Quantum cluster algebras and quantum nilpotent algebras

Definition of Quantum Nilpotent Algebras.

Let K be an arbitrary base field. A skew polynomial extension of a K-algebra A,A↦A[x;σ,δ],is a generalization of the classical polynomial algebra A[x]. It is defined using an algebra automorphism σ of A and a skew-derivation δ. The algebra A[x;σ,δ] is isomorphic to A[x] as a vector space, and the variable x commutes with the elements of A as follows:xa=σ(a)x+δ(a) for all a∈A.

For every nilpotent Lie algebra n of dimension m, there exists a chain of ideals of nn=nm⊳nm−1⊳…⊳n1⊳n0={0}such that dim(nk/nk−1)=1 and [n,nk]⊆nk−1 for 1≤k≤m. Choosing an element xk in the complement of nk−1 in nk for each 1≤k≤m leads to the following presentation of the universal enveloping algebra u(n) as an iterated skew polynomial extension:u(n)≅K[x1][x2;id,δ2]…[xm;id,δm],where all of the derivations δ2,…,δm are locally nilpotent.

Definition 1: An iterated skew polynomial extensionR=K[x1][x2;σ2,δ2]⋯[xN;σN,δN][1]is called a quantum nilpotent algebra if it is equipped with a rational action of a K-torus H by K-algebra automorphisms satisfying the following conditions:

• (a) The elements x1,…,xN are H-eigenvectors.

• (b) For every 2≤k≤N, δk is a locally nilpotent σk-derivation of K[x1]⋯[xk−1;σk−1,δk−1].

• (c) For every 1≤k≤N, there exists hk∈H such that σk=(hk⋅) and the hk-eigenvalue of xk, to be denoted by λk, is not a root of unity.

The universal enveloping algebras of finite dimensional nilpotent Lie algebras satisfy all of the conditions in the definition except for the last part of the third one. More precisely, in that case, one can take H={1}, conditions (a) and (b) are satisfied, and in condition (c), we have λk=1. Thus, condition (c) is the main feature that separates the class of quantum nilpotent algebras from the class of universal enveloping algebras of nilpotent Lie algebras. The torus H is needed to define the eigenvalues λk.

The algebras in Definition 1 are also known as Cauchon–Goodearl–Letzter extensions. The axiomatics came from the works of Goodearl and Letzter (13) and Cauchon (14), which investigated in this generality the stratification of the prime spectrum of an algebra into strata associated with its H-prime ideals.

The Gelfand-Kirillov dimension of the algebra in Eq. 1 equals N.

Example 1: For two positive integers m and n, and q∈K∗, define the algebra of quantum matrices Rq[Mm×n] as the algebra with generators t_ij, 1≤i≤m and 1≤j≤n, and relationstijtkj=qtkjtij, for  i<k,tijtil=qtiltij, for  j<l,tijtkl=tkltij, for  i<k, j>l,tijtkl−tkltij=(q−q−1)tiltkj, for  i<k, j<l.It is an iterated skew polynomial extension, whereRq[Mm×n]=K[x1][x2;σ2,δ2]…[xN;σN,δN],N=mn, and x(i−1)n+j=tij. It is easy to write explicit formulas for the automorphisms σk and skew derivations δk from the above commutation relations, and to check that each δk is locally nilpotent. The torus H=(K*)m+n acts on Rq[Mm×n] by algebra automorphisms by the rule(ξ1,…,ξm+n)⋅tij≔ξiξm+j−1tijfor all (ξ1,…,ξm+n)∈(K∗)m+n. Definehij≔(1,…,1,q−1,1,…,1,q,1,…,1)∈H,where q−1 and q reside in positions i and m+j, respectively. Then σ(i−1)n+j=(hij⋅) andhij⋅tij=q−2tij.Thus, for all q∈K∗ that are not roots of unity, the algebras Rq[Mm×n] are examples of quantum nilpotent algebras.

UFDs.

The notion of UFDs plays an important role in algebra and number theory. Its noncommutative analog was introduced by Chatters (15). A nonzero, noninvertible element p of a domain R (a ring without zero divisors) is called prime if pR=Rp and the factor R/Rp is a domain. A noetherian domain R is called a UFD if every nonzero prime ideal of R contains a prime element. Such rings possess the unique factorization property for all of their nonzero normal elements—the elements u∈R with the property that Ru=uR. If the ring R is acted upon by a group G, then one can introduce an equivariant version of this property. Such an R is called a G-UFD if every nonzero G-invariant prime ideal of R contains a prime element that is a G-eigenvector.

It was shown by Launois et al. (16) that every quantum nilpotent algebra R is a noetherian H-UFD. An H-eigenvector of such a ring R will be called a homogeneous element because this corresponds to the homogeneity property with respect to the canonically induced grading of R by the character lattice of H. In particular, we will use the more compact term of homogeneous prime element of R instead of a prime element of R that is an H-eigenvector.

Sequences of Prime Elements.

Next, we classify the set of all homogeneous prime elements of the chain of subalgebras{0}⊂R1⊂R2⊂…⊂RN=Rof a quantum nilpotent algebra R, where R_k is the subalgebra of R generated by the first k variables x1,…,xk.

We will denote by Z and Z≥0 the sets of all integers and nonnegative integers, respectively. Given two integers l≤k, set [l,k]≔{l,l+1,…,k}.

For a function η:[1,N]→Z, define the predecessor function p:[1,N]→[1,N]⊔{−∞} and successor function s:[1,N]→[1,N]⊔{+∞} for its level sets byp(k)={max{l<k|η(l)=η(k)},if such l exists,−∞,otherwise,ands(k)={min{l>k|η(l)=η(k)},if such l exists,+∞,otherwise.

Theorem 1.

Let R be a quantum nilpotent algebra of dimension N. There exist a function η:[1,N]→Z and elementsck∈Rk−1 for all 2≤k≤N  with  δk≠0such that the elements y1,…,yN∈R, recursively defined byyk≔{yp(k)xk−ck,if  δk≠0xk,if  δk=0,are homogeneous and have the property that for every k∈[1,N], the homogeneous prime elements of Rk are precisely the nonzero scalar multiples of the elementsyl for l∈[1,k] with s(l)>k.In particular, yk is a homogeneous prime element of Rk, for all k∈[1,N]. The sequence y1,…,yN and the level sets of a function η with these properties are both unique.

Example 2: Given two subsets I={i1<⋯<ik}⊂[1,m] and J={j1<⋯<jk}⊂[1,n], define the quantum minor ΔI,J∈Rq[Mm×n] byΔI,J=∑σ∈Sk(−q)ℓ(σ)ti1jσ(1)…tikjσ(k),where Sk denotes the symmetric group and ℓ:Sk→Z≥0 denotes the standard length function.

For the algebra of quantum matrices Rq[Mm×n], the sequence of prime elements from Theorem 1 consists of solid quantum minors; more precisely,y(i−1)n+j=Δ[i−min(i,j)+1,i],[j−min(i,j)+1,j]for all 1≤i≤m and 1≤j≤n. Furthermore, the function η:[1,mn]→Z can be chosen asη((i−1)n+j)≔j−i.

Definition 2: The cardinality of the range of the function η from Theorem 1 is called the rank of the quantum nilpotent algebra R and is denoted by rk(R).

For example, the algebra of quantum matrices Rq[Mm×n] has rank m+n−1.

Embedded Quantum Tori.

An N×N matrix q≔(qkl) with entries in K is called multiplicatively skewsymmetric ifqklqlk=qkk=1 for 1≤l,k≤N.Such a matrix gives rise to the quantum torus tq which is the K-algebra with generators Y1±1,…,YN±1 and relationsYkYl=qklYlYk for 1≤l,k≤N.

Let {e1,…,eN} be the standard basis of the lattice ZN. For a quantum nilpotent algebra R of dimension N, define the eigenvalues λkl∈K:hk⋅xl=λklxl for 1≤l<k≤N.[2]There exists a unique group bicharacter Ω:ZN×ZN→K* such thatΩ(ek,el)={1,if k=l,λkl,if k>l,λlk−1,if k<l.Set e−∞≔0. Define the vectorse¯k≔ek+ep(k)+⋯∈ZN,[3]noting that only finitely many terms in the sum are nonzero. Then {e¯1,…,e¯N} is another basis of ZN.

Theorem 2.

For each quantum nilpotent algebra R, the sequence of prime elements from Theorem 1 defines an embedding of the quantum torus Tq associated with the N×N multiplicatively skewsymmetric matrix with entriesqkl≔Ω(e¯k,e¯l), 1≤k,l≤Ninto the division ring of fractions Fract(R) of R such that Yk±1↦yk±1, for all 1≤k≤N.

Symmetric Quantum Nilpotent Algebras.

Definition 3: A quantum nilpotent algebra R as in Definition 1 will be called symmetric if it can be presented as an iterated skew polynomial extension for the reverse order of its generators,R=K[xN][xN−1;σN−1∗,δN−1∗]⋯[x1;σ1∗,δ1∗],in such a way that conditions (a)–(c) in Definition 1 are satisfied for some choice of hN∗,…,h1∗∈H.

A quantum nilpotent algebra R is symmetric if and only if it satisfies the Levendorskii–Soibelman type straightening lawxkxl−λklxlxk=∑nl+1,…,nk−1∈Z≥0ξnl+1,…,nk−1xl+1nl+1…xk−1nk−1for all l<k (where the ξ• are scalars) and there exist hk∗∈H such that hk∗⋅xl=λlk−1xl for all l>k. The defining commutation relations for the algebras of quantum matrices Rq[Mm×n] imply that they are symmetric quantum nilpotent algebras.

Denote by ΞN the subset of the symmetric group S_N consisting of all permutations that have the property thatτ(k)=maxτ([1,k−1])+1[4]

orτ(k)=min τ([1,k−1])−1[5]for all 2≤k≤N. In other words, ΞN consists of those τ∈SN such that τ([1,k]) is an interval for all 2≤k≤N. For each τ∈ΞN, a symmetric quantum nilpotent algebra R has the presentationR=K[xτ(1)][xτ(2);στ(2)″,δτ(2)″]⋯[xτ(N);στ(N)″,δτ(N)″],[6]where στ(k)″≔στ(k) and δτ(k)″≔δτ(k) if Eq. 4 is satisfied, whereas στ(k)″≔στ(k)∗ and δτ(k)″≔δτ(k)∗ if Eq. 5 holds. This presentation satisfies the conditions (a)–(c) in Definition 1 for the elements hτ(k)″∈H, given by hτ(k)″≔hτ(k) in case of Eq. 4 and hτ(k)″≔hτ(k)∗ in case of Eq. 5. The use of the term symmetric in Definition 3 is motivated by the presentations in Eq. 6 parametrized by the elements of the subset ΞN of the symmetric group S_N.

Proposition.

For every symmetric quantum nilpotent algebra R, the hk∗-eigenvalues of x_k, to be denoted by λk∗, satisfyλk∗=λl∗for all 1≤k,l≤N such that η(k)=η(l) and s(k)≠+∞, s(l)≠+∞. They are related to the eigenvalues λl byλk∗=λl−1for all 1≤k,l≤N such that η(k)=η(l) and s(k)≠+∞, p(l)≠−∞.

Construction of Exchange Matrices.

Our construction of a quantum cluster algebra structure on a symmetric quantum nilpotent algebra R of dimension N will have as the set of exchangeable indicesex≔{k∈[1,N]|s(k)≠+∞}.[7]We will impose the following two mild conditions:

• (A) The field K contains square roots λkl of the scalars λkl for 1≤l<k≤N such that the subgroup of K* generated by all of them contains no elements of order 2.

• (B) There exist positive integers d_n, n∈range(η), for the function η from Theorem 1 such that(λk∗)dη(l)=(λl∗)dη(k) for all k,l∈ex.

Remark 1: With the exception of some two-cocycle twists, for all of the quantum nilpotent algebras R coming from the theory of quantum groups, the subgroup of K* generated by {λkl|1≤l<k≤N} is torsion-free. For all such algebras, condition (A) only requires that K contains square roots of the scalars λk l.

All symmetric quantum nilpotent algebras that we are aware of satisfyλk∗=qmk for  1≤k≤Nfor some nonroot of unity q∈K* and positive integers m1,…,mN. The Proposition implies that all of them satisfy condition (B).

The set ex has cardinality N−rk(R), where rk(R) is the rank of the quantum nilpotent algebra R. By an N×ex matrix, we will mean a matrix of size N×(N−rk(R)) whose rows and columns are indexed by the sets [1,N] and ex, respectively. The set of such matrices with integer entries will be denoted by MN×ex(Z).

Theorem 4.

For every symmetric quantum nilpotent algebra R of dimension N satisfying conditions (A) and (B), there exists a unique matrix B˜=(blk)∈MN×ex(Z) whose columns satisfy the following two conditions:

• i) Ω(∑l=1Nblkel¯,e¯n)={λn∗,if k=n1,if k≠n,for all k∈ex and n∈[1,N] (a system of linear equations written in a multiplicative form);

• ii) The products y1b1k…yNbNk are fixed under H for all k∈ex (a homogeneity condition).

The second condition can be written in an explicit form using the fact that y_k is an H-eigenvector and its eigenvalue equals the product of the H-eigenvalues of xk,…,xpnk(k), where n_k is the maximal nonnegative integer n such that pn(k)≠−∞. This property is derived from Theorem 1.

Example 3: It follows from Example 2 that in the case of the algebras of quantum matrices Rq[Mm×n],ex={(i−1)n+j|1≤i<m,1≤j<n}.The bicharacter Ω:Zmn×Zmn→K∗ is given byΩ(e(i−1)n+j,e(k−1)n+l)=qδjlsign(k−i)+δiksign(l−j)for all 1≤i,k≤m and 1≤j,l≤n. Furthermore,λs∗=q2 for  1≤s≤mn.After an easy computation, one finds that the unique solution of the system of equations in Theorem 4 is given by the matrix B˜=(b(i−1)n+j,(k−1)n+j)∈Mmn×ex(Z) with entriesb(i−1)n+j,(k−1)n+l={±1,if i=k,l=j±1or j=l,k=i±1or i=k±1,j=l±1,0,otherwisefor all i,k∈[1, m] and j,l∈[1,n].

Cluster Algebra Structures.

Let us consider a symmetric quantum nilpotent algebra R of dimension N. When Theorem 1 is applied to the presentation of R from Eq. 6 associated with the element τ∈ΞN, we obtain a sequence of prime elementsyτ,1,…,yτ,N∈R.Similarly, applying Theorem 4 to the presentation from Eq. 6, we obtain the integer matrixB˜τ∈MN×ex(Z).For τ=id, we recover the original sequence y1,…,yN and matrix B˜.

Theorem 5.

Every symmetric quantum nilpotent algebra R of dimension N satisfying the conditions (A) and (B) possesses a canonical structure of quantum cluster algebra for which no frozen cluster variables are inverted. Its initial seed has:

• i) Cluster variables ζ1y1,…,ζNyN for some ζ1,…,ζN∈K*, among which the variables indexed by the set ex from Eq. 7 are exchangeable and the rest are frozen;

• ii) Exchange matrix B˜ given by Theorem 4.

Furthermore, this quantum cluster algebra aways coincides with the corresponding upper quantum cluster algebra.

After an appropriate rescaling, each of the generators x_k of such an algebra R given by Eq. 1 is a cluster variable. Moreover, for each element τ of the subset ΞN of the symmetric group S_N, R has a seed with cluster variables obtained by reindexing and rescaling the sequence of prime elements yτ,1,…,yτ,N. The exchange matrix of this seed is the matrix B˜τ.

The base fields of the algebras covered by this theorem can have arbitrary characteristic. We refer the reader to theorem 8.2 of ref. 17 for a complete statement of the theorem, which includes additional results and gives explicit formulas for the scalars ζ1,…,ζN and the necessary reindexing and rescaling of the sequences yτ,1,…,yτ,N.

Define the following automorphism of the lattice ZN:g=l1e1+⋯+lNeN ↦ g¯≔l1e¯1+⋯+lNe¯Nfor all l1,…,lN∈Z in terms of the vectors e¯1,…,e¯N from Eq. 3. The construction of seeds for quantum cluster algebras in the work of Berenstein and Zelevinsky (11) requires assigning quantum frames to all of them. The quantum frame M:ZN→Fract(R) associated with the initial seed for the quantum cluster algebra structure in Theorem 5 is uniquely reconstructed from the rulesM(ek)=ζkyk for 1≤k≤NandM(f+g)=Ω(f¯,g¯)M(f)M(g) for f,g∈ZN.Analogous formulas describe the quantum frames associated with the elements τ of the set ΞN.

Example 4: The cluster variables in the initial seed from Theorem 5 for Rq[Mm×n] areΔ[i−min(i,j)+1,i],[j−min(i,j)+1,j],where 1≤i≤m and 1≤j≤n. The ones with i=m or j=n are frozen. This example and Example 3 recover the quantum cluster algebra structure of Geiß et al. (12) on Rq[Mm×n].

We finish the section by raising two questions concerning the line of Theorem 5:

• 1. If a symmetric quantum nilpotent algebra has two iterated skew polynomial extension presentations that satisfy the assumptions in Definition 1 and Definition 3, and these two presentations are not obtained from each other by a permutation in ΞN, how are the corresponding quantum cluster algebra structures on R related?

• 2. What is the role of the quantum seeds of a symmetric quantum nilpotent algebra R indexed by ΞN among the set of all quantum seeds? Is there a generalization of Theorem 5 that constructs a larger family of quantum seeds using sequences of prime elements in chains of subalgebras?

For the first question, we expect that the two quantum cluster algebra structures on R are the same (i.e., the corresponding quantum seeds are mutation-equivalent) if the maximal tori for the two presentations are the same and act in the same way on R. However, proving such a fact appears to be difficult due to the generality of the setting. The condition on the tori is natural in light of theorem 5.5 of ref. 18, which proves the existence of a canonical maximal torus for a quantum nilpotent algebra. Without imposing such a condition, the cluster algebra structures can be completely unrelated. For example, every polynomial algebraR=K[x1,…,xN]over an infinite field K is a symmetric quantum nilpotent algebra with respect to the natural action of (K*)N. The quantum cluster algebra structure on R constructed in Theorem 5 has no exchangeable indices, and its frozen variables are x1,…,xN. Each polynomial algebra has many different presentations associated with the elements of its automorphism group, and the corresponding cluster algebra structures are not related in general.

Quantized Universal Enveloping Algebra.

Let g be a finite dimensional complex simple Lie algebra of rank r with Cartan matrix (cij). For an arbitrary field K and a nonroot of unity q∈K*, following the notation of Jantzen (19), one defines the quantized universal enveloping algebra uq(g) with generatorsKi±1,Ei,Fi,  1≤i≤rand relationsKi−1Ki=KiKi−1=1, KiKj=KjKi,KiEjKi−1=qicijEj, KiFjKi−1=qi−cijFj,EiFj−FjEi=δi,jKi−Ki−1qi−qi−1,∑n=01−cij(−1)n[1−cijn]qi(Ei)nEj(Ei)1−cij−n=0,  i≠j,together with the analogous relation for the generators F_i. Here, {d1,…,dr} is the collection of relatively prime positive integers such that the matrix (dicij) is symmetric and qi≔qdi. The algebra uq(g) is a Hopf algebra with coproductΔ(Ki)=Ki⊗Ki, Δ(Ei)=Ei⊗1+Ki⊗Ei,Δ(Fi)=Fi⊗Ki−1+1⊗Fi.

The quantum Schubert cell algebras u±[w], parametrized by the elements w of the Weyl group W of g, were introduced by De Concini et al. (20) and Lusztig (21). In the work of Geiß et al. (12), the term quantum unipotent groups was used. These algebras are quantum analogs of the universal enveloping algebras u(n±∩​w(n∓)), where n± are the nilradicals of a pair of opposite Borel subalgebras of g. The torus H≔(K*)r acts on uq(g) byh⋅Ki±1=Ki±1, h⋅Ei=ξiEi, h⋅Fi=ξi−1Fi,[8]for all h=(ξ1,…,ξr)∈H and 1≤i≤r. The subalgebras u±[w] are preserved by this action.

Denote by α1,…,αr the set of simple roots of g and by s1,…,sr∈W the corresponding set of simple reflections. All algebras u±[w] are symmetric quantum nilpotent algebras for all base fields K and nonroots of unity q∈K*. In fact, with each reduced expressionw=si1…siN,one associates a presentation of u±[w] that satisfies Definition 1 and Definition 3 as follows. Consider the Weyl group elementsw≤k≔si1…sik, 1≤k≤N, and w≤0≔1.In terms of these elements, the roots of the Lie algebra n+∩​w(n−) areβ1=αi1, β2=w≤1(αi2), …, βN≔w≤N−1(αiN).As in the works of Jantzen (19) and Lusztig (21), one associates with those roots the Lusztig root vectors Eβ1,Fβ1,…,EβN,FβN∈uq(g). The quantum Schubert cell algebra u−[w], defined as the subalgebra of uq(g) generated by Fβ1,…,FβN, has an iterated skew polynomial extension presentation of the formu−[w]=K[Fβ1][Fβ2;σ2,δ2]…[FβN;σN,δN],[9]for which conditions (a)–(c) of Definition 1 are satisfied with respect to the action of Eq. 8. Moreover, this presentation satisfies the condition for a symmetric quantum nilpotent algebra in Definition 3 because of the Levendorskii–Soibelman straightening law (19) in uq(g). The opposite algebra u+[w], generated by Eβ1,…,EβN, has analogous properties and is actually isomorphic (19) to u−[w].

The algebra Rq[Mm×n] is isomorphic to one of the algebras u−[w] for g=slm+n and a certain choice of w∈Sm+n.

Quantized Function Algebras.

The irreducible finite dimensional modules of uq(g) on which the elements K_i act diagonally via powers of the scalars q_i are parametrized by the set P+ of dominant integral weights of g. The module corresponding to such a weight λ will be denoted by V(λ).

Let G be the connected, simply connected algebraic group with Lie algebra g. The Hopf subalgebra of uq(g)* spanned by the matrix coefficients cf,yλ of all modules V(λ) [where f∈V(λ)* and y∈V(λ)] is denoted by Rq[G] and called the quantum group corresponding to G. The weight spaces V(λ)wλ are one-dimensional for all Weyl group elements w. Considering the fundamental representations V(ϖ1),…,V(ϖr), and a normalized covector and vector in each of those weight spaces, one defines, following Berenstein and Zelevinsky (11), the quantum minorsΔw,vi=Δwϖi,vϖi∈Rq[G], 1≤i≤r, w,v∈W.

The subalgebras of Rq[G] spanned by the elements of the form cf,yλ, where y is a highest or lowest weight vector of V(λ) and f∈V(λ)*, are denoted by R±. They are quantum analogs of the base affine space of G. With the help of the Demazure modules Vw+(λ)=uq+(g)V(λ)wλ [where uq+(g) is the unital subalgebra of uq(g) generated by E1,…,Er], one defines the idealIw+=Span{cf,yλ|λ∈P+, f|Vw+(λ)=0, y∈V(λ)λ}of R+. Analogously one defines an ideal Iw− of R−. For all pairs of Weyl group elements (w,v), the quantized coordinate ring of the double Bruhat cell (9)Gw,v≔B+wB+∩​B−vB−,where B± are opposite Borel subgroups of G, is defined byRq[Gw,v]≔(Iw+R−+R+Iv−)[(Δw,1i)−1,(Δvw0,w0i)−1],where the localization is taken over all 1≤i≤r and w₀ denotes the longest element of the Weyl group W.

To connect cluster algebra structures on the two kinds of algebras (quantum Schubert cells and quantum double Bruhat cells), we use certain subalgebras Sw+ of (R+/Iw+)[(Δw,1i)−1,1≤i≤r], which were defined by Joseph (22). They are the subalgebras generated by the elements(Δw,1i)−1(cf,yϖi+Iw+),for 1≤i≤r, f∈V(ϖi)*, and y a highest weight vector of V(ϖi). These algebras played a major role in the study of the spectra of quantum groups (22, 23) and the quantum Schubert cell algebras (24). In ref. 23, Yakimov constructed an algebra antiisomorphismφw:Sw+→u−[w].An earlier variant of it for uq(g) equipped with a different coproduct appeared in the work of Yakimov (24).

Cluster Structures on Quantum Schubert Cell Algebras.

Denote by 〈.,.〉 the Weyl group invariant bilinear form on the vector space ℝα1⊕⋯⊕ℝαr normalized by 〈αi,αi〉=2 for short roots αi. Let ||γ||2≔〈γ,γ〉.

Fix a Weyl group element w, and consider the quantum Schubert cell algebra u−[w]. A reduced expression w=si1…siN gives rise to the presentation in Eq. 9 of u−[w] as a symmetric quantum nilpotent algebra. The result of the application of Theorem 1 to it is as follows. The function η can be chosen asη(k)=ik for all 1≤k≤N.The predecessor function p is the function k↦k−, which plays a key role in the works of Fomin and Zelevinsky (1, 9),k−≔{max{l<k|il=ik},if such l exists,−∞,otherwise.The successor function s is the function k↦k+ in the works of Fomin and Zelevinsky (1, 9). The sequence of prime elements y1,…,yN consists of scalar multiples of the elementsφw(Δw≤pnk(k)−1,1ik(Δw≤k,1ik)−1), 1≤k≤N,where n_k denotes the maximal nonnegative integer n such that pn(k)≠−∞. The presentation in Eq. 9 of u−[w] as a symmetric quantum nilpotent algebra satisfies the conditions (A) and (B) if q∈K, in which case Theorem 5 produces a canonical cluster algebra structure on u−[w]. Among all of the clusters in Theorem 5, indexed by the elements of the subset ΞN of the symmetric group S_N, the one corresponding to the longest element of S_N is closest to the combinatorial setting of (10, 11). It goes with the reverse presentation of u−[w],u−[w]=K[FβN][FβN−1;σN−1∗,δN−1∗]…[Fβ1;σ1∗,δ1∗].The transition from the original presentation in Eq. 9 to the above one amounts to interchanging the roles of the predecessor and successor functions. As a result, the set of exchangeable indices for the latter presentation isexw≔{k∈[1,N]|k−≠−∞}.[10]

Theorem 6.

Consider an arbitrary finite dimensional complex simple Lie algebra g, a Weyl group element w∈W, a reduced expression of w, an arbitrary base field K, and a nonroot of unity q∈K* such that q∈K. The quantum Schubert cell algebra u−[w] possesses a canonical quantum cluster algebra structure for which no frozen cluster variables are inverted and the set of exchangeable indices is ex_w. Its initial seed consists of the cluster variablesq||(w−w≤k−1)ϖik||2/2φw(Δw≤k−1,1ik(Δw,1ik)−1),1≤k≤N. The exchange matrix B˜ of this seed has entries given bybkl={1,if k=p(l)−1,if k=s(l)cikil,if p(k)<p(l)<k<l−cikil,if p(l)<p(k)<l<k0,otherwisefor all 1≤k≤N and l∈exw. Furthermore, this quantum cluster algebra equals the corresponding upper quantum cluster algebra. For all k∈[1,N], m∈Z≥0 such that sm(k)∈[1,N], the elementsq||(w≤sm(k)−w≤k−1)ϖik||2/2   ×φw≤sm(k)(Δw≤k−1,1ik(Δw≤sm(k),1ik)−1)are cluster variables of u−[w].