⁱⁿ. Similarly we have

e⁻_i =X(P^Ei).

6 Main results

We are now ready to give an alternative proof of the main theorem in [13].

Theorem 6.1. We have the following unitary equivalences of positive representations restricted to the Borel partU^qqe(bR):

Pλ^b≃Pf^b, (6.1)

Pf^b⊗Pf^b≃Pf^b⊗ M, (6.2)

for some multiplicity moduleM ≃L²(R^N)whereUqqe(b_R) acts trivially. This means that the positive representations restricted toU^qqe(bR) is closed under taking tensor product.

Recall that given an embedding of

U^q(g)֒→ XDⁱⁱⁱⁱ_2,2 ⊂ XDⁱⁱ_2,1⊗ XDⁱⁱ_2,1

into the quantum torus algebra associated to a disk with 2 punctures and 2 marked points given by the coproduct, we show in [14] that the reducedRoperator can be decomposed into

R=R4· R3· R2· R1, (6.3)

where

R4=gb_iN(e⁺i_N⊗f^N,+)...gb_i2(e⁺i₂⊗f^2,+)gb_i1(e⁺i₁⊗f^1,+), R3=gb_iN(e⁻_iN⊗f^N,+)...gb_i2(e⁻_i2⊗f^2,+)gb_i1(e⁻_i1⊗f^1,+), R2=gb_i1(e⁺i₁⊗f^1,−)gb_i2(e⁺i₂⊗f^2,−)...gb_iN(e⁺i_N ⊗f^N,−), R1=gb_i1(e⁻_i1⊗f^1,−)gb_i2(e⁻_i2⊗f^2,−)...gb_iN(e⁻_iN ⊗f^N,−).

AlsoR² commute with R³. By the correspondence in Lemma 3.9, the 4 factors corre- spond to sequences of quiver mutations associated to flipping of triangulation in different configurations. In this paper, we show explicitly the following fact, which is only implied implicitly in [14].

Theorem 6.2. The quiver mutations µ^q_R

k:=µ^qm₁· · ·µ^qm_M

induced byR^k, k = 1,3, corresponding to the flip of triangulation preserve the Fi-paths (see Figure 6). More precisely, for an amalgamationQ^ii′ of two quivers withi^′=i(k= 1) ori^′=i(k= 3)alongF^out=Fin^′ , ifP^Fi,PF^′_iare theFipaths of the left and right triangle respectively, andPb^Fi are the Fipaths of the bottom triangle of the mutated quiver, then

µ^q_R

k(X(b PbF_i)) = (X(PF_iPF^′_i)), k= 1,3, (6.4) where Xbj belongs to the quantum torus algebraXb^ii′ of the mutated quiver, amalgamated alongEb=Fbout^′ (see Figure 6).

F_in

F_out F_in^′

F_out^′ E^′ E

Qⁱ Q^i′

≃_µR

k

b E^′ b

F_out^′ Eb b

F_in^′

Fb_in Fb_out Qⁱ Q^i′

Figure 6: Configurations of the mutationsµR_k,k= 1,3, with theFipaths schemat- ically shown in red, and the irrelevant extremal nodes grayed out.

According to the result of [14], the quiver mutations corresponding toRkusing Lemma 3.9 are of the type as in Figure 7, and in the doubly-laced case also as in Figure 8. (The case of typeG2 was treated explicitly in [14] already and will be omitted.)

· · · ·

· · · ·

... ...

k⁻ k k⁺

l l⁺

≃_µk

· · · ·

· · · ·

... ...

l k l⁺

k⁻ k⁺

Figure 7: Mutation at simply-laced part.

· · · ·

· · · ·

... ...

k⁻ k k⁺

l⁻ l l⁺

≃_µlµkµl

· · · ·

· · · ·

... ...

k⁻ k k⁺

l⁻ l l⁺

Figure 8: Mutation at doubly-laced part.