Poisson structure and second quantization of quantum cluster algebras
Fang Li Jie Pan
math.RT math.QA math.RA 13F60, 46L65, 17B63


























TheoremTheorem[section]
Lemma[Theorem]Lemma
Definition[Theorem]Definition
Corollary[Theorem]Corollary
Proposition[Theorem]Proposition
Example[Theorem]Example
Remark[Theorem]Remark
Conjecture[Theorem]Conjecture
Problem[Theorem]Problem
Observation[Theorem]Observation
*Thmr1Theorem 2.5
*Thmr2Theorem 4.5
*Thmr3Theorem 5.5
*Thmr4Theorem 6.1

Proof[1][Proof]

  #1



























 
 
 


 




Poisson structure and second quantization of quantum cluster algebras]
Poisson structure and second quantization of 
quantum cluster algebras


Fang Li
Department of Mathematics, Zhejiang University (Yuquan Campus), Hangzhou, Zhejiang 310027, P. R. China
fangli@zju.edu.cn


Jie Pan
Department of Mathematics, Zhejiang University (Yuquan Campus), Hangzhou, Zhejiang 310027, P. R. China
panjie_zhejiang@qq.com

Mathematics Subject Classification(2010): 13F60, 46L65, 17B63
Keywords: quantum cluster algebra, compatible Poisson structure, second quantization





[
    Jie Pan
    version of December 30, 2023
================================



Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the second quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its secondly quantized cluster algebras. Based on this observation, we find that a quantum cluster algebra possesses dual quantum cluster algebras such that their second quantization is essentially the same.

As an example, we give the secondly quantized cluster algebra A_p,q(SL(2)) of Fun_(SL_q(2)) in 5.2.1 and show that it is a non-trivial second quantization, which may be realized as a parallel supplement to two parameters quantization of the general quantum group. Furthermore, we obtain a class of quantum cluster algebras with coefficients which possess a non-trivial second quantization. Its one special kind is quantum cluster algebras with almost principal coefficients with an additional condition.

Finally, we prove that the compatible Poisson structures of a quantum cluster algebra without coefficients is always a locally standard Poisson structure. Following this, it is shown that the second quantization of a quantum cluster algebra without coefficients is in fact trivial.









§ INTRODUCTION AND PRELIMINARIES


The introduction of quantum cluster algebras <cit.> is an important development of the theory of cluster algebras, which establishes a connection between cluster theory and the theory of quantum groups.

The theory of quantum groups comes from theoretic physics. It first appears in the inverse scattering method used to construct and solve quantum integrable systems and then is developed by Drinfield, Jimbo, etc. However, up to now, there is no unified definition for a quantum group.
Considering that the structures of quantum torus for each seed of a quantum cluster algebra are some kind of quantum groups, we can regard a quantum cluster algebra as a family of quantum groups with mutation actions.


It is often a complicated work to explore the quantization of a concrete Lie algebra or an associative algebra. Moreover, the complexity of some problems faced by mathematicians and physicists leads to the demand of two-parameters and multi-parameters quantum groups.

But the analogue of quantum groups with two or even multiple parameters had not been set up in the theory of cluster algebras until recently. It is still an open problem how to properly define quantum cluster algebras with two or even multiple parameters. For example, it is natural to consider to give the definition by simply adding one parameter to quantum cluster algebras in the form as follows:

    X_iX_j=p^δ_ijq^λ_ijX^e_i+e_j.

There has been interesting researches given in <cit.> and <cit.> focusing on this aspect. In <cit.>, the authors generalizes original quantum cluster algebras to such quantum cluster algebras with more than one parameters, and then proves that a very large class of quantum nilpotent algebras, defined axiomatically, admit this quantum cluster algebra structures. And in <cit.>, such quantum cluster algebras, called Toroidal cluster algebras, are found a profound relationship with quantum affine algebras under the sense of categorification.

However, on the other hand, as pointed out in <cit.> and <cit.>, the parameters δ_ij and λ_ij in (<ref>) are independent, which means a two-parameters quantum cluster algebra so defined there is essentially determined by two parallel one-parameter quantum cluster algebras. In a sense, we may think such defined quantum cluster algebras with multiple parameters are “ trivial ". Hence, the new question is:

How to give a more non-trivial definition of quantum cluster algebras with two or multiple parameters in the above sense?

From our observation, the key lies in that how to meaningfully connect δ_ij with λ_ij to make them not independent to each other. It is the main aim of this paper. We will focus on the correspondence between the quantization of a cluster algebra and the compatible Poisson structure on it(<cit.>,<cit.>), and lift this fact to non-commutative level so as to define a kind of quantum algebras with two parameters, which are not trivial in the above sense. We call such algebras as secondly quantized cluster algebras. We can continue this way to discuss the possibility of higher quantization, which will be mentioned in the sequel.


Poisson geometry and its related Poisson algebra came from mechanics in nineteenth century named after the famous mathematical physicist S.D.Poisson. Due to its connections with classical mechanics, symplectic geometry, basing on integrable system and the correspondence between the quantization of a cluster algebra and the compatible Poisson structure on the cluster algebra, we will be able to find the background of (secondly) quantized cluster algebras in these research areas.

For n⩽ m∈ℕ, denote 𝕋_n the n-regular tree with vertices t∈𝕋_n.
Let ℱ be the field of rational functions over ℚ in m independent variables.

(1)  A seed at vertex t∈𝕋_n is a pair Σ=(𝒳̃(t),B̃(t)) such that

∙  𝒳̃(t)=(x_1;t,x_2;t,⋯,x_m;t) is an m-tuple satisfying that the elements form a free generating set of ℱ;

∙  B̃(t) is an m× n integer matrix such that the principal part is skew-symmetrizable, i.e. there is a positive diagonal matrix D satisfying DB(t) is skew-symmetric, where B(t) is the first n rows of B̃(t).

(2)  For any k∈[1,n], define the mutation μ_k at direction k satisfying that μ_k(Σ)=Σ^'=(𝒳̃^',B̃^'),

    μ_k(x_k;t)=∏_i=1^mx_i;t^[b_ik^t]_++∏_i=1^mx_i;t^[-b_ik^t]_+/x_k;t

where [a]_+=max{a,0} for a∈ℝ. And

    𝒳̃^'=(𝒳̃(t))\{x_k;t})⋃{μ_k(x_k;t)}.


    B̃^'=μ_k(B̃(t))=(b_ij^')_m× n
 satisfying that

    b_ij^'={[                               -b_ij(t)                   if   i=k    or   j=k; b_ij(t)+sgn(b_ik(t))[b_ik(t)b_kj(t)]_+                              otherwise ] .


It can be proved that μ_k is an involution.

Given seeds Σ(t)=(𝒳̃(t),B̃(t)) at t∈𝕋_n so that Σ(t^')=μ_k(Σ(t)) for any t-t^' in 𝕋_n connected by an edge labeled k∈[1,n], then the ℤ[x_n+1^±1,⋯,x_m^± 1]-subalgebra of ℱ generated by all variables in ⋃_t∈𝕋_n𝒳(t) is called the cluster algebra A(Σ) (or simply A) associated with Σ.


Besides, we also introduce the concept of quantum cluster algebras. For a vertex t_0∈𝕋_n, let Λ(t_0)=(λ_ij)_m× m be a skew-symmetric integer matrix satisfying

    B̃(t_0)^⊤Λ(t_0)=
    [ D O ]_n× mC1

Then (B̃(t_0),Λ(t_0)) is called a compatible pair. Let {e_i}_i=1^m be the standard basis for ℤ^m. Define a skew-symmetric bilinear form Λ_t_0:ℤ^m×ℤ^m→ℤ satisfying that

    Λ_t_0(e,f)=∑_i,j=1^ma_ib_jΛ_t_0(e_i,e_j)=∑_i,j=1^ma_ib_jλ_ij,

where e=∑_i=1^ma_ie_i,f=∑_j=1^mb_je_j.

Give a set of variables

    X̃(t_0)={X_t_0^e_1,⋯,X_t_0^e_n,X^e_n+1,⋯,X^e_m}

called the (extended) cluster at t_0, where X_t_0^e_i,i∈[1,n] are called the cluster variables at t_0 while X^e_i,i∈[n+1,m] are called frozen variables.

For the Laurent polynomial ring ℤ[q^±1/2] with a formal variable q, define a ℤ[q^±1/2]-algebra T_t_0 generated by X(t_0) satisfying the following relations:

    X_t_0^e_iX_t_0^e_j=q^1/2λ_ijX_t_0^e_i+e_j,∀ i,j∈[1,m]

We call T_t_0 the quantum torus at t_0. Denoted by ℱ_q the skew-field of fractions of T_t_0.

In general, for any e∈ℤ^m, let X_t_0^e denote the variable corresponding to e. Due to the bilinearity of Λ_t_0 and the fact that e is generated by {e_i| i∈[1,m]}, we obtain that

    X_t_0^eX_t_0^f=q^1/2Λ_t_0(e,f)X_t_0^e+f



[<cit.>]
(i)  Given a fixed t_0∈𝕋_n, we denote Σ(t_0)=(X̃(t_0),B̃(t_0),Λ(t_0)) an initial quantum seed.

(ii)  Let t∈𝕋_n be an adjacent vertex of t_0, i.e. t-t_0 is an edge in 𝕋_n labeled k∈[1,n]. Let b_k(t_0) be the k-th column of B̃(t_0). Define the mutation μ_k at direction k satisfying that

    X_t^e_k=μ_k(X_t_0^e_k)=X_t_0^-e_k+[b_k(t_0)]_++X_t_0^-e_k+[-b_k(t_0)]_+

such that

    X̃(t)=(X̃(t_0)\{X^e_k(t_0)})⋃{X_t^e_k}.

where B̃(t)=μ_k(B̃(t_0)) is the same as that according to (<ref>).
And, Λ(t)=μ_k(Λ(t_0))=(λ_ij(t))_m× m where

    λ_ij(t)={[ -λ_kj(t_0)+∑_l=1^m[b_lk(t_0)]_+λ_lj(t_0)                              if   i=k≠ j; -λ_ik(t_0)+∑_l=1^m[b_lk(t_0)]_+λ_il(t_0)                              if   j=k≠ i;                                λ_ij(t_0)                                otherwise ] .



It can be proved that the seed Σ(t)=(X̃(t),B̃(t),Λ(t)) at t still satisfies the relation (<ref>) and (<ref>) and μ_k is an involution.

It can be seen that in the quantum case, because of the relation (<ref>), B̃(t) is always of full column rank n.

(.<cit.>.) Given seeds Σ(t)=(X̃(t),B̃(t),Λ(t)) at t∈𝕋_n, if Σ(t) and Σ(t^') can do mutation to each other for any adjacent pair of vertices t-t^' in 𝕋_n, then the ℤ[q^±1/2][X^± e_n+1,⋯,X^± e_m]-subalgebra of ℱ_q generated by all variables in ⋃_t∈𝕋_nX(t) is called the quantum cluster algebra A_q(Σ) (or simply A_q) associated with Σ.


We call Λ(t) the deformation matrix of this quantum cluster algebra A_q at t∈𝕋_n.


Now we explain the relationship between cluster algebras in non-quantum and quantum cases from their definitions.

For a quantum cluster algebra A_q, let q→ 1. Then we obtain a non-quantum cluster algebra A, which is called the correspondent classical version (briefly, CCV) of A_q; conversely, A_q is called the correspondent quantum version (briefly, CQV) of A. Under this relationship, the (extended) clusters X̃(t)={X_t^e_1, ⋯, X_t^e_m} in A_q and the (extended) clusters 𝒳̃(t)={x_1;t, ⋯, x_m;t} in A correspond to each other.

Note that since the rank of B̃(t) of A_q is always n as mentioned above, only those cluster algebras A whose B̃(t) are of rank n have the correspondent quantum versions.

Due to our motivation for this work, we will first discuss Poisson structures on quantum cluster algebras. So here, we recall the concepts and notations of Poisson structures.

A Poisson structure on an associative k-algebra  is a triple (,·,{-,-}) where (,{-,-}) is a Lie k-algebra i.e. satisfying Jacobi identity such that the Leibniz rule holds: for any a,b,c∈,

    {a,bc}={a,b}c+b{a,c}.

Algebra  together with a Poisson structure on it is called a Poisson algebra. Denote the Hamiltonian of a∈ by

    ham(a)={a,-}∈ End_k(,).

Then the Leibniz rule is equivalent to that ham(a) is a derivation of  as an associative algebra for any a∈.

Let  be an associative algebra. [a,b]=ab-ba is called the commutator of a and b, for any a,b∈. And for any λ∈ k, (,·,λ[-,-]) is a Poisson algebra called a standard Poisson structure on (,·).



As we know, so far only the Poisson structure of (non-quantum) cluster algebras has been studied, e.g. see <cit.> and <cit.>. We recall the following notions from <cit.>:

(1)  For a cluster algebra A, one of its extended cluster 𝒳̃=(x_1,⋯,x_m) is said to be log-canonical with respect to a Poisson structure (A,·,{-,-}) if {x_i,x_j}=ψ_ijx_ix_j, where ψ_ij∈ℤ for any i,j∈[1,m].

(2)  A Poisson structure {-,-} on a cluster algebra A is called compatible with A if all clusters in A are log-canonical with respect to { -,-}. In this case, the matrix Ψ=(ψ_ij)_m× m is called the Poisson matrix associated to cluster 𝒳̃ (with respect to the Poisson structure).

From <cit.>, we know that a compatible Poisson structure is given on a cluster algebra A via a family of Ψ(t) (t∈𝕋_n) as Poisson matrices such that following mutation formula of Ψ(t) holds for each adjacent vertex pair (t, t') in 𝕋_n connected by an edge labeled k:


    ψ_ij(t^')={[ -ψ_kj(t)+∑_l=1^m[b_lk(t)]_+ψ_lj(t)                         if  i=k≠ j; -ψ_ik(t)+∑_l=1^m[b_lk(t)]_+ψ_il(t)                         if  j=k≠ i;                            ψ_ij(t)                          otherwise ] .

And for a compatible Poisson structure, there is always B̃^⊤(t)Ψ(t)=(D0) for any t∈𝕋_n.

We similarly define the compatibility of non-commutative Poisson structures on a quantum cluster algebra.


(1)  For a quantum cluster algebra A_q, one of its extended cluster X̃(t)=(X_1,⋯,X_m) at t∈𝕋_n is said to be log-canonical with respect to a Poisson structure (A_q,·,{-,-}) if {X_i,X_j}=ω_ijX^e_i+e_j, where ω_ij∈ℤ[q^±1/2] for any i,j∈[1,m]. In this case, the matrix Ω(t)=(ω_ij)_m× m is called the Poisson matrix associated to the cluster X̃(t) (with respect to the Poisson structure).

(2)  A Poisson structure {-,-} on a quantum cluster algebra A_q is called compatible with A_q if all clusters in A_q are log-canonical with respect to { -,-}.


Trivially, Poisson matrices are always skew-symmetric for either commutative or non-commutative cases.


The paper is organized as follows.

In Section 2 we give the mutation formula of Poisson matrices in a quantum cluster algebra A_q (Theorem <ref>) and the equivalent characterization for two adjacent clusters to be log-canonical with a Poisson structure on A_q (Theorem <ref>).

In Section 3, we introduce in Definition <ref> the concept of the second quantization of a quantum cluster algebra A_q based on the correspondence between Poisson matrices and (second) deformation matrices, which means the correspondence between compatible Poisson structures and (secondly) quantized cluster algebras. And Proposition <ref> is given here to show that a second deformation matrix must satisfy (<ref>) which is the same as (<ref>). Following this proposition, we find in Theorem <ref> that a quantum cluster algebra possesses dual quantum cluster algebras such that their second quantization are essentially the same.

In Section 4, we prove Lemma <ref>, in which the condition (C4) for all seeds is transformed to the condition Λ̂B̃=cD̃ for an arbitrary seed.

In Section 5, we introduce the cluster decomposition of a quantum cluster algebra. Then with the help of Lemma <ref>, the compatibility of a Poisson structure comes down to the cluster indecomposable case, see Proposition <ref>. And so does second quantization (Theorem <ref>).

As an example, we give the secondly quantized cluster structure A_p,q(SL(2)) of Fun_(SL_q(2)) in 5.2.1 and show that it is a non-trivial second quantization, which may be realized as a parallel supplement to two parameters quantization of the general quantum group. After that, via a cluster extension, we obtain a class of quantum cluster algebras which possess a non-trivial second quantization (Theorem <ref>). One special kind of it is quantum cluster algebras with almost principal coefficients with an additional condition (Corollary <ref>).

In Section 6, it is shown that a Poisson structure is compatible with a quantum cluster algebra without coefficients if and only if it is a locally standard Poisson structure (Theorem <ref>). Therefore the second quantization of a quantum cluster algebra without coefficients is always trivial (Corollary <ref>)






§ COMPATIBLE POISSON STRUCTURES ON QUANTUM CLUSTER ALGEBRAS AND MUTATION OF POISSON MATRICES


In <cit.>, compatible Poisson structures on cluster algebras are characterized and moreover, such structures are constructed on Grassmannians. In this section and Section 5, we will discuss Poisson structures compatible with a quantum cluster algebra.


Here and in the following, we always assume that in a quantum cluster algebra A_q, the initial quantum seed at t_0 is (X̃,B̃,Λ), where X̃=(X_1,X_2,…,X_m) (denote X_i=X^e_i,i∈[1,m]) with the first n variables mutable, B̃ is an m × n skew-symmetrizable integer matrix with skew-symmetizer D and Λ is an m × m skew-symmetric matrix such that (B̃,Λ) is a compatible pair.

First of all, notice that if {-,-} is trivial, i.e.{X,Y}=0 for any X,Y∈ A_q, then ω_ij are all 0, thus it is naturally compatible with A_q. Therefore in the following we only consider about nontrivial Poisson structures.

For a quantum cluster algebra A_q, if its quantum seed X̃ and μ_k(X̃) are log-canonical with a nontrivial Poisson structure {-,-} and the Poisson matrices associated to them are Ω=(ω_ij)_m× m and Ω^'=(ω_ij^')_m× m respectively, then

(1)  for any j≠ k, where j∈[1,m] while k∈ [1,n], we have

    [     ∑_b_tk>0(ω_tjq^1/2λ_jt∑_h=1^[b_tk]_+q^∑_i=t^m([b_ik]_+-δ_ik)λ_ji-hλ_jt)-ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[b_ik]_+; =∑_b_tk<0(ω_tjq^1/2λ_jt∑_h=1^[-b_tk]_+q^∑_i=t^m([-b_ik]_+-δ_ik)λ_ji-hλ_jt)-ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[-b_ik]_+ ]

where δ_ij equals 1 when i=j and 0 otherwise.

(2)  the mutation formula of Poisson matrices Ω in direction k is given as follows:

    ω_ij^'=
    {[ q^1/2(λ_jk-∑_t=1^m[b_tk]_+λ_jt)H                      if   i=k≠ j;                          -ω_ki^'                      if   j=k≠ i;                             ω_ij                        otherwise ] .

where H denotes the left or right side of (<ref>).




(1)  For any k∈[1,n], let X̃^'=μ_k(X̃)=(X_1,⋯,X_k-1,X_k^',⋯,X_m). By the assumption, X̃^' is log-canonical with respect to {-,-}. Therefore {X_k^',X_j}=ω_kj^'X^'^e_k+e_j for some ω_kj^'∈ℤ[q^±1/2] for any j≠ k∈[1,m].

By the exchange relations for quantum cluster algebras, we obtain that

{X_k^',X_j}
={X^-e_k+∑_b_ik>0b_ike_i, X_j} + {X^-e_k-∑_b_ik<0b_ike_i, X_j}

    =q^1/2∑_l>hλ_lh([b_lk]_+-δ_lk)([b_hk]_+-δ_hk){∏_i=1^mX_i^[b_ik]_+-δ_ik, X_j}
    +q^1/2∑_l>hλ_lh([-b_lk]_+-δ_lk)([-b_hk]_+-δ_hk){∏_i=1^mX_i^[-b_ik]_+-δ_ik,X_j},

and

{∏_i=1^mX_i^[b_ik]_+-δ_ik,X_j}

=∑_b_tk>0∏_i=1^t-1X_i^[b_ik]_+-δ_ik{X_t^[b_tk]_+,X_j}∏_i=t+1^mX_i^[b_ik]_+-δ_ik-∏_i=1^k-1X_i^[b_ik]_+{X_k^-1,X_j}∏_i=k+1^mX_i^[b_ik]_+ 

=∑_b_tk>0∑_h=0^[b_tk]_+-1∏_i=1^t-1X_i^[b_ik]_+-δ_ikX_t^h{X_t,X_j}X_t^[b_tk]_+-h-1∏_i=t+1^mX_i^[b_ik]_+-δ_ik

.                       -∏_i=1^k-1X_i^[b_ik]_+X_k^-1{X_k,X_j}X_k^-1∏_i=k+1^mX_i^[b_ik]_+      (say ∏_i=1^0M = 1,∀ M)

=∑_b_tk>0∑_h=0^[b_tk]_+-1ω_tj∏_i=1^t-1X_i^[b_ik]_+-δ_ikX_t^hX^e_t+e_jX_t^[b_tk]_+-h-1∏_i=t+1^mX_i^[b_ik]_+-δ_ik
-ω_kj∏_i=1^k-1X_i^[b_ik]_+X_k^-1X^e_k+e_jX_k^-1∏_i=k+1^mX_i^[b_ik]_+

=∑_b_tk>0∑_h=0^[b_tk]_+-1ω_tjq^1/2λ_jt∏_i=1^t-1X_i^[b_ik]_+-δ_ikX_t^h+1X_jX_t^[b_tk]_+-h-1∏_i=t+1^mX_i^[b_ik]_+-δ_ik
-ω_kjq^1/2λ_kj∏_i=1^k-1X_i^[b_ik]_+X_k^-1X_j∏_i=k+1^mX_i^[b_ik]_+

=(∑_b_tk>0∑_h=0^[b_tk]_+-1ω_tjq^1/2λ_jt+([b_tk]_+-h-1)λ_jt+∑_i=t+1^mλ_ji([b_ik]_+-δ_ik)-ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[b_ik]_+)∏_i=1^mX_i^[b_ik]_+-δ_ikX_j

=(∑_b_tk>0∑_h=1^[b_tk]_+ω_tjq^1/2λ_jt+∑_i=t^mλ_ji([b_ik]_+-δ_ik)-hλ_jt-ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[b_ik]_+)∏_i=1^mX_i^[b_ik]_+-δ_ikX_j.



And similarly,

    {∏_i=1^mX_i^[-b_ik]_+-δ_ik,X_j}=(∑_b_tk<0∑_h=1^[-b_tk]_+ω_tjq^1/2λ_jt+∑_i=t^mλ_ji([-b_ik]_+-δ_ik)-hλ_jt-ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[-b_ik]_+)∏_i=1^mX_i^[-b_ik]_+-δ_ikX_j

Therefore,

    [                                                                                                            {X_k^',X_j}=      q^1/2∑_l>hλ_lh([b_lk]_+-δ_lk)([b_hk]_+-δ_hk)(∑_b_tk>0∑_h=1^[b_tk]_+ω_tjq^1/2λ_jt+ ∑_i=t^mλ_ji([b_ik]_+-δ_ik)-hλ_jt;                                                                                                                                                                                -ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[b_ik]_+)∏_i=1^mX_i^[b_ik]_+-δ_ikX_j;                                                                                                                         +q^1/2∑_l>hλ_lh([-b_lk]_+-δ_lk)([-b_hk]_+-δ_hk)(∑_b_tk<0∑_h=1^[-b_tk]_+ω_tjq^1/2λ_jt+ ∑_i=t^mλ_ji([-b_ik]_+-δ_ik)-hλ_jt;                                                                                                                                                                              -ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[-b_ik]_+)∏_i=1^mX_i^[-b_ik]_+-δ_ikX_j ]

=(∑_b_tk>0∑_h=1^[b_tk]_+ω_tjq^1/2λ_jt+∑_i=t^mλ_ji([b_ik]_+-δ_ik)-hλ_jt-ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[b_ik]_+)X^-e_k+∑_b_ik>0b_ike_iX_j

    +(∑_b_tk<0∑_h=1^[-b_tk]_+ω_tjq^1/2λ_jt+∑_i=t^mλ_ji([-b_ik]_+-δ_ik)-hλ_jt-ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[-b_ik]_+)X^-e_k-∑_b_ik<0b_ike_iX_j.    ()


On the other hand,

    ω_kj^'X^'^e_k+e_j
    =ω_kj^'q^1/2λ_jk^'X_k^'X_j
    =ω_kj^'q^1/2(∑_t=1^m[b_tk]_+λ_jt-λ_jk)(X^-e_k+∑_b_ik>0b_ike_i+X^-e_k-∑_b_ik<0b_ike_i)X_j.


Since {X_k^',X_j}=ω_kj^'X^'^e_k+e_j and the fact that
cluster Laurent monomials in a cluster X are ℤ[q^±1/2]-linear independent, we compare the coefficients of the corresponding cluster Laurent monomials in () and the right-side of (<ref>), it follows that

∑_b_tk>0∑_h=1^[b_tk]_+ω_tjq^1/2λ_jt+∑_i=t^mλ_ji([b_ik]_+-δ_ik)-hλ_jt-ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[b_ik]_+= ω_kj^'q^1/2(∑_t=1^m[b_tk]_+λ_jt-λ_jk), 

also, ∑_b_tk<0∑_h=1^[-b_tk]_+ω_tjq^1/2λ_jt+∑_i=t^mλ_ji([-b_ik]_+-δ_ik)-hλ_jt-ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[-b_ik]_+= ω_kj^'q^1/2(∑_t=1^m[b_tk]_+λ_jt-λ_jk).

So (<ref>) is satisfied.

(2)  It can also be seen from above equations that when (<ref>) is satisfied, ω_kj^'=q^1/2(λ_jk-∑_t=1^m[b_tk]_+λ_jt)H. Similar for ω_ik^'. And as X_i,X_j do not change in mutation at direction k when i,j≠ k, ω_ij^'=ω_ij.



A cluster algebra can be regarded as a quantum cluster algebra with q=1 or Λ=0. Then we can see in this case that above mutation formula of Poisson matrices coincides with the mutation formula of Poisson matrices for a cluster algebra in (<ref>).


In the sequel, we will always use r/s=u/v to represent rv=su no matter s or v equals to 0 or not.

Under the same condition as that of Theorem <ref>, let u, v, j∈ [1,m], k∈[1,n] and j≠ k.

(i) If b_uk≠0, then ω_uj/ω_kj=q^1/2λ_u j-q^1/2λ_ju/q^1/2λ_kj-q^1/2λ_jk.

(ii) If b_ukb_vk≠0, then ω_uj/ω_vj=q^1/2λ_uj-q^1/2λ_ju/q^1/2λ_vj- q^1/2λ_jv.


In the proof of Theorem <ref>, if we choose {p_1,⋯,p_m} another permutation of [1,m] instead of [1,m] in (<ref>), we will finally obtain an equation different from (<ref>) as

    {X^-e_k+∑_b_ik>0b_ike_i,X_j}
    =q^1/2∑_l>hλ_p_lp_h([b_p_lk]_+-δ_p_lk)([b_p_hk]_+-δ_p_hk){∏_i=1^mX_p_i^[b_p_ik]_+-δ_p_ik,X_j}

Denote p^-1(k)=i∈ [1,m] when p_i=k. We have



{∏_i=1^mX_p_i^[b_p_ik]_+-δ_p_ik,X_j}

=∑_b_p_tk>0∏_i=1^t-1X_p_i^[b_p_ik]_+-δ_p_ik{X_p_t^[b_p_tk]_+,X_j}∏_i=t+1^mX_p_i^[b_p_ik]_+-δ_p_ik-∏_i=1^p^-1(k)-1X_p_i^[b_p_ik]_+{X_k^-1,X_j}∏_i=p^-1(k)+1^mX_p_i^[b_p_ik]_+ 

=∑_b_p_tk>0∑_h=0^[b_p_tk]_+-1∏_i=1^t-1X_p_i^[b_p_ik]_+-δ_p_ikX_p_t^h{X_p_t,X_j}X_p_t^[b_p_tk]_+-h-1∏_i=t+1^mX_p_i^[b_p_ik]_+-δ_p_ik

.                                                         -∏_i=1^p^-1(k)-1X_p_i^[b_p_ik]_+X_k^-1{X_k,X_j}X_k^-1∏_i=p^-1(k)+1^mX_p_i^[b_p_ik]_+

=∑_b_p_tk>0∑_h=0^[b_p_tk]_+-1ω_p_tj∏_i=1^t-1X_p_i^[b_p_ik]_+-δ_p_ikX_p_t^hX^e_p_t+e_jX_p_t^[b_p_tk]_+-h-1∏_i=t+1^mX_p_i^[b_p_ik]_+-δ_p_ik

.                                                        
-ω_kj∏_i=1^p^-1(k)-1X_p_i^[b_p_ik]_+X_k^-1X^e_k+e_jX_k^-1∏_i=p^-1(k)+1^mX_p_i^[b_p_ik]_+

=(∑_b_p_tk>0ω_p_tjq^1/2λ_jp_t∑_h=1^[b_p_tk]_+q^∑_i=t^m([b_p_ik]_+-δ_p_ik)λ_jp_i-hλ_jp_t-ω_kjq^1/2λ_kj+∑_i=p^-1(k)+1^mλ_jp_i[b_p_ik]_+)∏_i=1^mX_p_i^[b_p_ik]_+-δ_p_ik. ()



Thus we replace () into the first term of the right-side of (<ref>) in the proof of Theorem <ref> and calculate as we did there, it follows that

    {X_k^',X_j}
    
    =     (∑_b_p_tk>0∑_h=1^[b_p_tk]_+ω_p_tjq^1/2λ_jp_t+∑_i=t^mλ_jp_i([b_p_ik]_+-δ_p_ik)-hλ_jp_t-ω_kjq^1/2λ_kj+∑_i=p^-1(k)+1^mλ_jp_i[b_p_ik]_+)X^-e_k+∑_b_ik>0b_ike_iX_j
        +(∑_b_tk<0∑_h=1^[-b_tk]_+ω_tjq^1/2λ_jt+∑_i=t^mλ_ji([-b_ik]_+-δ_ik)-hλ_jt-ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[-b_ik]_+)X^-e_k-∑_b_ik<0b_ike_iX_j.

Again, because of (<ref>) and comparing coefficients of cluster Laurent monomials in X due to their ℤ[q^±1/2]-linear independence, analogue to the proof of Theorem <ref> (1), we get that


    [ ∑_b_p_tk>0(ω_p_tjq^1/2λ_jp_t∑_h=1^[b_p_tk]_+q^∑_i=t^m([b_p_ik]_+-δ_p_ik)λ_jp_i-hλ_jp_t)-ω_kjq^1/2λ_kj+∑_i=p^-1(k)+1^mλ_jp_i[b_p_ik]_+;                       =∑_b_tk<0(ω_tjq^1/2λ_jt∑_h=1^[-b_tk]_+q^∑_i=t^m([-b_ik]_+-δ_ik)λ_ji-hλ_jt)-ω_kjq^1/2λ_kj+∑_i=k+1^mλ_ji[-b_ik]_+ ]


Now we can prove (i) in the case b_uk>0. The other case is similar.

In (<ref>), choose the permutation (p_1,⋯,p_m) to be (1,⋯,û,⋯,k̂,⋯,m,u,k) and (1,⋯,û,⋯,k̂, ⋯,m,k,u) respectively (û means the absence of u, etc.), we get two equations whose right-sides are the same as that of (<ref>). Subtracting these two equations, we have:

    ω_u jq^1/2λ_ju∑_h=1^b_u kq^(b_u k-h)λ_ju-λ_jk-ω_k jq^1/2λ_kj
    =ω_u jq^1/2λ_ju∑_h=1^b_u kq^(b_u k-h)λ_ju-ω_k jq^1/2λ_kj+b_u kλ_ju

Therefore, we have

    ω_u j/ω_kj
    =q^1/2λ_kj(1-q^b_u kλ_ju)/q^1/2λ_ju∑_h=1^b_u kq^(b_u k-h)λ_ju(q^-λ_jk-1)
    =q^-1/2λ_j u(1-q^λ_ju)(1-q^b_u kλ_ju)/q^-1/2λ_jk(1-q^b_u kλ_ju)(q_-λ_jk-1)
    =q^1/2λ_uj-q^1/2λ_ju/q^1/2λ_kj-q^1/2λ_jk.


Next, we prove (ii) in the case b_u kb_vk>0. We only proof the case b_u k,b_vk>0, the other case is similar.

Similarly, in (<ref>), choose the permutation (p_1,⋯,p_m) to be
(1,⋯,û,⋯,v̂,⋯,m,u,v) and (1,⋯,û,⋯,v̂, ⋯,m,v,u)
respectively, we get two equations whose right-sides are the same as that of (<ref>). Subtracting these two equations, we have:


    ω_u jq^1/2λ_ju∑_h=1^b_u kq^(b_u k-h)λ_ju+b_v kλ_jv+ω_v jq^1/2λ_jv∑_h=1^b_vkq^(b_vk-h)λ_jv



    =ω_u jq^1/2λ_ju∑_h=1^b_u kq^(b_u k-h)λ_ju+ω_v jq^1/2λ_jv∑_h=1^b_vkq^(b_vk-h)λ_jv+b_u kλ_ju.

Thus,

    ω_u j/ω_vj
    =q^1/2λ_jv∑_h=1^b_vkq^(b_vk-h)λ_jv(q^b_u kλ_ju-1)/q^1/2λ_ju∑_h=1^b_u kq^(b_u k-h)λ_ju(q^b_v kλ_jv-1)
    =q^-1/2λ_j u(1-q^b_vkλ_jv)(q_b_u kλ_ju-1)(1-q^λ_ju)/q^-1/2λ_j v(1-q^b_u kλ_ju)(q_b_v kλ_jv-1)(1-q^λ_jv)
    =q^1/2λ_u j-q1/2λ_ju/q^1/2λ_vj-q^1/2λ_jv.


Moreover, consider the case for b_ukb_vk<0. If λ_kj≠ 0 or b_uv≠ 0, then we can obtain the result we want by the first part of this Lemma. If λ_kj=b_uv=0, take a mutation at direction k. After mutation, λ^'_uj=λ_uj, λ_vj^'=λ_vj, ω_uj^'=ω_uj and ω_vj^'=ω_vj, but b_uv^'=b_ukb_kv≠ 0. Hence again by the first part (i) of this lemma, we have
ω_u j/ω_vj=q^1/2λ_u j-q1/2λ_ju/q^1/2λ_vj-q^1/2λ_jv.



It follows from Lemma <ref> that


  * If λ_kj≠ 0, then ω_u j=a(q^1/2λ_u j-q^1/2λ_ju), where a∈ℤ[q^±1/2] for any b_u k≠ 0.

  * If λ_vj≠ 0 and b_vk≠ 0, then ω_u j=a(q^1/2λ_u j-q^1/2λ_ju), where a∈ℤ[q^±1/2] for any b_u k≠ 0.



The following theorem turns (<ref>) equivalently into the collection of three conditions (C2'), (C3'), (C4'), which are easier to deal with for us.

If X̃ is log-canonical with a Poisson structure {-,-} on a quantum cluster algebra A_q and {X_i,X_j}=ω_ijX^e_i+e_j for any i,j∈[1,m], then μ_k(X̃) is log-canonical with it if and only if the following conditions hold for any j∈[1,m],k∈[1,n],k≠ j:



  (C2') For any u∈ [1,m], if b_uk≠0, then ω_uj/ω_kj=q^1/2λ_u j-q^1/2λ_ju/q^1/2λ_kj-q^1/2λ_jk.

  (C3') For any u,v∈ [1,m], if b_ukb_vk≠ 0, then ω_uj/ω_vj=q^1/2λ_uj-q^1/2λ_ju/q^1/2λ_vj- q^1/2λ_jv.

  (C4') ∑_t:λ_tj=0ω_tjb_tk=0.



For the necessary part, Lemma <ref> claim the first two conditions. Combining these with equations (<ref>), we can reach the third one case by case:

Case 1: λ_kj≠ 0. Then ω_tj=0 if b_tk≠ 0 and λ_tj=0, thus ∑_λ_tj=0ω_tjb_tk=0.

Case 2: λ_kj= 0 and there are u,v∈[1,m] such that b_uk>0,b_vk<0,λ_ujλ_vj≠ 0. Like case 1, ω_tj=0 if b_tk≠ 0 and λ_tj=0.

Case 3: λ_kj= 0, there is u such that b_uk>0,λ_uj≠ 0 and for any v such that b_vk<0, λ_vj=0. Then ω_tj=0 if b_tk>0 and λ_tj=0 and equations (<ref>) can be simplified as

    a_u(1-q^∑_t=1^m[b_tk]_+λ_jt)=∑_tω_tj[-b_tk]_+

Because ∑_t=1^m[b_tk]_+λ_jt=∑_tb_tkλ_jt=0, we have ∑_tω_tj[-b_tk]_+=0. Therefore ∑_λ_tj=0ω_tjb_tk=0.

Case 4: λ_kj= 0, there is v such that b_vk<0,λ_vj≠ 0 and for any u such that b_uk>0, λ_uj=0. Similar to case 3.

Case 5: λ_kj= 0 and for any u such that b_uk≠ 0, λ_uj=0. Then (<ref>) looks like

    ∑_tω_tj[b_tk]_+=∑_tω_tj[-b_tk]_+

thus ∑_λ_tj=0ω_tjb_tk=0.

The sufficent part can be seen by direct calculations. Once these conditions are true, the formula (<ref>) holds.






§ PHILOSOPHY OF SECOND QUANTIZATION FOR QUANTUM CLUSTER ALGEBRAS


For the family of Poisson matrices Ψ(t) (t∈𝕋_n) of a Poisson structure on a cluster algebra A, it is interesting to note that their mutation formula in (<ref>) is the same as that in (<ref>) for the deformation matrices Λ(t) of a quantum cluster algebra A_q. From this fact, the relation between the Poisson structures of a cluster algebra and the quantization of this algebra can be given below.

For a quantum cluster algebra A_q=A_q(Σ) with seeds Σ(t)=(X(t),B̃(t),Λ(t)) at t∈𝕋_n, whose CCV is the cluster algebra A, let Ψ(t)=Λ(t), then we obtain a compatible Poisson structure on A with Poisson matrices Ψ(t). This compatible Poisson structure on A is given by the family of the deformation matrices Λ(t) of A_q.

Conversely, assume Ψ(t) (t∈𝕋_n) are the family of Poisson matrices of a compatible Poisson structure of a cluster algebra A. Let Λ(t)=Ψ(t), according to Theorem 3.2 (i) and (iii) in <cit.>, (B(t), Λ(t)) is a compatible pair satisfying the condition (<ref>). Then we obtain a quantum cluster algebra A_q as the CQV of A.

By the above discussion, we have the following statement:
 Assume A is a cluster algebra with exchange matrices B̃(t) which are of full column rank. Then, we have the following one-by-one correspondence:

.  {Compatible Poisson structures of A} 

.                                     ⟷  {Quantizations of A}={quantum cluster algebras as CQV's of A}

via

.  {Poisson matrices of A } = {Deformation matrices of A_q }.


Motivated by this observation, for a quantum cluster algebra A_q, if we have a (non-trivial) compatible Poisson structure on it, would it be possible to find an algebra as a further quantization of A_q whose like-compatible pairs are correspondent to the family of Poisson matrices of the Poisson structure of A_q? We will call this possible further quantization of A_q as the second quantization of A_q.

This is the reason we want to find out the (non-trivial) compatible Poisson structure on a quantum cluster algebra A_q.

In this section, we would like to give the exact definition of the so-called second quantization of A_q. 


We introduce the q-analog of an integer a which is [a]_q=q^a-q^-a/q-q^-1∈ℕ(q^± 1) for q∈ℂ. Given a deformation matrix Λ and a Poisson matrix Ω, we can define an m× m skew-symmetric matrix W(t)=(W_ij) as

    W_ij={[ ω_ijλ_ij/[λ_ij]_q^1/2               λ_ij≠ 0;                  ω_ij               λ_ij=0. ] .

We call W(t) the second deformation matrix at t.

From this definition we know that any two of Ω(t), Λ(t) and W(t) can determine the other one. And the three conditions in Theorem <ref> can also be restated for any j∈[1,m],k∈[1,n],k≠ j as following:


  (C2) For any u∈ [1,m], if b_uk≠0, then W_uj/W_kj=λ_u j/λ_kj.

  (C3) For any u,v∈ [1,m], if b_ukb_vk≠ 0, then W_uj/W_vj=λ_uj/λ_vj.

  (C4) ∑_t:λ_tj=0W_tjb_tk=0.



For t∈𝕋_n, let B̃(t)=(b_ij) be an m× n integer matrix with m⩾ n, Λ(t)=(λ_ij) an m× m integer skew-symmetric matrix and W(t)=(W_ij) an m× m skew-symmetric integer matrix. The triple (B̃(t),Λ(t),W(t)) is called compatible if (B̃(t),Λ(t)) is a compatible pair satisfying (C1) and any triple mutation equivalent to (B̃(t),Λ(t),W(t)) satisfies (C2),(C3) and (C4).

Recall that by Theorem <ref>, the latter condition including (C2),(C3) and (C4) is equivalent to that the Poisson structure induced by Ω(t) is compatible with the quantum cluster algebra A_q associated to the compatible pair (B̃(t),Λ(t)).

As usual we define the extended cluster at t∈𝕋_n to be a set of variables

    Ỹ(t)={Y_t^e_1,Y_t^e_2,⋯,Y_t^e_n,Y^e_n+1,⋯,Y^e_m},

where e_i∈ℤ^m are the standard basis. And the set of first n variables is called the cluster at t and denoted by Y(t). For p,q∈ℂ, let 𝒯_t be the ℤ[p^±1/2,q^±1/2]-algebra generated by Ỹ(t) satisfying the relation

    Y_t^e_iY_t^e_j=p^1/2W_ijq^1/2λ_ijY_t^e_i+e_j,∀ i,j∈[1,m],

We call 𝒯_t the II-quantum torus, or say, (p,q)-quantum torus at t.

Denote by ℱ_p,q the skew-field of fractions of 𝒯_t. Thus, 𝒯_t is a subalgebra of ℱ_p,q.

We can see that

    Y_t^e_iY_t^e_j=p^W_ijq^λ_ijY_t^e_j Y_t^e_i,∀ i,j∈[1,m].

We call Σ(t)=(Ỹ(t),B̃(t),Λ(t),W(t)) a II-quantum seed at t for the compatible triple (B̃(t),Λ(t),W(t)).

Let Σ(t) and Σ(t^') be two II-quantum seeds at t and t^' respectively. Denote by b_i the i-column of B̃(t) as a vector. Assume t and t^' are adjacent vertices by an edge labeled k in 𝕋_n. Σ(t^') is defined from Σ(t) by a mutation in direction k if Σ(t^')=μ_k(Σ(t))=(μ_k(Ỹ(t)),μ_k(B̃(t)),μ_k(Λ(t)),μ_k(W(t))), where

    Ỹ(t')=μ_k(Ỹ(t))=(Ỹ(t)∖{Y_t^e_k})⋃{μ_k(Y_t^e_k)},    Y_t'^e_k=μ_k(Y_t^e_k)=Y_t^-e_k+[b_k(t)]_++Y_t^-e_k+[-b_k(t)]_+

and

    W_ij(t^')={[ -W_kj(t)+∑_l=1^m[b_lk(t)]_+W_lj(t)                         if  i=k≠ j; -W_ik(t)+∑_l=1^m[b_lk(t)]_+W_il(t)                         if  j=k≠ i;                            W_ij(t)                          otherwise ] .

while the mutations of matrices B̃(t) and Λ(t) are the same as those we introduced before in (<ref>) and (<ref>) respectively.




For any t∈𝕋_n and k∈[1,n], let Σ(t)=(Ỹ,B̃,Λ,W) be a II-quantum seed at t, then μ_k(Σ(t))=(Ỹ^',B̃^',Λ^',W^') is also a II-quantum seed. And the Poisson structure associated to Σ(t) and μ_k(Σ(t)) are the same.



The compatibility of (B̃,Λ,W) is mutation invariant by definition, so what left is to prove

    Y^'^e_kY^e_j=p^W^'_kjq^λ^'_kjY^e_jY^'^e_k,∀ j∈[1,m]∖{k}.


Since Y^'^e_k=Y^-e_k+[b_k]_++Y^-e_k+[-b_k]_+, we have

    [                                                            Y^'^e_kY^e_j=                                    (Y^-e_k+[b_k]_++Y^-e_k+[-b_k]_+)Y^e_j;                                                                        =  p^∑_i=1^m[b_ik]_+W_ij-W_kjq^∑_i=1^m[b_ik]_+λ_ij-λ_kjY^e_jY^-e_k+[b_k]_+;                                                                          +p^∑_i=1^m[b_ik]_-W_ij-W_kjq^∑_i=1^m[b_ik]_-λ_ij-λ_kjY^e_jY^-e_k+[b_k]_-;                                                                        =                                            p^W^'_kjq^λ^'_kjY^e_jY^'^e_k. ]


Let Ω be the Poisson matrix associated to Σ(t). We need to verify that the Poisson matrix Ω^' associated to μ_k(Σ(t)) is exactly the Poisson matrix Ω^'' obtained from Ω by mutation at direction k. If i,j≠ k, then ω_ij^''=ω_ij=ω_ij^'. Next assume i=k≠ j, the case i≠ k=j is the same.

If any l∈ S={u| u=k  or  b_uk≠ 0}, λ_lj=0, we have λ_kj^'=-λ_kj+∑_l=1^m[b_lk]_+λ_lj=0. Then W_ij=ω_ij for any l∈ S and W_kj^'=W_kj. Therefore following (<ref>)

    ω^''_kj=-ω_kj+∑_l=1^m[b_lk]_+ω_lj=-W_kj+∑_l=1^m[b_lk]_+W_lj=W_kj^'=W_kj=ω_kj.


If there is v∈ S such that λ_vj≠ 0, then by Remark <ref> ω_ij=a[λ_ij]_q^1/2 for any i∈ S, where a∈ℤ. So W_ij=aλ_ij and W_kj^'=aλ_ij. Again by (<ref>), it can be checked that

    ω_kj^''=a[λ_kj^']_q^1/2,

hence W_kj^''=aλ_kj^'=W_kj^'.

Therefore Ω^'=Ω^'', which means the Poisson structures induced by Ω and by Ω^' are the same.



  For a quantum cluster algebra A_q with a compatible Poisson structure { , },
assign II-quantum seeds Σ(t) to every vertex t in 𝕋_n so that for any t and t^' adjacent by an edge labeled k, Σ(t^') is obtained from Σ(t) by a mutation in direction k by Definition <ref>. Denote by A_p,q=A_p,q(Σ) the ℤ[p^±1/2,q^±1/2][Y^± e_n+1,⋯,Y^± e_m]-subalgebra of ℱ_p,q generated by ⋃_t∈𝕋_nY(t). We call A_p,q the secondly quantized cluster algebra associated to {Σ(t)}_t∈𝕋_n, or say, the second quantization of A_q.


Trivially, if p tends to 1 or q tends to 1 or p tends to q, then the secondly quantized cluster algebra A_p,q degenerates to the quantum cluster algebras A_q, A_p, A_q with deformation matrix Λ, W, Λ+W respectively.

As we said before, any two of Ω(t), Λ(t) and W(t) determine the other one.
Hence, if we are given a secondly quantized cluster algebra A_p,q with deformation matrices Λ(t)=(λ_i,j) and second deformation matrices W(t)=(W_ij), then we can obtain the Poisson matrices Ω(t) of a compatible Poisson structure on A_q via:

    ω_ij={[ W_ij[λ_ij]_q^1/2/λ_ij,                λ_ij≠ 0;                  W_ij,                λ_ij=0. ].

Therefore, when Λ(t) is fixed, we have the following correspondance:
 Assume A_q is a quantum cluster algebra with the compatible pairs (B̃(t),Λ(t)). Then, we have the following one-by-one correspondence:


{Compatible Poisson structures of A_q} 


.                   ⟷  {Second Quantizations of A_q}={Secondly quantized cluster algebras A_p,q of A_q}

via

.  {Poisson matrices of A_q }  ⟷  {Second deformation matrices of A_p,q}.



We say a matrix B has a matrix decomposition B=B_1⊕ B_2 if

    B=[ B_1   O;   O B_2 ]


For a quantum cluster algebra A_q, if for a certain seed (Ỹ,B̃,Λ,W), there are the matrix decompositions W=⊕_i=1^rW_i and Λ=⊕_i=1^rΛ_i such that

    W_i=a_iΛ_i,  for some   a_i∈ℤ  ∀ i=1,⋯,r,

and let I_i be the index set of W_i, then

(i)
    X_kX_l={[ p^1/2W_klq^1/2λ_klX_kX_l=(p^a_iq)^1/2λ_klX_kX_l                                        k,l∈ I_i;                                          X_lX_k                              k∈ I_i,l∈ I_j,i≠ j ].


(ii)in this case, (<ref>) and (i) always hold for any seed.

Hence, the secondly quantized cluster algebra A_p,q is essentially a quantum cluster algebra. Thus under this condition, we say the secondly quantized cluster algebra A_p,q to be trivial. Otherwise, it is called non-trivial.

For example W=aΛ,a∈ℤ, then there is a canonical ℤ[p^±1/2]-algebra isomorphism A_p,q≅ A_q⊗_ℤ[q^±1/2]ℤ[p^±1/2,q^±1/2] sending Y^e_i_t to X^e_i_t, q to p^-aq and p to p, therefore the quantum cluster algebra A_q can be embedded into the secondly quantized cluster algebra A_p,q, which means that their cluster algebraic structures coincide.


Let (X̃(t),B̃(t),Λ(t)) be a seed of a quantum cluster algebra A_q at t∈𝕋_n and {-,-} a compatible Poisson structure on A_q. Then the second deformation matrix W(t) satisfies that

    B̃(t)^⊤W(t)=c(D  O),
    C1^*

where c∈ℤ[q^±1/2] and D is the skew-symmetrizer of B̃(t).


Here, since we only discuss with the seed of A_q at the vertex t, we will omit t for clusters, exchange matrices, deformation matrices and etc.. Let B̃_l (l∈ [1,n]) denote the l-th column of B̃.
Let k∈[1,n],j∈[1,m] and k≠ j.

If λ_kj≠ 0 or b_lkλ_lj≠ 0 for some l, then according to Remark <ref>, ω_ij=a[λ_ij]_q^1/2 for any i such that b_ik≠ 0 or i=k, where a∈ℤ[q^±1/2]. Hence w_ij=aλ_ij for any i such that b_ik≠ 0 or i=k and B̃_k^⊤W_j=∑_i=1^mb_ikW_ij=a∑_i=1^mb_ikλ_ij=0, where W_j represents the j-th column of W.

Otherwise, if λ_ij=0 for all i such that b_ik≠ 0 or i=k. Then we know from Lemma <ref> that B̃_k^⊤W_j=∑_i=1^mb_ikW_ij=∑_i=1^mb_ikω_ij=0.

Therefore, B̃^⊤W=(M  O), where M is an n× n diagonal matrix. Moreover, B̃^⊤WB̃ is a skew-symmetric matrix, which means MB is skew-symmetric, i.e. M=cD.



Hence given a compatible triple (B̃,Λ,W), (B̃,W) is also a compatible pair in the meaning of the condition (C1). Moreover, by Definition <ref>, it can be checked that (B̃,W,aΛ) is a compatible triple for certain a∈ℤ to make sure the corresponding Poisson matrix having elements in ℤ[q^±1/2]. Therefore, (B̃,Λ) and (B̃,W) induce two quantization A_q and A^'_p for a cluster algebra A associated to B̃. In general, A_q and A^'_p are different as quantum cluster algebras. Then (B̃,Λ,W) and (B̃,W,aΛ) respectively induces a second quantization A_p,q and A_q,p^' for A_q and A^'_p. However, according to the definition of secondly quantized algebras, there is a cluster isomorphism ψ from A_p,q⊗_ℤ[q^±1/2]ℤ[q^±1/2a] to A_q,p^' fixing variables and sending p to q, q to p^a.

We will call the quantum cluster algebra A^'_p a dual quantum cluster algebra of A_q on the compatible pair (B̃,Λ,W).

Therefore, we have the following two ways of quantization induced by a triple (B̃,Λ,W):



In summary, we have the following conclusion:

For a cluster algebra A and a compatible triple (B̃,Λ,W) from its two various Poisson structures, the secondly quantized cluster algebras A_p.q and A'_q,p from A_q and its dual A'_p determined by the compatible pairs (B̃,Λ) and (B̃,W) respectively with the compatible triples (B̃,Λ,W) and (B̃,W,aΛ)  (a∈Z) are essentially the same under the cluster isomorphism ψ:

    ψ:  A_p,q⊗_ℤ[q^±1/2]ℤ[q^±1/2a]⟶ A_q,p^'

fixing variables and sending p to q, q to p^a.



The basic idea of this paper is to lift the Poisson matrices of (quantum) cluster algebras to the deformation matrices, so as to complete the first or second quantization of this algebra. Based on this idea, we can similarly continue to obtain the third quantization of a secondly quantized cluster algebra, even in general, to obtain the (n+1)-th quantization of an n-th quantized cluster algebra. The whole progress would be much like that in this paper but with more parameters. The further quantization may go on until it is faced to no non-trivial quantization, since all matrices R satisfying B̃^⊤R=cD consists of a finite-dimension linear space.

Particularly, in Section 6, we will prove that the second quantization is trivial in the case without coefficients but may be admitted to be non-trivial in the case with coefficients.

Recently, before this work (the first version) has be finished in the beginning of this year, we found toroidal cluster algebras defined in <cit.> and <cit.>. According to Proposition <ref>, a secondly quantized cluster algebra in this paper is a toroidal cluster algebra associated to an initial seed containing a compatible triple.

In <cit.>, Laurent phenomena was proved to be true for toroidal cluster algebras in skew-symmetrizable case as well as positivity for cluster variable in skew-symmetric case. Moreover, the exchange graph of a toroidal cluster algebra is incident to that of its corresponding (quantum) cluster algebra. Therefore, the corresponding properties of secondly quantized cluster algebras naturally follow as they are a special kind of toroidal cluster algebras.





§ A LEMMA FOR DECOMPOSABILITY OF QUANTUM CLUSTER ALGEBRAS

First of all, let us focus on the condition (C4), which looks similar to the condition for a Poisson bracket being compatible with a cluster algebra in <cit.>. Note that in (C4'): ∑_λ_tj=0ω_tjb_tk=0, the sum is given only for those terms whose corresponding λ_tj=0. So, we define a new matrix Ω̂={ω̂_ij} from Ω where ω̂_ij=ω_ij if λ_ij=0 and ω̂_ij=0 otherwise.

For any i≠ j∈[1,m], we say that i is connected to j (with respect to B) if b_ij≠ 0. We define the relation i∼ j if either i=j or i is connected to j.


In a quantum cluster algebra A_q with a Poisson structure {-,-}, let (X̃^',B̃^',Λ^') be a seed mutation equivalent to the initial seed (X̃,B̃,Λ). Assume X̃ and X̃^' are log-canonical with respective to {-,-} with Poisson matrices Ω and Ω^' respectively. Then ∑_λ_tj^'=0ω_tj^'b_tk^'=0 for any j≠ k if and only if Ω̂B̃=cD̃, where c∈ℤ[q^±1/2],
D̃=
[ D; O ]_m× n for D
a skew-symmetrizer of B. And, in this case Ω̂^'B̃^'=cD̃.


This proof is similar to that in <cit.>, but more annoying.

First assume (X̃^',B̃^',Λ^')=μ_k(X̃,B̃,Λ). For any j∈[1,m],h∈[1,n],j≠ h, we write ∑_λ_tj=0ω_tjb_th=0 and ∑_λ_th=0ω_thb_th=a_h. We will show that ∑_λ_tj^'=0ω_tj^'b_th^'=0 for j≠ h if and only if Ω̂B̃=cD̃, and then in this case Ω̂^'B̃^'=cD̃. Then by induction this lemma holds for any cluster mutation equivalent to the initial cluster.

First We prove that ∑_λ_tj^'=0ω_tj^'b_th^'=0 for j≠ h if and only if Ω̂B̃=cD̃ case by case.

Case 1: k≠ h,j and there is l∼ k such that λ_lj≠ 0. Because λ_lj^'=λ_lj≠ 0 and b_lk^'=-b_lk≠ 0, we have ω_tj=0 if λ_tj=0 and t∼ k, and ω_kj^'=0 if λ_kj^'=0 according to Lemma <ref>. Hence we have that


    ∑_λ_tj^'=0ω_tj^'b_th^'
    =∑_ω_tj^'b_th^'
    =∑_ω_tj^'b_th^'
    =∑_ω_tj(b_th+[b_tk]_+b_kh+b_tk[-b_kh]_+)


    =∑_λ_tj=0ω_tjb_th+∑_λ_tj=0ω_tjb_kh[b_tk]_++∑_λ_tj=0ω_tjb_tk[b_kh]_+
    =∑_λ_tj=0ω_tjb_kh[b_tk]_+
    =0.





Case 2: k≠ h,j and for any l∼ k,λ_lj=0. In this case, we can calculate that the mutation formula of Ω is

    ω_ij^'={[                      ω_ij                      i≠ k; -ω_kj+∑_t=1^m[b_tk]_+ω_tj                       i=k ] .

So we have

   ∑_λ_tj^'=0ω_tj^'b_th^'
 = ω_kj^'b_kh^'+∑_ ω_tj^'b_th^'

=    (-ω_kj+∑_t=1^m[b_tk]_+ω_tj)(-b_kh)+∑_ω_tj(b_th+[b_tk]_+b_kh+b_tk[-b_kh]_+)

=   ∑_λ_tj=0ω_tjb_th+[-b_kh]_+∑_λ_tj=0ω_tjb_tk

=   0.




Case 3: k=h. We have

    ∑_λ_tj^'=0ω_tj^'b_th^'
    =∑_ω_tj^'b_th^'
    =∑_ω_tj(-b_th)
    =-∑_λ_tj=0ω_tjb_th
    =0.


Case 4: k=j. For i∈[1,m], if there is l∼ j so that λ_il≠ 0, then λ_ij^'=0 induces ω_ij^'=0. Otherwise if λ_il=0 for any l∼ j, then ω_ij^'=-ω_ij+∑_t=1^m[b_tj]_+ω_it. Therefore


∑_λ_tj^'=0ω_tj^'b_th^'
   =∑_ω_tj^'b_th^'

   =∑_(-ω_tj+∑_p=1^m[b_pj]_+ω_tp)(b_th+[b_tj]_+b_jh+b_tj[-b_jh]_+)

   =-∑_ω_tjb_th+ ∑_∑_p=1^m[b_pj]_+ω_tpb_th

   -∑_ω_tj[b_tj]_+b_jh+ ∑_∑_p=1^m[b_pj]_+ω_tp[b_tj]_+b_jh

   -∑_ω_tjb_tj[-b_jh]_++ ∑_∑_p=1^m[b_pj]_+ω_tpb_tj[-b_jh]_+.         ()

Let us deal with these terms one by one. We have:

    ∑_ω_tjb_th
    =∑_λ_tj=0ω_tjb_th
    =0.


This is because ω_tj=0 if λ_tj=0 and there is l∼ j satisfying λ_tl≠ 0.

   ∑_∑_p=1^m[b_pj]_+ω_tpb_th
 = ∑_p=1^m∑_[b_pj]_+ω_tpb_th

=   ∑_p=1^m[b_pj]_+(∑_ω_tpb_th)- ∑_λ_jp=0[b_pj]_+ω_jpb_jh
 = a_h[b_hj]_++∑_λ_tj=0[b_tj]_+ω_tjb_jh

=    a_h[b_hj]_++∑_[b_tj]_+ω_tjb_jh.

And,

   ∑_∑_p=1^m[b_pj]_+ω_tp[b_tj]_+b_jh

=   ∑_∑_[b_pj]_+ω_tp[b_tj]_+b_jh+ ∑_∑_[b_pj]_+ω_tp[b_tj]_+b_jh

=   0.

The last equality holds because the first term equals its opposite number.

    ∑_∑_[b_pj]_+ω_tp[b_tj]_+b_jh
    =∑_∑_[b_tj]_+ω_pt[b_pj]_+b_jh


    =-∑_∑_[b_pj]_+ω_tp[b_tj]_+b_jh;


and if b_pj≠ 0, then p∼ j so λ_tp=0, which together with λ_l^'p≠ 0 induces ω_tp=0.


    ∑_ω_tjb_tj[-b_jh]_+
    =∑_λ_tj=0ω_tjb_tj[-b_jh]_+
    =a_j[-b_jh]_+;



    ∑_∑_p=1^m[b_pj]_+ω_tpb_tj[-b_jh]_+
    =∑_p=1^m∑_[b_pj]_+ω_tpb_tj[-b_jh]_+
    =∑_p=1^m[b_pj]_+[-b_jh]_+(∑_λ_tp=0ω_tpb_tj)
    =0.


Substituting (<ref>),(<ref>),(<ref>),(<ref>) and (<ref>) into (), we get that

    ∑_λ_tj^'=0ω_tj^'b_th^'=a_h[b_hj]_+-a_j[-b_jh]_+,

which equals 0 if and only if a_h=cd_h for all h∈[1,n], where c∈ℤ[q^±1/2].



Now we will prove that when Ω̂B̃=cD̃, it holds Ω̂^'B̃^'=cD̃.

Case 1: k≠ h and there is l-k such that λ_lh≠ 0. Similar to the discussion of Case 1 above,

    [                                              ∑_λ_th^'=0ω_th^'b_th^'                            =∑_ω_th(b_th+[b_tk]_+b_kh+b_tk[-b_kh]_+);                                                                     =∑_λ_th=0ω_thb_th+∑_λ_th=0ω_thb_kh[b_tk]_++∑_λ_th=0ω_thb_tk[b_kh]_+;                                                                                                                       =∑_λ_th=0ω_thb_th;                                                                                                                                    =a_h ]


Case 2: k≠ h and for any l-k,λ_lh=0.

    [                                                      ∑_λ_th^'=0ω_th^'b_th^'                                                =ω_kh^'b_kh^'+∑_ω_th^'b_th^';                                                                             =(-ω_kh+∑_t=1^m[b_tk]_+ω_th)(-b_kh)+∑_ω_th(b_th+[b_tk]_+b_kh+b_tk[-b_kh]_+);                                                                                                             =∑_λ_th=0ω_thb_th+[-b_kh]_+∑_λ_th=0ω_thb_tk;                                                                                                                                                    =a_h ]


Case 3: k=h. We have


   ∑_λ_th^'=0ω_th^'b_th^' 
= ∑_(-ω_th+∑_p=1^m[b_ph]_+ω_tp)(-b_th) 
= ∑_ω_tj(-b_th)

=   ∑_λ_th=0ω_thb_th-∑_∑_p=1^m[b_ph]_+ω_tp)b_th 
=  a_h-∑_p=1^m[b_ph]_+(∑_λ_tp=0ω_tpb_th)

=   a_h.


In summary, we obtain Ω̂^'B̃^'=cD̃.


It can be seen from Theorem <ref> and Lemma <ref> that if we regard a cluster algebra as a quantum cluster algebra with Λ=0, then the compatible conditions (C2) and (C3) in Theorem <ref> naturally hold, while (C4) is exactly the compatible condition showed in <cit.> and <cit.>.































§ DECOMPOSABILITY OF QUANTUM CLUSTER ALGEBRAS AND NON-TRIVIAL SECOND QUANTIZATION




 §.§ Decomposability of quantum cluster algebras
.

First we want to show that the compatibility problem can come down to the cluster indecomposable case in the following sence


(1)  B̃ is said to be decomposable if there is an non-trivial partition of [1,m], [1,m]=I_1 I_2 so that b_ij=0 whenever i∈ I_1,j∈ I_2. In this case we denote the composition as B̃=B̃_I_1B̃_I_2. Otherwise, B̃ is called indecomposable.

(2)  A quantum cluster algebra A_q is said to be cluster indecomposable if there is an indecomposable matrix B̃ in a seed. Otherwise, A_q is called cluster decomposable.

In fact, it is easy to see that once a quantum cluster algebra A_q is cluster indecomposable (or cluster decomposable), all B̃ matrices of A_q are indecomposable (or decomposable, respectively).


Let (X̃,B̃,Λ) be the initial seed of a quantum cluster algebra A_q and {-,-} a Poisson bracket on A_q. If B̃ has the decomposition B̃=_i=1^rB̃_I_i with indecomposables B̃_I_i for i∈[1,r], then {-,-} is compatible with A_q if and only if {-,-}|_A_q,I_i is compatible with A_q,I_i (i∈[1,r]) and ∑_λ_tj=0ω_tjb_tk=0 holds for every k∈ [1,n],j∉ I(k), where A_q,I_i is the quantum cluster subalgebra of A_q generated by (B̃_I_i,Λ_I_i) and I(k) is the one of I_i containing k.



If {-,-} is compatible with A_q, it follows from the definition of compatibility that {-,-}|_A_q,I_i is compatible with A_q,I_i. And Lemma <ref> requests that ∑_λ_tj=0ω_tjb_tk=0 for any k∈ [1,n],j≠ k. In particular, equation holds when j∉ I(k).On the other hand, we learn from above discussion that {-,-} being compatible with A_q is equivalent to the three conditions of Lemma <ref> holding for every seed. And Lemma <ref> transforms the third condition to ∑_λ_tj=0ω_tjb_tk=cd_kδ_jk for any j∈[1,m],k∈[1,n]. Because of the decomposition of A_q, we know from the compatibility of {-,-}|_A_q,I_i and A_q,I_i, i∈[1,r] that the first two conditions as well as ∑_λ_tj=0ω_tjb_tk=cd_kδ_jk,j∈ I(k) always hold. The remaining are ∑_λ_tj=0ω_tjb_tk=0 for all j∉ I(k).


Let A_q,I_1, A_q,I_2 be two quantum cluster algebras with initial seeds (X̃_I_1,B̃_I_1,Λ_I_1) and (X̃_I_2,B̃_I_2,Λ_I_2) respectively, and let Θ be an |I_1|×|I_2| integer matrix satisfying

    {[ B̃_I_1^⊤Θ        =0;   ΘB̃_I_2       =0. ].

Define A_q,I_1_Θ A_q,I_2 to be the algebra equal to A_q,I_1⊗_[q^±1/2] A_q,I_2 as a [q^±1/2]-module
with twist multiplication:

    (a⊗ b)(c⊗ d)=∑_i,jk_il_jq^1/2r̅_i^⊤Θs̅_jaX̃_I_1^s̅_j⊗X̃_I_2^r̅_id

for b=∑_ik_iX̃_I_2^r̅_i,c=∑_jl_jX̃_I_1^s̅_j, where r̅_i,s̅_j are exponential column vectors.

Let A_q,I_i be quantum cluster algebras with initial seeds (X̃_I_i,B̃_I_i,Λ_I_i) for i∈ [1,r]. Then we can obtain the algebra _i=1^r A_q,I_i since there is the associativity for multiplication in the sense that


(A_q,I_1_Θ_1 A_q,I_2)_Θ^' A_q,I_3=A_q,I_1_Θ^'' (A_q,I_2_Θ_3A_q,I_3),


where

    Θ^'=[ Θ_2; Θ_3 ],  Θ^''=[ Θ_1 Θ_2 ].


Following the above notations, the algebra A_q=_i=1^r A_q,I_i is a quantum cluster algebra with quantum cluster subalgebras A_q,I_i for i∈ [1,r].


Let A_q,I_1⋃ I_2 be a quantum cluster algebra with initial seed (X̃_I_1⋃ I_2,B̃_I_1⋃ I_2,Λ_I_1⋃ I_2), where

    B̃_I_1⋃ I_2=[ B̃_I_1      O;      O B̃_I_2 ],  Λ_I_1⋃ I_2=[ Λ_I_1     Θ;  -Θ^⊤ Λ_I_2 ],

(B̃_I_1⋃ I_2,Λ_I_1⋃ I_2) is compatible because of (<ref>). Then clearly there is an isomorphism


η:   A_q,I_1_Θ A_q,I_2⟶ A_q,I_1⋃ I_2


via sending a⊗ b to ab. Moreover by induction on r, we have A_q=_i=1^r A_q,I_i≅ A_q,⋃_iI_i as a quantum cluster algebra.


From this proposition, we call A_q=_i=1^r A_q,I_i a cluster decomposition of A_q,⋃_iI_i.


Because of the Proposition <ref>, a Poisson structure {-,-} is compatible with A_q if and only if {-,-}|_A_q,I_i is compatible with A_q,I_i for any i and ∑_λ_tj=0ω_tjb_tk=0 for any k∈[1,n],j∉ I(k). Since the latter one is easy to be checked, in the sequel we only need to consider a cluster indecomposable quantum cluster algebra A_q.


Let A_q be a quantum cluster algebra with initial seed (X̃,B̃,Λ) and {-,-} a compatible Poisson bracket on A_q. Assume Ω is the Poisson matrix of the initial cluster with respect to {-,-}, B̃ has the decomposition B̃=⊕_i=1^rB̃_I_i with indecomposables B̃_I_i for i∈[1,r], and A_q,I_i is the quantum cluster indecomposable subalgebra of A_q determined by (B̃_I_i,Λ_I_i). Then,

(i)  A_q has a cluster decomposition A_q≅_i A_q,I_i;

(ii) Assume W is the second deformation matrix of the second quantization of A_q at the initial quantum seed induced by {-,-}. Then the second quantization of A_q is non-trivial if and only if at least one of the following statements holds:

(1)  there is at least one i∈ [1,r] such that the second quantization of A_q,I_i is non-trivial;

(2)  there is k∈ℤ and i,j∈[1,r] such that W|_I_i× I_i= kΩ|_I_i× I_i but W|_I_i× I_j≠ kΩ|_I_i× I_j, .



(i)  Denote Θ_i=Λ|_(⋃_j⩽ iI_j)× I_i+1 for i∈[1,r-1]. Because (B̃,Λ) is compatible,


A_q,I_1_Θ_1⋯_Θ_r-1A_q,I_r


is well defined. And η in previous proof shows an isomorphism between A_q and _i A_q,I_i.

(ii)  Recall that the second quantization induced by {-,-} is trivial if there are matrix decompositions W=⊕_i=1^sW_i and Λ=⊕_i=1^sΛ_i with W_i and Λ_i indecomposable such that W_i=a_iΛ_i for some a_i∈ℤ. Hence the second quantization is non-trivial if above decomposition of W and Λ have different size or there is i∈[1,s] such that W_i≠ kΛ_i for any k∈ℤ. The only two possible cases are that it induces non-trivial second quantization on A_q,I_i for some i or otherwise there is i,j∈[1,r] such that W|_I_i× I_j≠ kΩ|_I_i× I_j, where k∈ℤ satisfying W|_I_i× I_i= kΩ|_I_i× I_i.



(1) When A_q degenerates to a cluster algebra A as q→ 1, (a⊗ b)(c⊗ d)=ac⊗ bd.

(2) When A_q is a quantum cluster algebra without coefficients, B is invertible. Hence by (<ref>), Θ=O. So we also have (a⊗ b)(c⊗ d)=ac⊗ bd.

Then in these cases (1) and (2),  is exactly the tensor product ⊗ and the above cluster decomposition coincides with A=⊗_iA_I_i or A_q=⊗_iA_q, I_i.




 §.§ Quantum cluster algebras with non-trivial second quantization




  §.§.§ The non-trivial example from a quantum algebra
.

Firstly, we give a simple example of a quantum cluster algebra which has a non-trivial second quantization.

The quantum coordinate algebra (or say, quantum matrix algebra) Fun_(SL_q(2)) (see <cit.>) is generated by a, b, c, d with relations:

    ab=q^-1ba, ac=q^-1ca, db=qbd, dc=qcd, bc=cb,ad-da=(q^-1-q)bc

and

    ad-q^-1bc=1,

where 0 q∈ℂ is a parameter.

In fact, Fun_(SL_q(2)) has a quantum cluster structure, see <cit.>. Below is our explanation.

Let ℙ=ℂ[b,c]. Here b commutes with c by (<ref>).
In the 1-regular tree T_1: @C=0.75cmt_0∙@-[r]    ∙ t_1, we assign the quantum seed Σ(t_0)=(X(t_0),B(t_0),Λ(t_0)) on the vertex t_0, where X(t_0)={a}, X_fr={b,c}, X_t_0^e_1=a, X_t_0^e_2=X^e_2=b, X_t_0^e_3=X^e_3=c; Λ(t_0)=([  0 -1 -1;  1  0  0;  1  0  0 ]), B(t_0)=([ 0; 1; 1 ]). It can be verified that Λ(t_0)^⊤B(t_0)=([ 2; 0; 0 ]), i.e, (B(t_0), Λ(t_0)) is a compatible pair.

Moreover, let X(t_1)={d}, X_t_1^e_1=d. Then, according to (<ref>), we have

    d=q^-1a^-1bc+a^-1=X^-e_1+[b_1(t_0)]_+_t_0+X^-e_1+[-b_1(t_0)]_+_t_0,

which exactly means the mutation of cluster variables, i.e. μ_1(X_t_0^e_1))=X_t_1^e_1. And by mutation of matrices, we have Λ(t_1)=μ_1(Λ(t_0))=([  0  1  1; -1  0  0; -1  0  0 ]), B(t_1)=μ_1(B(t_0))=([  0; -1; -1 ]).

Following (<ref>), the relations of quantum tori:

    X_*^e_iX_*^e_j=q^1/2Λ_*(e_i,e_j)X^e_i+e_j,    ∀ i,j=1,2,3, *=t_0, t_1

hold.

Therefore, through the above discussion, Fun_(SL_q(2)) can be realized as the ℤ[q^±1/2]ℙ-quantum cluster algebra, i.e. Fun_(SL_q(2))=A_q(Σ(t_0)).

Now we can give the second quantization of Fun_(SL_q(2)) according to its compatible Poisson structures. By (C1*), we have B̃(t_0)^⊤W(t_0)=c(D O) for some c∈Z[q^±1/2], then it can be induced that the second deformation matrix W(t_0) must be of the form

    W(t_0)=[    0 -w_1 -w_2;  w_1    0    0;  w_2    0    0 ]

where w_1+w_2≠ 0. Moreover, it can be checked easily that the matrix W(t_0) in (<ref>) satisfies (C2), (C3) and (C4). So, such matrix W(t_0) is what we need. Next, by mutation formula, we have W(t_1)=[    0  w_1  w_2; -w_1    0    0; -w_2    0    0 ].

According to definition, the second quantization induced by (B(t_0),Λ(t_0),W(t_0)) is trivial if and only if W(t_0)=hΛ(t_0) for some constant h, which means w_1=w_2. Therefore, when w_1+w_2≠ 0 and w_1≠ w_2, the obtained second quantization A_p,q of Fun_(SL_q(2)) is non-trivial.

In this case, the relations of quantum tori are

    X_*^e_iX_*^e_j=p^1/2W_*(e_i,e_j)q^1/2Λ_*(e_i,e_j)X^e_i+e_j,    ∀ i,j=1,2,3, *=t_0, t_1.


Following these relations and X_t_0^e_1=a,  X_t_0^e_2=b,  X_t_0^e_3=c,  X_t_1^e_1=d, the secondly quantized cluster algebra A_p,q of Fun_(SL_q(2)) can be realized as the Z[q^±1/2]-algebra generated by a,b,c,d satisfying the relations as follows:

    ab=r^-1ba, ac=s^-1ca, db=rbd, dc=scd, bc=cb,ad-da=[(rs)^-1/2-(rs)^1/2]bc

and

    ad-(rs)^-1/2bc=1,

where r=p^w_1q,s=p^w_2q. Now, also write A_p,q as A_p,q(SL(2)).


We know from <cit.> that the 2-parameters quantum coordinate algebra Fun_C(GL_r,s(2)) is generated by t_ij, det_r,s^± 1 with relations:

    t_11t_12=r^-1t_12t_11,   t_11t_21=st_21t_11,   t_21t_22=r^-1t_22t_21,   t_12t_22=st_22t_12,


    t_12t_21=rst_21t_12,   t_11t_22-t_22t_11=(s-r)t_21t_12,   det_r,sdet_r,s^-1=det_r,s^-1det_r,s=1,


    det_r,st_ij=(rs)^i-jt_ij(det_r,s),   det_r,s=t_11t_22-st_21t_12=t_22t_11-rt_21t_12=t_11t_22-r^-1t_12t_21.


If we consider this 2-parameters quantum algebra from GL_r,s(2) to SL_r,s(2), then we have det_r,s=1.
Replacing it into the relation det_r,st_ij=(rs)^i-jt_ij(det_r,s), we get r=s^-1, that is, 2-parameter quantum algebra Fun(GL_r,s(2)) is degenerated into one parameter quantum algebra Fun(GL_r(2)). It means that this method of 2-parameters quantization Fun(GL_r,s(2)) of Fun(GL_r(2)) has no effect on the special quantum linear group SL_r(2).

On the other hand, the second quantization A_p,q(SL(2)) in the above example is for the special quantum linear group SL_q(2). So, we can say that the second quantization A_p,q(SL(2)) provides a way to realize two-parameters quantization of the special quantum linear group SL_q(2), as a parallel supplement to the method of two parameters quantization of the general quantum group.




  §.§.§ Non-trivial secondly quantized cluster algebras via cluster extensions
.

In this part, we will present a class of quantum cluster algebras with non-trivial second quantization via cluster extensions.



For m>l⩾ n and a∈, let

    B̃=
    [ B; L;   ]_m× n  Λ=
    [ Λ_0   O;   O   O ]_m× m  
    W=
    [ aΛ_0    O;    O    P ]_m× m

In these matrices, the sizes of blocks are correspondent, B is an l× n symmetrizable integer matrix with skew-symmetrizer D, L is an integer matrix and P is a non-zero skew-symmetric integer matrix. Assume B̃^⊤Λ=(DO), L^⊤P=O and under any sequence of mutations, the sub-matrix L of B̃ always maintains column sign coherent.

Then (B̃,Λ,W) is compatible and determines a quantum cluster algebra A_q. The corresponding second quantization A_p,q of A_q is non-trivial.



B̃^⊤W=a(B^⊤Λ_0   L^⊤P)=a(DO). Because of the column sign coherence of L, [L]_+^⊤P=O, which ensures the zero blocks of W remain after any sequence of mutations. Hence it can be verified that for any sequences of mutations μ_I, there is always

    μ_I(Λ)=[ μ_I(Λ_0)        O;        O        O ],  μ_I(W)=[ μ_I(aΛ_0)         O;         O         P ].

Then according to the definition of the compatible triple, we see easily that (B̃,Λ,W) is compatible. And by definition the second quantization is non-trivial.


As a concrete example if we let

    B̃=
    [  0  1; -1  0;  1  0;  1  0;  1  0 ],    Λ=
    [  0  1  0  0  0; -1  0  0  0  0;  0  0  0  0  0;  0  0  0  0  0;  0  0  0  0  0 ],    
    W=
    [  0  a  0  0  0; -a  0  0  0  0;  0  0  0 -b  b;  0  0  b  0 -b;  0  0 -b  b  0 ],

then (B̃,Λ,W) is compatible for any a,b∈ℤ and it raises a non-trivial second quantization A_p,q.


For any quantum seed Σ=(X̃,B̃,Λ) of a quantum cluster algebra A_q, we call a quantum seed Σ'=(X̃^',B̃^',Λ^') a cluster extension of Σ if m'>m and X̃^'=X̃⋃{X_m+1,⋯,X_m^'} with X_m+1,⋯,X_m^' as extra frozen variables, in which the first m rows of B̃^' is B̃ and Λ^'=[ Λ O; O O ].
The quantum cluster algebra A_q associated to Σ is a specialization of the quantum cluster algebra A'_q associated to Σ^' with X_m+1=⋯=X_m^'=1. In this case, we also call the quantum cluster algebra A'_q a cluster extension of A_q.

Assume A_q is a quantum cluster algebra with the cluster decomposition A_q=_i^rA_q,I_i and A_q,I_i^' is a cluster extension of A_q,I_i respectively for i=1,⋯,r. Then _i^rA_q,I_i^' is a cluster extension of A_q.


This result can be verified by comparing initial B-matrices.



So we can focus on the indecomposable case as follows.


Let Σ=(X̃,B̃,Λ) be an arbitrary quantum seed of a quantum cluster indecomposable algebra A_q. Then there is a cluster extension Σ^'=(X̃^',B̃^',Λ^') of Σ such that the cluster extension A'_q of A_q admits a non-trivial second quantization A'_p,q.[We are grateful to Zhuoheng He for the discussion about the above matrix equation. ]


Choose a fixed seed Σ_t_0=(X̃_t_0,B̃_t_0,Λ_0) in A_q. Let  be the cluster algebra with principal coefficients having [ B_t_0;   I_n ] as initial exchange matrix. Then there is a seed in  having [ B; C ] as exchange matrix with principal part B. For any choice of the initial seed Σ_t_0 of A_q, there may be more than one C-matrices C such that [ B; C ] is an extended exchange matrix of a seed of . We just choose arbitrary one.

Let Σ'=(X̃^',B̃^',Λ^') be a cluster extension of Σ such that B̃^'=[  B̃;   C; C^' ],
where C^' consists of some s(>n+1) (maybe, repeated) rows from C.

Since C^' consists of several rows from C, according to the definition of mutations of exchange matrices B̃, μ_k⋯μ_1(C') still consists of the corresponding rows of μ_k⋯μ_1(C) after any sequence of mutations μ_1, ⋯, μ_k. Therefore, C^' inherits column sign coherence from C, and thus, [   C; C^' ]
maintains column sign coherence after any sequence of mutations.

In order to apply Lemma <ref>, we need that there is a non-zero skew-symmetric integer (n+s)× (n+s)-matrix P=(p_ij) satisfying that

    [   C; C^' ]^⊤P=O_n× (n+s).


Consider this matrix equation (<ref>) for P. We claim that it has a non-zero skew-symmetric matrix solution P.

Let p̅_i be the i-th column of P, that is, P=(p̅_1 ⋯ p̅_n+s). Then (<ref>) can be written as

    [   C; C^' ]^⊤(p̅_1 ⋯ p̅_n+s)=O,

which is equivalent to the system of linear equations

    [ [     C^⊤ (C^')^⊤ ]                   O                   O;                   O                   ⋱                   O;                   O                   O [     C^⊤ (C^')^⊤ ] ]_n(n+s)× (n+s)^2[   p̅_1;      ⋮; p̅_n+s ]_(n+s)^2× 1=O_n(n+s)× 1

This system contains n(n+s) linear equations.

Additionally, in order to make P a skew-symmetric matrix, we need also the following system of linear equations:

    {[                         p_ii                         = 0,             if   i=1,…, n+s;;                    p_ij+p_ji                         = 0, if   i≠j, for  i,j=1,…, n+s. ].

This system contains (n+s)+1/2(n+s)(n+s-1) linear equations.

Combining (<ref>) with (<ref>), we obtain a system of r linear equations, denoted as (I), whose solution {p_ij:  i,j=1,…,n+s} always forms a skew-symmetric matrix P=(p_ij), where

    r=n(n+s)+(n+s)+1/2(n+s)(n+s-1)=(n+s)(3/2n+1/2s+1/2).


It is easy to see that (n+s)^2-r>0 when s>n+1. Then, in the system (I) of linear equations, the number of undetermined elements is always larger than the number of linear equations in this case. Moreover, note that the system (I) is homogeneous. Hence, the system (I) has non-zero solution, say {p^0_ij:  i,j=1,…,n+s}. Thus, as a solution of the matrix equation (<ref>), we can obtain a non-zero skew-symmetric (integer) matrix P_0=(p^0_ij).

Then, by Lemma <ref>, B̃^', Λ^' and W=
[   Λ   O;   O P_0 ]_(m+n+s)× (m+n+s) are compatible and moreover, they induce a non-trivial second quantization of the quantum cluster algebra A'_q associated to Σ^'.


We call a quantum cluster algebra A_q to be with almost principal coefficients if its initial extended exchange matrix has the form

    B̃_t_0=[ B_t_0;   I_n;     J ],

where J consists of some (maybe, repeated) rows from I_n.

Particularly, in the above theorem and its proof, when Σ_t_0=(X_t_0,B_t_0,Λ_t_0) is a quantum seed of a quantum cluster algebra A^o_q without coefficients, let Σ^'_t_0=(X̃^'_t_0,B̃^'_t_0,Λ^'_t_0) be a cluster extension of Σ_t_0 such that B̃^'_t_0=[ B_t_0;   I_n;     J ]_m× n,
where J_s× n (s>n+1) consists of some (maybe, repeated) rows from I_n. Then, the quantum cluster algebra A_q' associated to Σ^'_t_0 is a quantum cluster algebra with almost principal coefficients.

As done in the proof of Theorem <ref>, let Σ^'=(X̃^',B̃^',Λ^') be any seed in A_q' mutation equivalent to Σ^'_t_0 with B̃^'=[   B;   C; C^' ],
where C is a C-matrix and C^'_s× n consists of some (maybe, repeated) rows from C.

Then by Theorem <ref>, A_q' admits non-trivial second quantization since C^' has more than n+1 rows. In this situation, m=n+n+s. So, s>n+1 if and only if m>3n+1. Therefore we have the following corollary:


Any quantum cluster algebra A'_q with almost principal coefficients having m× n extended exchange matrices always admits a non-trivial second quantization A'_p,q when m>3n+1.










§ SECOND QUANTIZATION OF QUANTUM CLUSTER ALGEBRAS WITHOUT COEFFICIENTS

In the following assume A_q is a quantum cluster algebra without coefficients and (X,B,Λ) is the initial seed. In this case, B and Λ are both invertible. Suppose B has decomposition B=B_1 B_2⋯ B_s, where B_r is indecomposable. Let the set of indices of B_r as submatrix be I_r and X(r)={X_j| j∈ I_r}. As we said before, the coefficients-free condition indicates the invertibility of B, and thus the invertibility of B_r for each r. Meanwhile, B^⊤Λ=D. Hence Λ has decomposition Λ=Λ_1Λ_2⋯Λ_s, where Λ_r is indecomposable and the set of indices of Λ_r is exactly I_r. Then (X(r),B_r,Λ_r) is a seed from which we get a quantum cluster subalgebra A_q,I_r. Moreover, it follows from the decompositions of B and Λ the decomposition of quantum cluster algebras A_q=A_q,I_1⊗ A_q,I_2⊗⋯⊗ A_q,I_s.


Let {-,-} be a Poisson bracket on a quantum cluster algebra A_q. For a seed Σ=(X̃,B̃,Λ), let B=B_1… B_s with B_r indecomposable. {-,-} is a locally standard Poisson structure if {X_i,X_j}=0 when i and j are from different I_r and {-,-} is of standard poisson structure on each X(r), i.e, {X_i,X_j}=a_r[X_i,X_j], where i,j∈ I_r,a_r∈ℤ[q^±1/2].



Let A_q be a quantum cluster algebra without coefficients. Then a Poisson structure {-,-} on A_q is compatible with A_q if and only if it is locally standard on A_q.


“If":
The compatibility of a Poisson structure on A_q can be verified directly using the definition of locally standard Poisson brackets.

“Only if": First let A_q be indecomposable. As Proposition <ref> says, B^⊤W=cD, where c∈ℤ[q^±1/2] and D is a skew-symmetrizer of B. So W=c(B^⊤)^-1D=c· aΛ, where aλ_ij∈ℤ[q^±1/2] for any i,j∈[1,m]. According to the definition, ω_ij=c[λ_ij]_q^1/2, i.e, {X_i,X_j}=c[X_i,X_j]. Therefore for any α_i,β_j∈ℤ^n, {∏_i=1^nX_i^α_i,∏_j=1^nX_j^β_j}=a[∏_i=1^nX_i^α_i,∏_j=1^nX_j^β_j], which means that {-,-} is standard.

Then according to Proposition <ref>, {-,-} is locally standard for any quantum cluster algebra without coefficients.















As showed in the last section, any compatible Poisson structure on A_q are locally standard. In particular, it is standard when restricted on each quantum cluster subalgebra A_q(i). Therefore without loss of generality, in the following we can assume A_q is indecomposable with a standard Poisson structure. Then in a compatible triple, using the proof of Theorem <ref>, we have:

    Ω=a[            0 [λ_12]_q^1/2            ⋯ [λ_1n]_q^1/2; [λ_21]_q^1/2            0            ⋯ [λ_2n]_q^1/2;            ⋮            ⋮            ⋱            ⋮; [λ_n1]_q^1/2 [λ_n2]_q^1/2            ⋯            0 ].

where a is an integer. And W_ij=aλ_ij for any i,j∈[1,n]. Therefore in this case, from (<ref>) we obtain that

    Y_t^e_iY_t^e_j=(p^aq)^1/2λ_ijY_t^e_i+e_j,∀ i,j∈[1,n].

So, the secondly quantized cluster algebra A_p,q is essentially a quantum cluster algebra with one parameter.



In this aspect, it is showed above that:

The second quantization of a quantum cluster algebra without coefficients is always trivial.








Acknowledgements: 

This project is supported by the National Natural Science Foundation of China(No.11671350) and the Zhejiang Provincial Natural Science Foundation of China (No. LY19A010023).

Fang Li thanks to Naihuan Jing and Zhaobing Fan for their encouragement and discussion at the beginning of this work.


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