Greedy bases in rank 2 quantum cluster algebras Kyungyong Leea, Li Lib, Dylan Rupelc,1, and Andrei Zelevinskyc,2 a Department of Mathematics, Wayne State University, Detroit, MI 48202; bDepartment of Mathematics and Statistics, Oakland University, Rochester, MI 48309; and cDepartment of Mathematics, Northeastern University, Boston, MA 02115
Edited by Bernard Leclerc, University of Caen, France, and accepted by the Editorial Board December 5, 2013 (received for review August 16, 2013)
We identify a quantum lift of the greedy basis for rank 2 coefficientfree cluster algebras. Our main result is that our construction does not depend on the choice of initial cluster, that it builds all cluster monomials, and that it produces bar-invariant elements. We also present several conjectures related to this quantum greedy basis and the triangular basis of Berenstein and Zelevinsky. standard monomial basis
| triangular basis
C
luster algebras were introduced by Fomin and Zelevinsky (1) as a combinatorial tool for understanding the canonical basis and positivity phenomena in the coordinate ring of an algebraic group. This was followed by the definition of quantum cluster algebras by Berenstein and Zelevinsky (2) as a similar combinatorial tool for understanding canonical bases of quantum groups. In connection with these foundational goals, it is important to understand the various bases of cluster algebras and quantum cluster algebras. In this paper, we confirm the existence of a quantum greedy basis in rank 2 quantum cluster algebras. It is independent of the choice of an initial cluster, contains all cluster monomials, and specializes at v = 1 to the greedy basis of a classical rank 2 cluster algebra introduced by Lee et al. (3). Another important basis in an acyclic quantum cluster algebra is the triangular basis of Berenstein and Zelevinsky (4). Like triangular bases, quantum greedy bases can be easily computed. This enables us to study and compare these bases computationally. The structure of the paper is as follows. Section 1 begins with a recollection of rank 2 commutative cluster algebras. We recall the construction of greedy bases here and in the process we provide an axiomatic description. In Sec. 2 we review the definition of rank 2 quantum cluster algebras. In Sec. 3 we present our main result: the existence of a quantum lift of the greedy basis using the axiomatic description. Section 4 recalls the definition of the triangular basis and Sec. 5 provides an overview of results and open problems on the comparison between various bases of rank 2 quantum cluster algebras.
The proof of this theorem in particular establishes a connection between denominator vectors of cluster variables and almost positive roots in a root system Φ ⊂ Z2 . Thus, we say Aðb; cÞ is of affine (respectively, wild) type if bc= 4 (respectively, bc ≥ 5). T ±1 is called universally Laurent An element x ∈ m∈Z Z xm± 1 ; xm+1 because the expansion of x in every cluster is Laurent. If for a nonzero element x the coefficients of the Laurent expansion in each cluster are positive integers, then x is called universally positive. A universally positive element in Aðb; cÞ is said to be indecomposable if it cannot be expressed as a sum of two universally positive elements. Sherman and Zelevinsky (6) studied in great detail the collection of indecomposable universally positive elements. One of their main results is the following: if the commutative cluster algebra Aðb; cÞ is of finite or affine type, then the indecomposable universally positive elements form a Z basis in Aðb; cÞ, moreover this basis contains the 2 set of all cluster monomials fxam1 xam+1 : a1 ; a2 ∈ Z≥0 ; m ∈ Zg. However, the situation becomes much more complicated in wild types. In particular, it was shown by Lee et al. (7) that for bc ≥ 5 the indecomposable universally positive elements of Aðb; cÞ are not linearly independent. The greedy basis of Aðb; cÞ introduced by Lee et al. (3) is a subset of the indecomposable universally positive elements which admits a beautiful combinatorial description. In this section we will describe an axiomatic characterization of the greedy basis. The elements of the greedy basis take on a particular form which is motivated by a well-known pattern in the initial cluster expansion of cluster monomials. An element x ∈ Aðb; cÞ is pointed at ða1 ; a2 Þ ∈ Z2 if it can be written in the form X cq 1 −a2 eð p; qÞxbp x = x−a 1 x2 1 x2 p;q≥0
with eð0; 0Þ = 1 and eðp; qÞ = eða1 ; a2 ; p; qÞ ∈ Z for all p; q ≥ 0. It is well-known that xa11 xa22 is the denominator of a cluster monomial im if and only if ða1 ; a2 Þ ∈ Z2 ∖Φim + (cf. 8, 9), where Φ+ is the set of positive imaginary roots, i.e.,
1. Rank 2 Cluster Algebras and Their Greedy Bases Fix positive integers b; c > 0. The commutative cluster algebra Aðb; cÞ is the Z subalgebra of Qðx1 ; x2 Þ generated by the cluster variables fxm gm∈Z , where the xm are rational functions in x1 and x2 defined recursively by the exchange relations b x + 1 if m is odd; xm+1 xm−1 = m xcm + 1 if m is even: It is a fundamental result of Fomin and Zelevinsky (1) that, although the exchange relations appear to produce rational functions, one always obtains a Laurent polynomial whose denominator is simply a monomial in x1 and x2 . They actually showed the following slightly stronger result. Theorem 1. (ref. 1, theorem 3.1, Laurent phenomenon) For ±1 any m ∈ Z we have Aðb; cÞ ⊂ Z½xm± 1 ; xm+1 . The cluster algebra Aðb; cÞ is of finite type if the collection of all cluster variables is a finite set. Fomin and Zelevinsky (5) went on to classify cluster algebras of finite type. Theorem 2. (ref. 5, theorem 1.4) The cluster algebra Aðb; cÞ is of finite type if and only if bc ≤ 3. 9712–9716 | PNAS | July 8, 2014 | vol. 111 | no. 27
Significance The quantum cluster algebras are a family of noncommutative rings introduced by Berenstein and Zelevinsky as the quantum deformation of the commutative cluster algebras. At the heart of their definition is a desire to understand bases of quantum algebras arising from the representation theory of nonassociative algebras. Thus a natural and important problem in the study of quantum cluster algebras is to study their bases with good properties. In this paper, we lay out a framework for understanding the interrelationships between various bases of rank two quantum cluster algebras. Author contributions: K.L., L.L., D.R., and A.Z. designed research; K.L., L.L., D.R., and A.Z. performed research; K.L., L.L., D.R., and A.Z. analyzed data; and K.L., L.L., and D.R. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. B.L. is a guest editor invited by the Editorial Board. 1
To whom correspondence should be addressed. Email: [email protected].
2
Deceased April 10, 2013.
www.pnas.org/cgi/doi/10.1073/pnas.1313078111
For ða1 ; a2 Þ ∈ Φim + , define the region ba2 p < a1 or Rgreedy = ðp; qÞ ∈ R2≥0 j q + b − ca1 S ca1 p+ c− q < a2 fð0; a1 Þ; ða2 ; 0Þg: ba2 In other words, Rgreedy is the region bounded by the broken line ð0; 0Þ; ða2 ; 0Þ; ða1 =b; a2 =cÞ; ð0; a1 Þ; ð0; 0Þ; with the convention that this region includes the closed segments ½ð0; 0Þ; ða2 ; 0Þ and ½ð0; a1 Þ; ð0; 0Þ but excludes the rest of the boundary (Fig. 1). We can define greedy elements in two different ways, either by axioms or by recurrence relations. Here we choose to define them using recurrence relations as follows. Theorem 3. (ref. 3, proposition 1.6) For each ða1 ; a2 Þ ∈ Z2 , there exists a unique element in Aðb; cÞ pointed at ða1 ; a2 Þ whose coefficients eð p; qÞ satisfy the following recurrence relation: eð0; 0Þ = 1, eð p; qÞ = 8 p X > ½a2 − cq+ + k − 1 k−1 > > ð−1Þ eðp − k; qÞ > > k < k=1 q > X > ½a1 − bp+ + ℓ − 1 > ℓ−1 > ð−1Þ eðp; q − ℓÞ > : ℓ
if ca1 q ≤ ba2 p; if ca1 q ≥ ba2 p;
ℓ=1
where we use the standard notation ½a+ = maxða; 0Þ. We define the greedy element pointed at ða1 ; a2 Þ, denoted x½a1 ; a2 , to be the unique element determined by Theorem 3. For ða1 ; a2 Þ ∈ Φim + , we give a direct characterization for the greedy element. Theorem 4. For each ða1 ; a2 Þ ∈ Φim + , the greedy element x½a1 ; a2 is the unique element in Aðb; cÞ pointed at ða1 ; a2 Þ whose coefficients satisfy the following two axioms: • (Support) eðp; qÞ = 0 for ð p; qÞ ∉ Rgreedy ; • (Divisibility)
X
‒ if a2 > cq; then ð1 + tÞa2 −cq
eði; qÞti ;
(a) The greedy elements x½a1 ; a2 for ða1 ; a2 Þ ∈ Z2 form a Z basis in Aðb; cÞ, which we refer to as the greedy basis. (b) The greedy basis is independent of the choice of an initial cluster. (c) The greedy basis contains all cluster monomials. (d) Greedy elements are universally positive and indecomposable. Our goal in this work is to generalize the above theorem to the setting of rank 2 quantum cluster algebras. The proof of Theorem 5 given in ref. 3 uses combinatorial objects called compatible pairs in an essential way (cf. ref. 3, theorem 1.11). Unfortunately this method has difficulties–limitations in generalizing to the study of quantum cluster algebras. More precisely, if one could simply assign a power of v to each compatible pair then the quantum greedy elements would have again been universally positive, which is unfortunately false in general (see the example at the end of Sec. 3). This justifies our approach to Theorem 5, with the exception of (d), which can easily be generalized to the quantum setting. 2. Rank 2 Quantum Cluster Algebras In this section we define our main objects of study, namely quantum cluster algebras, and recall important fundamental facts related to these algebras. We restrict attention to rank 2 quantum cluster algebras where we can describe the setup in very concrete terms. We follow (as much as possible) the notation and conventions of refs. 3, 4. Consider the following quantum torus: E D T = Z v ± 1 X1± 1 ; X2± 1 : X2 X1 = v2 X1 X2 (this setup is related to the one in ref. 10, which uses the formal variable q instead of v by setting q = v−2 ). There are many choices for quantizing cluster algebras; to rigidify the situation we require the quantum cluster algebra to be invariant under a certain involution. The bar involution is the Z-linear antiautomorphism of T determined by f ðvÞ = f ðv−1 Þ for f ∈ Z½v ± 1 and
i
X
eðp; iÞti : ‒ if a1 > bp; then ð1 + tÞa1 −bp
i
The key step in the proof of Theorem 4 is based on an observation made in ref. 3, sec. 2: the first (respectively, second)
fX1a1 X2a2 = f X2a2 X1a1 = v2a1 a2 f X1a1 X2a2 ða1 ; a2 ∈ ZÞ: An element which is invariant under the bar involution is said to be bar-invariant. Let F be the skew field of fractions of T . The quantum cluster algebra Av ðb; cÞ is the Z½v ± 1 subalgebra of F generated by the quantum cluster variables fXm gm∈Z defined recursively from the quantum exchange relations Xm+1 Xm−1 =
vb Xmb + 1 vc Xmc + 1
if m is odd; if m is even:
By a simple induction one can easily check the following (quasi-) commutation relations between neighboring cluster variables Xm ; Xm+1 in Av ðb; cÞ: Xm+1 Xm = v2 Xm Xm+1 ðm ∈ ZÞ:
Fig. 1.
Lee et al.
Support region of (quantum) greedy elements.
SPECIAL FEATURE
recurrence relation in Theorem 3 is equivalent to the vanishing of the (p, q)th coefficient of the Laurent expansion of x½a1 ; a2 with respect to the cluster fx0 ; x1 g (respectively, fx2 ; x3 g). It is straightforward to check that such vanishing implies the Support and Divisibility axioms in Theorem 4. The following theorem summarizes the results from ref. 3. Theorem 5. (ref. 3, theorem 1.7)
[1]
It then follows that all cluster variables are bar-invariant, therefore Av ðb; cÞ is also stable under the bar involution. Moreover, Eq. 1 implies that each cluster fXm ; Xm+1 g generates a quantum torus PNAS | July 8, 2014 | vol. 111 | no. 27 | 9713
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2 2 2 Φim + d ða1 ; a2 Þ ∈ Z>0 : ca1 − bca1 a2 + ba2 ≤ 0 :
±1 T m = Z v ± 1 Xm± 1 ; Xm+1 : Xm+1 Xm = v2 Xm Xm+1 : It is easy to see that the bar involution does not depend on the choice of an initial quantum torus T m . The appropriate quantum analogs of cluster monomials for Av ðb; cÞ are the (bar-invariant) quantum cluster monomials which are certain elements of a quantum torus T m , more precisely they are a2 Xmða1 ;a2 Þ = va1 a2 Xma1 Xm+1 ða1 ; a2 ∈ Z≥0 ; m ∈ ZÞ:
The following quantum analog of the Laurent phenomenon was proven by Berenstein and Zelevinsky (2). Theorem 6 (2). For any m ∈ Z we have Av ðb; cÞ ⊂ T m . Moreover, T Av ðb; cÞ = T m: m∈Z
A nonzero elementh of Av ðb; cÞiis called universally positive if it T ±1 . A universally positive element lies in m∈Z Z≥0 v ± 1 Xm± 1 ; Xm+1 in Av ðb; cÞ is indecomposable if it cannot be expressed as a sum of two universally positive elements. It is known as a very special case of the results of refs. 11–15 that cluster monomials are universally positive Laurent polynomials. Explicit combinatorial expressions for these positive coefficients can be obtained from the results of refs. 16–18. 3. Quantum Greedy Bases This section contains the main result of the paper. Here we introduce the quantum greedy basis and present its nice properties. Analogous to the construction of greedy elements, the elements of the quantum greedy basis take on the following particular form. An element X ∈ Av ðb; cÞ is said to be pointed at ða1 ; a2 Þ ∈ Z2 if it has the form X ðbp−a1 ;cq−a2 Þ X= eð p; qÞX1 p;q≥0
with eð0; 0Þ = 1 and eð p; qÞ ∈ Z½v ± 1 for all p and q. The next theorem is an analog of Theorem 3; to state our result precisely we need more notation. Let w denote a formal invertible variable. For n ∈ Z, k ∈ Z≥0 define the bar-invariant quantum numbers and bar-invariant quantum binomial coefficients by wn − w−n = wn−1 + wn−3 + ⋯ + w−n+1 ; ½nw = w − w−1 " # n ½n ½n − 1w ⋯½n − k + 1w = w : ½kw ½k − 1w ⋯½1w k w h i n Recall that k will always be a Laurent polynomial in w. Hence w
taking w = v or v we obtain Laurent polynomials in v as well. Theorem 7. For each ða1 ; a2 Þ ∈ Z2 , there exists a unique element in Av ðb; cÞ pointed at ða1 ; a2 Þ whose coefficients eðp; qÞ satisfy the following recurrence relation: eð0; 0Þ = 1, b
c
eðp; " # 8 pqÞ = X ½a2 − cq+ + k − 1 > k−1 > > ð−1Þ eðp − k; qÞ > > < k k=1
if ca1 q ≤ ba2 p; vb
" # q > X > ½a1 − bp+ + ℓ − 1 > ℓ−1 > ð−1Þ eðp; q − ℓÞ > : ℓ ℓ=1 vc
if ca1 q ≥ ba2 p:
We define the quantum greedy element pointed at ða1 ; a2 Þ, denoted X½a1 ; a2 , to be the unique element determined by Theorem 7. 9714 | www.pnas.org/cgi/doi/10.1073/pnas.1313078111
The following theorem is analogous to Theorem 4. Theorem 8. For each ða1 ; a2 Þ ∈ Φim + , the quantum greedy element X½a1 ; a2 is the unique element in Av ðb; cÞ pointed at ða1 ; a2 Þ whose coefficients satisfy the following two axioms: • (Support) eðp; qÞ = 0 for ðp; qÞ ∉ Rgreedy ; • (Divisibility) Let t denote a formal variable which commutes with v. Then X a2 −cq
− ∏ 1 + vbða2 −cq+1−2jÞ t eði; qÞti ; j=1
i
a1 −bp
− ∏
X
1 + vcða1 −bp+1−2jÞ t eðp; iÞti :
j=1
i
Theorem 8 follows from an argument similar to the proof of Theorem 4. Our main theorem below states that quantum greedy elements possess all of the desired properties described in Theorem 5 except the positivity (d). Theorem 9. (Main Theorem) (a) The quantum greedy elements X½a1 ; a2 for ða1 ; a2 Þ ∈ Z2 form a Z½v ± 1 basis in Av ðb; cÞ, which we refer to as the quantum greedy basis. (b) The quantum greedy basis is bar-invariant and independent of the choice of an initial cluster. (c) The quantum greedy basis contains all cluster monomials. (d) If X½a1 ; a2 is universally positive, then it is indecomposable. (e) The quantum greedy basis specializes to the commutative greedy basis by the substitution v = 1. Full proofs of Theorems 7, 8, and 9 will appear elsewhere. The hardest part is to show the existence of quantum greedy elements, i.e., that the recursions in Theorem 7 terminate. The main idea in the proof is to realize X½a1 ; a2 as a degeneration of certain elements in Av ðb; cÞ whose supports are contained in the closure of Rgreedy . We end this section with an example where positivity of quantum greedy elements fails. Let ðb; cÞ = ð2; 3Þ and consider the pointed coefficient eð2; 1Þ in the Laurent expansion of X½3; 4. Using Theorem 7, we see that eð2; 1Þ is not in Z≥0 ½v ± 1 : 2 1 − eð0; 1Þ = eð1; 1Þ − eð0; 1Þ eð2; 1Þ = eð1; 1Þ 2 v2 1 v2 1 4 3 = eð1; 0Þ − eð0; 1Þ = eð0; 0Þ − eð0; 0Þ 1 v3 1 v2 1 v3 = v6 + v2 + v−2 + v−6 − v6 + 1 + v−6 = v2 − 1 + v−2 : Further computation shows that the greedy elements X½3; 5; X½5; 4; X½5; 7; X½5; 8; X½7; 5; X½7; 10; X½7; 11; etc: are not positive. Similarly, positivity can be seen to fail for a large class of quantum greedy elements when ðb; cÞ = ð2; 5Þ; ð3; 4Þ; ð4; 6Þ: 4. Triangular Bases The construction of the triangular basis begins with the standard monomial basis. For every ða1 ; a2 Þ ∈ Z2 , we define the standard monomial M½a1 ; a2 (which is denoted Eð−a1 ;−a2 Þ in ref. 4) by setting ½a
½−a1 +
M½a1 ; a2 = va1 a2 X3 1 + X1
½−a2 +
X2
½a
X0 2 + :
[2]
It is known (2) that the elements M½a1 ; a2 form a Z½v ± 1 basis of Av ðb; cÞ. Lee et al.
C½a1 ; a2 − M½a1 ; a2 ∈ L: Using the universal acyclicity of rank 2 quantum cluster algebras, we apply the following theorem. Theorem 10. (ref. 4, theorem 1.6]) The triangular basis does not depend on the choice of initial cluster and it contains all cluster monomials. Similar to the support condition for (quantum) greedy elements, we make the following conjecture about the supports of triangular elements. Conjecture 11. Let ða1 ; a2 Þ ∈ Φim + . For 0 ≤ p ≤ a2 , 0 ≤ q ≤ a1 , the pointed coefficient eðp; qÞ of X ðbp−a1 ;cq−a2 Þ in C½a1 ; a2 is nonzero if and only if bp2 + bcpq + cq2 ≤ ca1 q + ba2 p; (Fig. 2). 5. Open Problems In this section we present open problems and conjectures relating to the triangular basis and quantum greedy basis of a rank 2 quantum cluster algebra. Our aim is to find Z½v ± 1 bases of Av ðb; cÞ satisfying the properties laid out in the following definition.
SPECIAL FEATURE
Fig. 3. Properties of the quantum greedy basis for bjc or cjb.
A basis B for Av ðb; cÞ is said to be strongly positive if the following hold: (1) (2) (3) (4)
each element of B is bar-invariant; B is independent of the choice of an initial cluster; B contains all cluster monomials; any product of elements from B can be expanded as a linear combination of elements of B with coefficients in Z≥0 ½v ± 1 .
Note that all elements in a strongly positive basis are universally positive. Indeed, consider multiplying a basis element B ∈ B ða ;a Þ by a cluster monomial X1 1 2 with a1 ; a2 sufficiently large to clear the denominator in the initial cluster expansion of B. By properties (3) and (4) of a strongly positive basis the product ða ;a Þ
BX1 1 2 is in Z≥0 ½v ± 1 ½X1 ; X2 and thus we see that a strongly positive basis element has nonnegative coefficients in its initial cluster expansion. It then follows that B is universally positive because it is independent of the choice of an initial cluster. Kimura and Qin (15) showed the existence of bases for acyclic skew-symmetric quantum cluster algebras [hence for Av ðb; cÞ with b = c] which satisfy (1), (3), and (4). In the rank 2 case, their bases are exactly the triangular bases, and hence satisfy (2) as well (see ref. 4, remark 1.8]). Conjecture 12. Given any strongly positive basis B, there exists a unique basis element pointed at ða1 ; a2 Þ for each ða1 ; a2 Þ ∈ Z2 . It is easy to see that property (3) of strongly positive bases together with Conjecture 12 implies that every element of B is pointed. We note that the conclusion of Conjecture 12 holds for both the quantum greedy basis and the triangular basis. In what follows, we write Y ≤ Z for Y ; Z ∈ Av ðb; cÞ if Z − Y is a Laurent polynomial with coefficients in Z≥0 ½v ± 1 . Conjecture 13. (a) There exists a unique strongly positive basis Bupper satisfying the basis element in the basis element in ≤ B pointed at ða1 ; a2 Þ Bupper pointed at ða1 ; a2 Þ for every strongly positive basis B and every ða1 ; a2 Þ ∈ Z2 . (b) The triangular basis is Bupper . Conjecture 14. Suppose that bjc or cjb. (a) There exists a unique strongly positive basis Blower satisfying the basis element in the basis element in ≤ B pointed at ða1 ; a2 Þ Blower pointed at ða1 ; a2 Þ
Fig. 2. Conjectured support region of triangular basis elements is the closed region OAC with a curved edge AC, where O = ð0,0Þ, A = ða2 ,0Þ, and C = ð0,a1 Þ. The support region of the (quantum or nonquantum) greedy element pointed at ða1 ,a2 Þ is the polygon OABC, where B = ða1 =b,a2 =cÞ. Note that the line BA (respectively, BC) is tangent to the curved edge AC at point A (respectively, C).
Lee et al.
for every strongly positive basis B and every ða1 ; a2 Þ ∈ Z2 . (b) The quantum greedy basis is Blower . In contrast with the example at the end of Sec. 3, we do not observe a failure of positivity when ðb; cÞ = ð1; 5Þ; ð1; 6Þ; ð2; 4Þ; PNAS | July 8, 2014 | vol. 111 | no. 27 | 9715
MATHEMATICS
The importance of this basis comes from its computability; however, this basis will not serve in our goals of understanding quantum cluster algebras. Indeed, it is easy to see that the standard monomials are not bar-invariant and do not contain all of the cluster monomials, moreover they are inherently dependent on the choice of an initial cluster. These drawbacks provide a motivation to consider the triangular basis (as defined below) constructed from the standard monomial basis with a built-in bar-invariance property. As the name suggests, the triangular basis is defined by a triangularity property relating it to the standard monomial basis. To describe this triangular relationship we introduce the “v-positive” lattice L = ⊕ vZ½vM½a1 ; a2 , where the summation runs over all ða1 ; a2 Þ ∈ Z2 . We now define a basis fC½a1 ; a2 : ða1 ; a2 Þ ∈ Z2 g by specifying how it relates to the standard monomial basis: (P1) Each C½a1 ; a2 is bar-invariant. (P2) For each ða1 ; a2 Þ ∈ Z2 , we have
Proposition 15. h i a1 ;a2 ra″ ;a″ 1
ð2; 6Þ; ð3; 3Þ; ð3; 6Þ. This provides the motivation for assuming the condition “bjc or cjb ” in Conjecture 14. Conjectures 12, 13, and 14 are trivially true for finite types ðbc < 4Þ, and are also known for affine types ðbc = 4Þ (19, 20). Next we study the expansion coefficients relating the various bases. Let us begin by introducing notation. For ða1 ; a2 Þ; ;a2 a1 ;a2 ; ra′ ;a′ ∈ ða′1 ; a′2 Þ ∈ Z2 we define expansion coefficients qaa′1 ;a′ 1
X½a1 ; a2 = M½a1 ; a2 +
X ða′1 ;a′2 Þ
Edited by Bernard Leclerc, University of Caen, France, and accepted by the Editorial Board December 5, 2013 (received for review August 16, 2013)
We identify a quantum lift of the greedy basis for rank 2 coefficientfree cluster algebras. Our main result is that our construction does not depend on the choice of initial cluster, that it builds all cluster monomials, and that it produces bar-invariant elements. We also present several conjectures related to this quantum greedy basis and the triangular basis of Berenstein and Zelevinsky. standard monomial basis
| triangular basis
C
luster algebras were introduced by Fomin and Zelevinsky (1) as a combinatorial tool for understanding the canonical basis and positivity phenomena in the coordinate ring of an algebraic group. This was followed by the definition of quantum cluster algebras by Berenstein and Zelevinsky (2) as a similar combinatorial tool for understanding canonical bases of quantum groups. In connection with these foundational goals, it is important to understand the various bases of cluster algebras and quantum cluster algebras. In this paper, we confirm the existence of a quantum greedy basis in rank 2 quantum cluster algebras. It is independent of the choice of an initial cluster, contains all cluster monomials, and specializes at v = 1 to the greedy basis of a classical rank 2 cluster algebra introduced by Lee et al. (3). Another important basis in an acyclic quantum cluster algebra is the triangular basis of Berenstein and Zelevinsky (4). Like triangular bases, quantum greedy bases can be easily computed. This enables us to study and compare these bases computationally. The structure of the paper is as follows. Section 1 begins with a recollection of rank 2 commutative cluster algebras. We recall the construction of greedy bases here and in the process we provide an axiomatic description. In Sec. 2 we review the definition of rank 2 quantum cluster algebras. In Sec. 3 we present our main result: the existence of a quantum lift of the greedy basis using the axiomatic description. Section 4 recalls the definition of the triangular basis and Sec. 5 provides an overview of results and open problems on the comparison between various bases of rank 2 quantum cluster algebras.
The proof of this theorem in particular establishes a connection between denominator vectors of cluster variables and almost positive roots in a root system Φ ⊂ Z2 . Thus, we say Aðb; cÞ is of affine (respectively, wild) type if bc= 4 (respectively, bc ≥ 5). T ±1 is called universally Laurent An element x ∈ m∈Z Z xm± 1 ; xm+1 because the expansion of x in every cluster is Laurent. If for a nonzero element x the coefficients of the Laurent expansion in each cluster are positive integers, then x is called universally positive. A universally positive element in Aðb; cÞ is said to be indecomposable if it cannot be expressed as a sum of two universally positive elements. Sherman and Zelevinsky (6) studied in great detail the collection of indecomposable universally positive elements. One of their main results is the following: if the commutative cluster algebra Aðb; cÞ is of finite or affine type, then the indecomposable universally positive elements form a Z basis in Aðb; cÞ, moreover this basis contains the 2 set of all cluster monomials fxam1 xam+1 : a1 ; a2 ∈ Z≥0 ; m ∈ Zg. However, the situation becomes much more complicated in wild types. In particular, it was shown by Lee et al. (7) that for bc ≥ 5 the indecomposable universally positive elements of Aðb; cÞ are not linearly independent. The greedy basis of Aðb; cÞ introduced by Lee et al. (3) is a subset of the indecomposable universally positive elements which admits a beautiful combinatorial description. In this section we will describe an axiomatic characterization of the greedy basis. The elements of the greedy basis take on a particular form which is motivated by a well-known pattern in the initial cluster expansion of cluster monomials. An element x ∈ Aðb; cÞ is pointed at ða1 ; a2 Þ ∈ Z2 if it can be written in the form X cq 1 −a2 eð p; qÞxbp x = x−a 1 x2 1 x2 p;q≥0
with eð0; 0Þ = 1 and eðp; qÞ = eða1 ; a2 ; p; qÞ ∈ Z for all p; q ≥ 0. It is well-known that xa11 xa22 is the denominator of a cluster monomial im if and only if ða1 ; a2 Þ ∈ Z2 ∖Φim + (cf. 8, 9), where Φ+ is the set of positive imaginary roots, i.e.,
1. Rank 2 Cluster Algebras and Their Greedy Bases Fix positive integers b; c > 0. The commutative cluster algebra Aðb; cÞ is the Z subalgebra of Qðx1 ; x2 Þ generated by the cluster variables fxm gm∈Z , where the xm are rational functions in x1 and x2 defined recursively by the exchange relations b x + 1 if m is odd; xm+1 xm−1 = m xcm + 1 if m is even: It is a fundamental result of Fomin and Zelevinsky (1) that, although the exchange relations appear to produce rational functions, one always obtains a Laurent polynomial whose denominator is simply a monomial in x1 and x2 . They actually showed the following slightly stronger result. Theorem 1. (ref. 1, theorem 3.1, Laurent phenomenon) For ±1 any m ∈ Z we have Aðb; cÞ ⊂ Z½xm± 1 ; xm+1 . The cluster algebra Aðb; cÞ is of finite type if the collection of all cluster variables is a finite set. Fomin and Zelevinsky (5) went on to classify cluster algebras of finite type. Theorem 2. (ref. 5, theorem 1.4) The cluster algebra Aðb; cÞ is of finite type if and only if bc ≤ 3. 9712–9716 | PNAS | July 8, 2014 | vol. 111 | no. 27
Significance The quantum cluster algebras are a family of noncommutative rings introduced by Berenstein and Zelevinsky as the quantum deformation of the commutative cluster algebras. At the heart of their definition is a desire to understand bases of quantum algebras arising from the representation theory of nonassociative algebras. Thus a natural and important problem in the study of quantum cluster algebras is to study their bases with good properties. In this paper, we lay out a framework for understanding the interrelationships between various bases of rank two quantum cluster algebras. Author contributions: K.L., L.L., D.R., and A.Z. designed research; K.L., L.L., D.R., and A.Z. performed research; K.L., L.L., D.R., and A.Z. analyzed data; and K.L., L.L., and D.R. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. B.L. is a guest editor invited by the Editorial Board. 1
To whom correspondence should be addressed. Email: [email protected].
2
Deceased April 10, 2013.
www.pnas.org/cgi/doi/10.1073/pnas.1313078111
For ða1 ; a2 Þ ∈ Φim + , define the region ba2 p < a1 or Rgreedy = ðp; qÞ ∈ R2≥0 j q + b − ca1 S ca1 p+ c− q < a2 fð0; a1 Þ; ða2 ; 0Þg: ba2 In other words, Rgreedy is the region bounded by the broken line ð0; 0Þ; ða2 ; 0Þ; ða1 =b; a2 =cÞ; ð0; a1 Þ; ð0; 0Þ; with the convention that this region includes the closed segments ½ð0; 0Þ; ða2 ; 0Þ and ½ð0; a1 Þ; ð0; 0Þ but excludes the rest of the boundary (Fig. 1). We can define greedy elements in two different ways, either by axioms or by recurrence relations. Here we choose to define them using recurrence relations as follows. Theorem 3. (ref. 3, proposition 1.6) For each ða1 ; a2 Þ ∈ Z2 , there exists a unique element in Aðb; cÞ pointed at ða1 ; a2 Þ whose coefficients eð p; qÞ satisfy the following recurrence relation: eð0; 0Þ = 1, eð p; qÞ = 8 p X > ½a2 − cq+ + k − 1 k−1 > > ð−1Þ eðp − k; qÞ > > k < k=1 q > X > ½a1 − bp+ + ℓ − 1 > ℓ−1 > ð−1Þ eðp; q − ℓÞ > : ℓ
if ca1 q ≤ ba2 p; if ca1 q ≥ ba2 p;
ℓ=1
where we use the standard notation ½a+ = maxða; 0Þ. We define the greedy element pointed at ða1 ; a2 Þ, denoted x½a1 ; a2 , to be the unique element determined by Theorem 3. For ða1 ; a2 Þ ∈ Φim + , we give a direct characterization for the greedy element. Theorem 4. For each ða1 ; a2 Þ ∈ Φim + , the greedy element x½a1 ; a2 is the unique element in Aðb; cÞ pointed at ða1 ; a2 Þ whose coefficients satisfy the following two axioms: • (Support) eðp; qÞ = 0 for ð p; qÞ ∉ Rgreedy ; • (Divisibility)
X
‒ if a2 > cq; then ð1 + tÞa2 −cq
eði; qÞti ;
(a) The greedy elements x½a1 ; a2 for ða1 ; a2 Þ ∈ Z2 form a Z basis in Aðb; cÞ, which we refer to as the greedy basis. (b) The greedy basis is independent of the choice of an initial cluster. (c) The greedy basis contains all cluster monomials. (d) Greedy elements are universally positive and indecomposable. Our goal in this work is to generalize the above theorem to the setting of rank 2 quantum cluster algebras. The proof of Theorem 5 given in ref. 3 uses combinatorial objects called compatible pairs in an essential way (cf. ref. 3, theorem 1.11). Unfortunately this method has difficulties–limitations in generalizing to the study of quantum cluster algebras. More precisely, if one could simply assign a power of v to each compatible pair then the quantum greedy elements would have again been universally positive, which is unfortunately false in general (see the example at the end of Sec. 3). This justifies our approach to Theorem 5, with the exception of (d), which can easily be generalized to the quantum setting. 2. Rank 2 Quantum Cluster Algebras In this section we define our main objects of study, namely quantum cluster algebras, and recall important fundamental facts related to these algebras. We restrict attention to rank 2 quantum cluster algebras where we can describe the setup in very concrete terms. We follow (as much as possible) the notation and conventions of refs. 3, 4. Consider the following quantum torus: E D T = Z v ± 1 X1± 1 ; X2± 1 : X2 X1 = v2 X1 X2 (this setup is related to the one in ref. 10, which uses the formal variable q instead of v by setting q = v−2 ). There are many choices for quantizing cluster algebras; to rigidify the situation we require the quantum cluster algebra to be invariant under a certain involution. The bar involution is the Z-linear antiautomorphism of T determined by f ðvÞ = f ðv−1 Þ for f ∈ Z½v ± 1 and
i
X
eðp; iÞti : ‒ if a1 > bp; then ð1 + tÞa1 −bp
i
The key step in the proof of Theorem 4 is based on an observation made in ref. 3, sec. 2: the first (respectively, second)
fX1a1 X2a2 = f X2a2 X1a1 = v2a1 a2 f X1a1 X2a2 ða1 ; a2 ∈ ZÞ: An element which is invariant under the bar involution is said to be bar-invariant. Let F be the skew field of fractions of T . The quantum cluster algebra Av ðb; cÞ is the Z½v ± 1 subalgebra of F generated by the quantum cluster variables fXm gm∈Z defined recursively from the quantum exchange relations Xm+1 Xm−1 =
vb Xmb + 1 vc Xmc + 1
if m is odd; if m is even:
By a simple induction one can easily check the following (quasi-) commutation relations between neighboring cluster variables Xm ; Xm+1 in Av ðb; cÞ: Xm+1 Xm = v2 Xm Xm+1 ðm ∈ ZÞ:
Fig. 1.
Lee et al.
Support region of (quantum) greedy elements.
SPECIAL FEATURE
recurrence relation in Theorem 3 is equivalent to the vanishing of the (p, q)th coefficient of the Laurent expansion of x½a1 ; a2 with respect to the cluster fx0 ; x1 g (respectively, fx2 ; x3 g). It is straightforward to check that such vanishing implies the Support and Divisibility axioms in Theorem 4. The following theorem summarizes the results from ref. 3. Theorem 5. (ref. 3, theorem 1.7)
[1]
It then follows that all cluster variables are bar-invariant, therefore Av ðb; cÞ is also stable under the bar involution. Moreover, Eq. 1 implies that each cluster fXm ; Xm+1 g generates a quantum torus PNAS | July 8, 2014 | vol. 111 | no. 27 | 9713
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2 2 2 Φim + d ða1 ; a2 Þ ∈ Z>0 : ca1 − bca1 a2 + ba2 ≤ 0 :
±1 T m = Z v ± 1 Xm± 1 ; Xm+1 : Xm+1 Xm = v2 Xm Xm+1 : It is easy to see that the bar involution does not depend on the choice of an initial quantum torus T m . The appropriate quantum analogs of cluster monomials for Av ðb; cÞ are the (bar-invariant) quantum cluster monomials which are certain elements of a quantum torus T m , more precisely they are a2 Xmða1 ;a2 Þ = va1 a2 Xma1 Xm+1 ða1 ; a2 ∈ Z≥0 ; m ∈ ZÞ:
The following quantum analog of the Laurent phenomenon was proven by Berenstein and Zelevinsky (2). Theorem 6 (2). For any m ∈ Z we have Av ðb; cÞ ⊂ T m . Moreover, T Av ðb; cÞ = T m: m∈Z
A nonzero elementh of Av ðb; cÞiis called universally positive if it T ±1 . A universally positive element lies in m∈Z Z≥0 v ± 1 Xm± 1 ; Xm+1 in Av ðb; cÞ is indecomposable if it cannot be expressed as a sum of two universally positive elements. It is known as a very special case of the results of refs. 11–15 that cluster monomials are universally positive Laurent polynomials. Explicit combinatorial expressions for these positive coefficients can be obtained from the results of refs. 16–18. 3. Quantum Greedy Bases This section contains the main result of the paper. Here we introduce the quantum greedy basis and present its nice properties. Analogous to the construction of greedy elements, the elements of the quantum greedy basis take on the following particular form. An element X ∈ Av ðb; cÞ is said to be pointed at ða1 ; a2 Þ ∈ Z2 if it has the form X ðbp−a1 ;cq−a2 Þ X= eð p; qÞX1 p;q≥0
with eð0; 0Þ = 1 and eð p; qÞ ∈ Z½v ± 1 for all p and q. The next theorem is an analog of Theorem 3; to state our result precisely we need more notation. Let w denote a formal invertible variable. For n ∈ Z, k ∈ Z≥0 define the bar-invariant quantum numbers and bar-invariant quantum binomial coefficients by wn − w−n = wn−1 + wn−3 + ⋯ + w−n+1 ; ½nw = w − w−1 " # n ½n ½n − 1w ⋯½n − k + 1w = w : ½kw ½k − 1w ⋯½1w k w h i n Recall that k will always be a Laurent polynomial in w. Hence w
taking w = v or v we obtain Laurent polynomials in v as well. Theorem 7. For each ða1 ; a2 Þ ∈ Z2 , there exists a unique element in Av ðb; cÞ pointed at ða1 ; a2 Þ whose coefficients eðp; qÞ satisfy the following recurrence relation: eð0; 0Þ = 1, b
c
eðp; " # 8 pqÞ = X ½a2 − cq+ + k − 1 > k−1 > > ð−1Þ eðp − k; qÞ > > < k k=1
if ca1 q ≤ ba2 p; vb
" # q > X > ½a1 − bp+ + ℓ − 1 > ℓ−1 > ð−1Þ eðp; q − ℓÞ > : ℓ ℓ=1 vc
if ca1 q ≥ ba2 p:
We define the quantum greedy element pointed at ða1 ; a2 Þ, denoted X½a1 ; a2 , to be the unique element determined by Theorem 7. 9714 | www.pnas.org/cgi/doi/10.1073/pnas.1313078111
The following theorem is analogous to Theorem 4. Theorem 8. For each ða1 ; a2 Þ ∈ Φim + , the quantum greedy element X½a1 ; a2 is the unique element in Av ðb; cÞ pointed at ða1 ; a2 Þ whose coefficients satisfy the following two axioms: • (Support) eðp; qÞ = 0 for ðp; qÞ ∉ Rgreedy ; • (Divisibility) Let t denote a formal variable which commutes with v. Then X a2 −cq
− ∏ 1 + vbða2 −cq+1−2jÞ t eði; qÞti ; j=1
i
a1 −bp
− ∏
X
1 + vcða1 −bp+1−2jÞ t eðp; iÞti :
j=1
i
Theorem 8 follows from an argument similar to the proof of Theorem 4. Our main theorem below states that quantum greedy elements possess all of the desired properties described in Theorem 5 except the positivity (d). Theorem 9. (Main Theorem) (a) The quantum greedy elements X½a1 ; a2 for ða1 ; a2 Þ ∈ Z2 form a Z½v ± 1 basis in Av ðb; cÞ, which we refer to as the quantum greedy basis. (b) The quantum greedy basis is bar-invariant and independent of the choice of an initial cluster. (c) The quantum greedy basis contains all cluster monomials. (d) If X½a1 ; a2 is universally positive, then it is indecomposable. (e) The quantum greedy basis specializes to the commutative greedy basis by the substitution v = 1. Full proofs of Theorems 7, 8, and 9 will appear elsewhere. The hardest part is to show the existence of quantum greedy elements, i.e., that the recursions in Theorem 7 terminate. The main idea in the proof is to realize X½a1 ; a2 as a degeneration of certain elements in Av ðb; cÞ whose supports are contained in the closure of Rgreedy . We end this section with an example where positivity of quantum greedy elements fails. Let ðb; cÞ = ð2; 3Þ and consider the pointed coefficient eð2; 1Þ in the Laurent expansion of X½3; 4. Using Theorem 7, we see that eð2; 1Þ is not in Z≥0 ½v ± 1 : 2 1 − eð0; 1Þ = eð1; 1Þ − eð0; 1Þ eð2; 1Þ = eð1; 1Þ 2 v2 1 v2 1 4 3 = eð1; 0Þ − eð0; 1Þ = eð0; 0Þ − eð0; 0Þ 1 v3 1 v2 1 v3 = v6 + v2 + v−2 + v−6 − v6 + 1 + v−6 = v2 − 1 + v−2 : Further computation shows that the greedy elements X½3; 5; X½5; 4; X½5; 7; X½5; 8; X½7; 5; X½7; 10; X½7; 11; etc: are not positive. Similarly, positivity can be seen to fail for a large class of quantum greedy elements when ðb; cÞ = ð2; 5Þ; ð3; 4Þ; ð4; 6Þ: 4. Triangular Bases The construction of the triangular basis begins with the standard monomial basis. For every ða1 ; a2 Þ ∈ Z2 , we define the standard monomial M½a1 ; a2 (which is denoted Eð−a1 ;−a2 Þ in ref. 4) by setting ½a
½−a1 +
M½a1 ; a2 = va1 a2 X3 1 + X1
½−a2 +
X2
½a
X0 2 + :
[2]
It is known (2) that the elements M½a1 ; a2 form a Z½v ± 1 basis of Av ðb; cÞ. Lee et al.
C½a1 ; a2 − M½a1 ; a2 ∈ L: Using the universal acyclicity of rank 2 quantum cluster algebras, we apply the following theorem. Theorem 10. (ref. 4, theorem 1.6]) The triangular basis does not depend on the choice of initial cluster and it contains all cluster monomials. Similar to the support condition for (quantum) greedy elements, we make the following conjecture about the supports of triangular elements. Conjecture 11. Let ða1 ; a2 Þ ∈ Φim + . For 0 ≤ p ≤ a2 , 0 ≤ q ≤ a1 , the pointed coefficient eðp; qÞ of X ðbp−a1 ;cq−a2 Þ in C½a1 ; a2 is nonzero if and only if bp2 + bcpq + cq2 ≤ ca1 q + ba2 p; (Fig. 2). 5. Open Problems In this section we present open problems and conjectures relating to the triangular basis and quantum greedy basis of a rank 2 quantum cluster algebra. Our aim is to find Z½v ± 1 bases of Av ðb; cÞ satisfying the properties laid out in the following definition.
SPECIAL FEATURE
Fig. 3. Properties of the quantum greedy basis for bjc or cjb.
A basis B for Av ðb; cÞ is said to be strongly positive if the following hold: (1) (2) (3) (4)
each element of B is bar-invariant; B is independent of the choice of an initial cluster; B contains all cluster monomials; any product of elements from B can be expanded as a linear combination of elements of B with coefficients in Z≥0 ½v ± 1 .
Note that all elements in a strongly positive basis are universally positive. Indeed, consider multiplying a basis element B ∈ B ða ;a Þ by a cluster monomial X1 1 2 with a1 ; a2 sufficiently large to clear the denominator in the initial cluster expansion of B. By properties (3) and (4) of a strongly positive basis the product ða ;a Þ
BX1 1 2 is in Z≥0 ½v ± 1 ½X1 ; X2 and thus we see that a strongly positive basis element has nonnegative coefficients in its initial cluster expansion. It then follows that B is universally positive because it is independent of the choice of an initial cluster. Kimura and Qin (15) showed the existence of bases for acyclic skew-symmetric quantum cluster algebras [hence for Av ðb; cÞ with b = c] which satisfy (1), (3), and (4). In the rank 2 case, their bases are exactly the triangular bases, and hence satisfy (2) as well (see ref. 4, remark 1.8]). Conjecture 12. Given any strongly positive basis B, there exists a unique basis element pointed at ða1 ; a2 Þ for each ða1 ; a2 Þ ∈ Z2 . It is easy to see that property (3) of strongly positive bases together with Conjecture 12 implies that every element of B is pointed. We note that the conclusion of Conjecture 12 holds for both the quantum greedy basis and the triangular basis. In what follows, we write Y ≤ Z for Y ; Z ∈ Av ðb; cÞ if Z − Y is a Laurent polynomial with coefficients in Z≥0 ½v ± 1 . Conjecture 13. (a) There exists a unique strongly positive basis Bupper satisfying the basis element in the basis element in ≤ B pointed at ða1 ; a2 Þ Bupper pointed at ða1 ; a2 Þ for every strongly positive basis B and every ða1 ; a2 Þ ∈ Z2 . (b) The triangular basis is Bupper . Conjecture 14. Suppose that bjc or cjb. (a) There exists a unique strongly positive basis Blower satisfying the basis element in the basis element in ≤ B pointed at ða1 ; a2 Þ Blower pointed at ða1 ; a2 Þ
Fig. 2. Conjectured support region of triangular basis elements is the closed region OAC with a curved edge AC, where O = ð0,0Þ, A = ða2 ,0Þ, and C = ð0,a1 Þ. The support region of the (quantum or nonquantum) greedy element pointed at ða1 ,a2 Þ is the polygon OABC, where B = ða1 =b,a2 =cÞ. Note that the line BA (respectively, BC) is tangent to the curved edge AC at point A (respectively, C).
Lee et al.
for every strongly positive basis B and every ða1 ; a2 Þ ∈ Z2 . (b) The quantum greedy basis is Blower . In contrast with the example at the end of Sec. 3, we do not observe a failure of positivity when ðb; cÞ = ð1; 5Þ; ð1; 6Þ; ð2; 4Þ; PNAS | July 8, 2014 | vol. 111 | no. 27 | 9715
MATHEMATICS
The importance of this basis comes from its computability; however, this basis will not serve in our goals of understanding quantum cluster algebras. Indeed, it is easy to see that the standard monomials are not bar-invariant and do not contain all of the cluster monomials, moreover they are inherently dependent on the choice of an initial cluster. These drawbacks provide a motivation to consider the triangular basis (as defined below) constructed from the standard monomial basis with a built-in bar-invariance property. As the name suggests, the triangular basis is defined by a triangularity property relating it to the standard monomial basis. To describe this triangular relationship we introduce the “v-positive” lattice L = ⊕ vZ½vM½a1 ; a2 , where the summation runs over all ða1 ; a2 Þ ∈ Z2 . We now define a basis fC½a1 ; a2 : ða1 ; a2 Þ ∈ Z2 g by specifying how it relates to the standard monomial basis: (P1) Each C½a1 ; a2 is bar-invariant. (P2) For each ða1 ; a2 Þ ∈ Z2 , we have
Proposition 15. h i a1 ;a2 ra″ ;a″ 1
ð2; 6Þ; ð3; 3Þ; ð3; 6Þ. This provides the motivation for assuming the condition “bjc or cjb ” in Conjecture 14. Conjectures 12, 13, and 14 are trivially true for finite types ðbc < 4Þ, and are also known for affine types ðbc = 4Þ (19, 20). Next we study the expansion coefficients relating the various bases. Let us begin by introducing notation. For ða1 ; a2 Þ; ;a2 a1 ;a2 ; ra′ ;a′ ∈ ða′1 ; a′2 Þ ∈ Z2 we define expansion coefficients qaa′1 ;a′ 1
X½a1 ; a2 = M½a1 ; a2 +
X ða′1 ;a′2 Þ
