Quantum Cluster Algebras and Quantum Nilpotent Algebras

Quantum cluster algebras and quantum nilpotent algebras

Kenneth R. Goodearla and Milen T. Yakimovb,1

aDepartment of Mathematics, University of California, Santa Barbara, CA 93106; and bDepartment of Mathematics, Louisiana State University, Baton Rouge, LA 70803

Edited by Bernard Leclerc, University of Caen, Caen, France, and accepted by the Editorial Board January 16, 2014 (received for review August 15, 2013) A major direction in the theory of cluster algebras is to construct quantum nilpotent algebra. The construction does not rely on (quantum) cluster algebra structures on the (quantized) coordinate any initial combinatorics of the algebras. On the contrary, the rings of various families of varieties arising in Lie theory. We prove construction itself produces intricate combinatorial data for prime that all algebras in a very large axiomatically defined class of non- elements in chains of subalgebras. When this is applied to special commutative algebras possess canonical quantum cluster algebra cases, we recover the Weyl group combinatorics that played a key structures. Furthermore, they coincide with the corresponding upper role in categorification earlier (7, 9, 10). Because of this, we expect quantum cluster algebras. We also establish analogs of these results that our construction will be helpful in building a unified cate- for a large class of Poisson nilpotent algebras. Many important fami- gorification of quantum nilpotent algebras. Finally, we also prove lies of coordinate rings are subsumed in the class we are covering, similar results for (commutative) cluster algebras using Poisson which leads to a broad range of applications of the general results prime elements. to the above-mentioned types of problems. As a consequence, we The results of the paper have many applications because a – prove the Berenstein Zelevinsky conjecture [Berenstein A, Zelevinsky number of important families of algebras arise as special cases in Adv Math – A (2005) 195:405 455] for the quantized coordinate rings our axiomatics. Berenstein et al. (10) proved that the coordinate of double Bruhat cells and construct quantum cluster algebra struc- rings of all double Bruhat cells in every complex simple Lie group tures on all quantum unipotent groups, extending the theorem of are upper cluster algebras. It was an important problem to decide Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the whether the latter coincide with the corresponding cluster algebras. case of symmetric Kac–Moody groups. Moreover, we prove that the We resolve this problem positively. On the quantum side, we prove upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) the Berenstein–Zelevinsky conjecture (11) on cluster algebra Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. structures for all quantum double Bruhat cells. Finally, we establish that the quantum Schubert cell algebras for all complex simple Lie noncommutative unique factorization domains | quantum groups | groups have quantum cluster algebra structures. Previously this was – quantum Schubert cell algebras known for symmetric Kac Moody groups due to Geiß et al. (12). Prime Elements of Quantum Nilpotent Algebras he theory of cluster algebras provides a unified framework Definition of Quantum Nilpotent Algebras. Let K be an arbitrary Tfor treating a number of problems in diverse areas of math- base field. A skew polynomial extension of a K-algebra A, ematics such as combinatorics, representation theory, topology, mathematical physics, algebraic and Poisson geometry, and dy- A↦A½x; σ; δ; namical systems (1–7). The construction of cluster algebras was invented by Fomin and Zelevinsky (1), who also obtained a num- is a generalization of the classical polynomial algebra A½x.Itis ber of fundamental results on them. This construction builds defined using an algebra automorphism σ of A and a skew- algebras in a novel way by producing infinite generating sets via derivation δ. The algebra A½x; σ; δ is isomorphic to A½x as a vec- a process of mutation rather than the classical approach using tor space, and the variable x commutes with the elements of A as generators and relations. follows: The main algebraic approach to clusteralgebrasreliesonrep- resentations of finite dimensional algebras and derived categories xa = σðaÞx + δðaÞ for all a ∈ A: (5, 8). In this paper, we describe a different algebraic approach based on noncommutative ring theory. For every nilpotent Lie algebra n of dimension m, there exists An important range of problems in the theory of cluster algebras a chain of ideals of n is to prove that the coordinate rings of certain algebraic varieties coming from Lie theory admit cluster algebra structures. The idea is Significance that once this is done, one can use cluster algebras to study canonical bases in such coordinate rings. Analogous problems deal Cluster algebras are used to study in a unified fashion phenom- with the corresponding quantizations. The approach via represen- ena from many areas of mathematics. In this paper, we present tation theory to this type of problem needs combinatorial data for a new approach to cluster algebras based on noncommutative quivers as a starting point. Such might not be available a priori. This ring theory. It deals with large, axiomatically defined classes of approach also differs from one family of varieties to another, which algebras and does not require initial combinatorial data. Because means that one needs to design an appropriate categorification of this, it has a broad range of applications to open problems on process in each case. constructing cluster algebra structures on coordinate rings and We prove that all algebras in a very general, axiomatically their quantum counterparts. defined class of quantum nilpotent algebras are quantum cluster algebras. The proof is based on constructing quantum clusters by Author contributions: K.R.G. and M.T.Y. designed research, performed research, and considering sequences of prime elements in chains of subalgebras wrote the paper. that are noncommutative unique factorization domains (UFDs). The authors declare no conflict of interest. These clusters are canonical relative to the mentioned chains of This article is a PNAS Direct Submission. B.L. is a guest editor invited by the Editorial Board. subalgebras, which are determined by the presentation of the 1To whom correspondence should be addressed. Email: [email protected].

9696–9703 | PNAS | July 8, 2014 | vol. 111 | no. 27 www.pnas.org/cgi/doi/10.1073/pnas.1313071111 Downloaded by guest on September 24, 2021 = ⊳ ⊳ ... ⊳ ⊳ = −1 n nm nm−1 n1 n0 f0g hijd 1; ...; 1; q ; 1; ...; 1; q; 1; ...; 1 ∈ H; SPECIAL FEATURE

= = ; ⊆ ≤ ≤ − such that dimðnk nk−1Þ 1and½n nk nk−1 for 1 k m. Choos- where q 1 and q reside in positions i and m + j, respectively. Then ing an element x in the complement of n − in n for each σ = · k k 1 k ði−1Þn+j ðhij Þ and 1 ≤ k ≤ m leads to the following presentation of the universal enveloping algebra UðnÞ as an iterated skew polynomial extension: −2 hij · tij = q tij: ≅ K ; ; δ ... ; ; δ ; UðnÞ ½x1½x2 id 2 ½xm id m Thus, for all q ∈ Kp that are not roots of unity, the algebras Rq½Mm × n are examples of quantum nilpotent algebras. where all of the derivations δ2; ...; δm are locally nilpotent. Definition 1: An iterated skew polynomial extension UFDs. The notion of UFDs plays an important role in algebra and number theory. Its noncommutative analog was introduced by = K ; σ ; δ ; σ ; δ [1] R ½x1½x2 2 2⋯½xN N N Chatters (15). A nonzero, noninvertible element p of a domain R (a ring without zero divisors) is called prime if pR = Rp and the is called a quantum nilpotent algebra if it is equipped with a rational = K K factor R Rp is a domain. A noetherian domain R is called a UFD action of a -torus H by -algebra automorphisms satisfying the if every nonzero prime ideal of R contains a prime element. Such following conditions: rings possess the unique factorization property for all of their (a) The elements x1; ...; xN are H-eigenvectors. nonzero normal elements—the elements u ∈ R with the property (b) For every 2 ≤ k ≤ N, δk is a locally nilpotent σk-derivation of that Ru = uR. If the ring R is acted upon by a group G, then one K ; σ ; δ ½x1⋯½xk−1 k−1 k−1. can introduce an equivariant version of this property. Such an R ≤ ≤ ∈ σ = · (c) For every 1 k N, there exists hk H such that k ðhk Þ and is called a G-UFD if every nonzero G-invariant prime ideal of λ the hk-eigenvalue of xk, to be denoted by k, is not a root R contains a prime element that is a G-eigenvector. of unity. It was shown by Launois et al. (16) that every quantum nilpotent The universal enveloping algebras of finite dimensional nilpo- algebra R is a noetherian H-UFD. An H-eigenvector of such a ring tent Lie algebras satisfy all of the conditions in the definition ex- R will be called a homogeneous element because this corresponds to the homogeneity property with respect to the canonically in- cept for the last part of the third one. More precisely, in that case, MATHEMATICS duced grading of R by the character lattice of H.Inparticular,we one can take H = f1g, conditions (a) and (b) are satisfied, and in will use the more compact term of homogeneous prime element of condition (c), we have λ = 1. Thus, condition (c) is the main k R instead of a prime element of R that is an H-eigenvector. feature that separates the class of quantum nilpotent algebras from the class of universal enveloping algebras of nilpotent Lie algebras. Sequences of Prime Elements. Next, we classify the set of all ho- λ The torus H is needed to define the eigenvalues k. mogeneous prime elements of the chain of subalgebras The algebras in Definition 1 are also known as Cauchon– – Goodearl Letzter extensions. The axiomatics came from the f0g ⊂ R1 ⊂ R2 ⊂ ... ⊂ RN = R works of Goodearl and Letzter (13) and Cauchon (14), which investigated in this generality the stratification of the prime spec- of a quantum nilpotent algebra R, where Rk is the subalgebra of ; ...; trum of an algebra into strata associated with its H-prime ideals. R generated by the first k variables x1 xk. Z Z The Gelfand-Kirillov dimension of the algebra in Eq. 1 We will denote by and ≥0 the sets of all integers and non- ≤ ; d equals N. negative integers, respectively. Given two integers l k,set½l k Example 1: For two positive integers m and n, and q ∈ Kp, define fl; l + 1; ...; kg. For a function η : ½1; N → Z, define the predecessor function p : the algebra of quantum matrices Rq½Mm × n as the algebra with ½1; N → ½1; N ⊔ f−∞g and successor function s : ½1; N → ½1; N ⊔ generators tij,1≤ i ≤ m and 1 ≤ j ≤ n, and relations f+ ∞g for its level sets by t t = qt t ; for i < k; ij kj kj ij maxfl < kjηðlÞ = ηðkÞg; if such l exists; pðkÞ = −∞; otherwise; tijtil = qtiltij; for j < l; and = ; < ; > ; tijtkl tkltij for i k j l minfl > kjηðlÞ = ηðkÞg; if such l exists; sðkÞ = −1 +∞; otherwise: tijtkl − tkltij = q − q tiltkj; for i < k; j < l:

It is an iterated skew polynomial extension, where Theorem 1. Let R be a quantum nilpotent algebra of dimension N. There exist a function η : ½1; N → Z and elements Rq½Mm × n = K½x1½x2; σ2; δ2 ...½xN ; σN ; δN ; c ∈ R − for all 2 ≤ k ≤ N with δ ≠ 0 = = k k 1 k N mn, and xði−1Þn+j tij. It is easy to write explicit formulas for the automorphisms σ and skew derivations δ from the above com- k k such that the elements y1; ...; yN ∈ R, recursively defined by mutation relations, and to check that each δk is locally nilpotent. * m+n The torus H = ðK Þ acts on Rq½Mm × n by algebra automor- − ; δ ≠ d ypðkÞxk ck if k 0 phisms by the rule yk xk; if δk = 0; ξ ; ...; ξ · dξ ξ−1 ð 1 m+nÞ tij i m+jtij are homogeneous and have the property that for every k ∈ ½1; N, the homogeneous prime elements of Rk are precisely the nonzero scalar ξ ; ...; ξ ∈ Kp m+n for all ð 1 m+nÞ ð Þ . Define multiples of the elements

Goodearl and Yakimov PNAS | July 8, 2014 | vol. 111 | no. 27 | 9697 Downloaded by guest on September 24, 2021 Cluster Structures on Quantum Nilpotent Algebras yl for l ∈ ½1; k with sðlÞ > k: Symmetric Quantum Nilpotent Algebras. Definition 3: A quantum

In particular, yk is a homogeneous prime element of Rk, for all nilpotent algebra R as in Definition 1 will be called symmetric if it k ∈ ½1; N. The sequence y1; ...; yN and the level sets of a function can be presented as an iterated skew polynomial extension for the η with these properties are both unique. reverse order of its generators, Example 2: Given two subsets I = fi1 < ⋯ < ikg ⊂ ½1; m and J = < < ⊂ ; Δ ∈ = K ; σ p ; δ p ; σ p ; δ p ; fj1 ⋯ jkg ½1 n, define the quantum minor I; J Rq½Mm × n by R ½xN xN−1 N−1 N−1 ⋯ x1 1 1 X ℓðσÞ Δ ; = −q t ...t ; in such a way that conditions (a)–(c) in Definition 1 are satisfied for I J ð Þ i1 jσð1Þ ik jσðkÞ p p σ∈ ; ...; ∈ Sk some choice of hN h1 H. A quantum nilpotent algebra R is symmetric if and only if it where Sk denotes the symmetric group and ℓ : Sk → Z≥0 denotes the satisfies the Levendorskii–Soibelman type straightening law standard length function. X n + n − For the algebra of quantum matrices Rq½Mm × n, the sequence of − λ = ξ l 1 ... k 1 xkxl klxlxk nl + 1;...;nk − 1 xl+1 xk−1 prime elements from Theorem 1 consists of solid quantum minors; nl + 1;...;nk − 1∈Z≥0 more precisely, < ξ p ∈ = Δ for all l k (where the • are scalars) and there exist hk H such yði−1Þn+j ½i−minði; jÞ+1; i; ½j−minði; jÞ+1; j p · = λ−1 > that hk xl lk xl for all l k. The defining commutation rela- ≤ ≤ ≤ ≤ η : ; → tions for the algebras of quantum matrices Rq½Mm × n imply that for all 1 i mand1 j n. Furthermore, the function ½1 mn they are symmetric quantum nilpotent algebras. Z canbechosenas Denote by ΞN the subset of the symmetric group SN consisting ηðði − 1Þn + jÞd j − i: of all permutations that have the property that Definition 2: The cardinality of the range of the function η from τðkÞ = max τ ½1; k − 1 + 1 [4] Theorem 1 is called the rank of the quantum nilpotent algebra R and is denoted by rkðRÞ. or For example, the algebra of quantum matrices Rq½Mm × n has rank m + n − 1. τðkÞ = min τ ½1; k − 1 − 1 [5] × qd Embedded Quantum Tori. An N N matrix ðqklÞ with entries in for all 2 ≤ k ≤ N. In other words, ΞN consists of those τ ∈ SN such K is called multiplicatively skewsymmetric if that τð½1; kÞ is an interval for all 2 ≤ k ≤ N. For each τ ∈ ΞN , a symmetric quantum nilpotent algebra R has the presentation q q = q = 1 for 1 ≤ l; k ≤ N: kl lk kk h i h i = K ; σ″ ; δ″ ; σ ; δ″ ; [6] R xτð1Þ xτð2Þ τð2Þ τð2Þ ⋯ xτðNÞ τ″ðNÞ τðNÞ Such a matrix gives rise to the quantum torus T q which is the K-algebra with generators Y ± 1; ...; Y ± 1 and relations 1 N σ″ dσ δ″ dδ 4 where τðkÞ τðkÞ and τðkÞ τðkÞ if Eq. is satisfied, whereas = ≤ ; ≤ : σ″ dσp δ″ dδp 5 YkYl qklYlYk for 1 l k N τðkÞ τðkÞ and τðkÞ τðkÞ if Eq. holds. This presentation satisfies the conditions (a)–(c) in Definition 1 for the elements Let fe ; ...; e g be the standard basis of the lattice ZN .Fora ″ ∈ ″ d 4 ″ d p 1 N hτðkÞ H, given by hτðkÞ hτðkÞ in case of Eq. and hτðkÞ hτ quantum nilpotent algebra R of dimension N, define the eigen- ðkÞ in case of Eq. 5. The use of the term symmetric in Definition 3 is values λ ∈ K: kl motivated by the presentations in Eq. 6 parametrized by the Ξ hk · xl = λklxl for 1 ≤ l < k ≤ N: [2] elements of the subset N of the symmetric group SN.

There exists a unique group bicharacter Ω : ZN × ZN → K* such Proposition. For every symmetric quantum nilpotent algebra R, the p λp that hk-eigenvalues of xk, to be denoted by k, satisfy 8 < ; = ; λ p = λ p 1 if k l k l Ω ; = λ ; > ; ðek elÞ : kl if k l λ−1; < : ≤ ; ≤ η = η ≠ + ∞ ≠ + ∞ lk if k l for all 1 k l N such that ðkÞ ðlÞ and sðkÞ , sðlÞ . They are related to the eigenvalues λl by Set e−∞d0. Define the vectors λ p = λ−1 d + + ∈ ZN ; [3] k l ek ek epðkÞ ⋯ for all 1 ≤ k; l ≤ N such that ηðkÞ = ηðlÞ and sðkÞ ≠ + ∞, pðlÞ ≠−∞. noting that only finitely many terms in the sum are nonzero. N Then fe1; ...; eN g is another basis of Z . Construction of Exchange Matrices. Our construction of a quantum cluster algebra structure on a symmetric quantum nilpotent alge- Theorem 2. For each quantum nilpotent algebra R, the sequence bra R of dimension N will have as the set of exchangeable indices of prime elements from Theorem 1 definesanembeddingof the quantum torus T q associated with the N × N multiplicatively exdfk ∈ ½1; NjsðkÞ ≠ +∞g: [7] skewsymmetric matrix with entries dΩ ; ; ≤ ; ≤ We will impose the following two mild conditions: qkl ðek elÞ 1 k l N pffiffiffiffiffi (A) The field K contains square roots λkl of the scalars λkl for ± 1↦ ≤ < ≤ K* into the division ring of fractions Fract(R) of R such that Yk 1 l k N such that the subgroup of generated by all of ± 1 ≤ ≤ yk , for all 1 k N. them contains no elements of order 2.

9698 | www.pnas.org/cgi/doi/10.1073/pnas.1313071111 Goodearl and Yakimov Downloaded by guest on September 24, 2021 ∈ η (B) There exist positive integers dn, n rangeð Þ, for the func- ; ...; ∈ : yτ; yτ; R SPECIAL FEATURE tion η from Theorem 1 such that 1 N λ p dηðlÞ = λ p dηðkÞ Similarly, applying Theorem 4 to the presentation from Eq. 6,we k l obtain the integer matrix for all k; l ∈ ex. ~ Bτ ∈ MN×exðZÞ: Remark 1: With the exception of some two-cocycle twists, for all of ~ the quantum nilpotent algebras R coming from the theory of quantum For τ = id, we recover the original sequence y1; ...; yN and matrix B. * groups, the subgroup of K generated by fλklj1 ≤ l < k ≤ Ng is torsion- free. For all such algebras, condition (A) only requires that K con- Theorem 5. Every symmetric quantum nilpotent algebra R of di- tains square roots of the scalars λkl . mension N satisfying the conditions (A) and (B) possesses a ca- All symmetric quantum nilpotent algebras that we are aware nonical structure of quantum cluster algebra for which no frozen of satisfy cluster variables are inverted. Its initial seed has: p ζ ; ...; ζ ζ ; ...; ζ ∈ K* λ = qmk for 1 ≤ k ≤ N i) Cluster variables 1y1 N yN for some 1 N , k among which the variables indexed by the set ex from Eq. 7 ∈ K* ; ...; are exchangeable and the rest are frozen; for some nonroot of unity q and positive integers m1 mN . ~ The Proposition implies that all of them satisfy condition (B). ii) Exchange matrix B given by Theorem 4. ex − The set has cardinality N rkðRÞ, where rkðRÞ is the rank of Furthermore, this quantum cluster algebra aways coincides with the quantum nilpotent algebra R.ByanN × ex matrix, we will × − the corresponding upper quantum cluster algebra. mean a matrix of size N ðN rkðRÞÞ whose rows and columns After an appropriate rescaling, each of the generators x of such are indexed by the sets 1; N and ex, respectively. The set of such k ½ an algebra R given by Eq. 1 is a cluster variable. Moreover, for each matrices with integer entries will be denoted by MN × exðZÞ. element τ of the subset ΞN of the symmetric group SN, R has a seed Theorem 4. For every symmetric quantum nilpotent algebra R of with cluster variables obtained by reindexing and rescaling the se- ; ...; dimension N satisfying conditions (A) and (B), there exists a unique quence of prime elements yτ;1 yτ;N . The exchange matrix of this ~ ~ = ∈ Z τ matrix B ðblkÞ MN×exð Þ whose columns satisfy the following seed is the matrix B . MATHEMATICS two conditions : The base fields of the algebras covered by this theorem can P λp; if k = n have arbitrary characteristic. We refer the reader to theorem 8.2 i) Ω N b e ; e = n for all k ∈ ex and n ∈ ½1; N l=1 lk l n 1; if k ≠ n; of ref. 17 for a complete statement of the theorem, which (a system of linear equations written in a multiplicative form); includes additional results and gives explicit formulas for the b1k ... bNk ∈ ex ii) The products y1 yN are fixed under H for all k (a scalars ζ ; ...; ζ and the necessary reindexing and rescaling of homogeneity condition). 1 N the sequences yτ;1; ...; yτ;N . ZN The second condition can be written in an explicit form using Define the following automorphism of the lattice : the fact that yk is an H-eigenvector and its eigenvalue equals the ; ...; g = l1e1 + ⋯ + lN eN ↦ gdl1e1 + ⋯ + lN eN product of the H-eigenvalues of xk xpnk ðkÞ, where nk is the n ≠−∞ maximal nonnegative integer n such that p ðkÞ . This prop- ; ...; ∈ Z ; ...; 3 erty is derived from Theorem 1. for all l1 lN in terms of the vectors e1 eN from Eq. . Example 3: It follows from Example 2 that in the case of the The construction of seeds for quantum cluster algebras in the work of Berenstein and Zelevinsky (11) requires assigning quantum algebras of quantum matrices R ½M × , q m n frames to all of them. The quantum frame M : ZN → FractðRÞ ex = fði − 1Þn + jj1 ≤ i < m; 1 ≤ j < ng: associated with the initial seed for the quantum cluster algebra structure in Theorem 5 is uniquely reconstructed from the rules The bicharacter Ω : Zmn × Zmn → Kp is given by Mðe Þ = ζ y for 1 ≤ k ≤ N k k k δ signðk−iÞ+δ signðl−jÞ Ω e − + ; e − + = q jl ik ði 1Þn j ðk 1Þn l and for all 1 ≤ i; k ≤ m and 1 ≤ j; l ≤ n. Furthermore, Mð f + gÞ = Ω f; g MðfÞMðgÞ for f; g ∈ ZN : p λ = q2 for 1 ≤ s ≤ mn: s Analogous formulas describe the quantum frames associated with the elements τ of the set ΞN . After an easy computation, one finds that the unique solution of the Example 4: The cluster variables in the initial seed from Theorem system of equations in Theorem 4 is given by the matrix B~ = ∈ Z 5 for Rq½Mm × n are ðbði−1Þn+j;ðk−1Þn+jÞ Mmn×exð Þ with entries 8 Δ ; >± ; = ; = ± ½i−minði; jÞ+1; i;½ j−minði; jÞ+1; j <> 1 if i k l j 1 or j = l; k = i ± 1 b − + ; − + = where 1 ≤ i ≤ m and 1 ≤ j ≤ n. The ones with i = morj= n are ði 1Þn j ðk 1Þn l > or i = k ± 1; j = l ± 1; :> frozen. This example and Example 3 recover the quantum cluster 0; otherwise algebra structure of Geiß et al. (12) on Rq½Mm × n. We finish the section by raising two questions concerning the for all i, k ∈ [1, m] and j, l ∈ [1, n]. line of Theorem 5: Cluster Algebra Structures. Let us consider a symmetric quantum 1. If a symmetric quantum nilpotent algebra has two iterated skew nilpotent algebra R of dimension N. When Theorem 1 is applied polynomial extension presentations that satisfy the assumptions to the presentation of R from Eq. 6 associated with the element in Definition 1 and Definition 3, and these two presentations are τ ∈ ΞN , we obtain a sequence of prime elements not obtained from each other by a permutation in ΞN , how are

Goodearl and Yakimov PNAS | July 8, 2014 | vol. 111 | no. 27 | 9699 Downloaded by guest on September 24, 2021 the corresponding quantum cluster algebra structures on R opposite Borel subalgebras of g. The torus HdðK*Þr acts on related? UqðgÞ by 2. What is the role of the quantum seeds of a symmetric quantum nilpotent algebra R indexed by Ξ among the set of all · ± 1= ± 1; · = ξ ; · = ξ−1 ; [8] N h Ki Ki h Ei iEi h Fi i Fi quantum seeds? Is there a generalization of Theorem 5 that constructs a larger family of quantum seeds using sequences = ξ ; ...; ξ ∈ ≤ ≤ ± for all h ð 1 rÞ H and 1 i r. The subalgebras U ½w of prime elements in chains of subalgebras? are preserved by this action. Denote by α ; ...; α the set of simple roots of g and by For the first question, we expect that the two quantum cluster 1 r s1; ...; sr ∈ W the corresponding set of simple reflections. All algebra structures on R are the same (i.e., the corresponding ± quantum seeds are mutation-equivalent) if the maximal tori for algebras U ½w are symmetric quantum nilpotent algebras for all K ∈ K* the two presentations are the same and act in the same way on base fields and nonroots of unity q . In fact, with each R. However, proving such a fact appears to be difficult due to reduced expression the generality of the setting. The condition on the tori is natural w = s ...s ; inlightoftheorem5.5ofref.18,whichprovestheexistenceof i1 iN a canonical maximal torus for a quantum nilpotent algebra. ± one associates a presentation of U ½w that satisfies Definition 1 Without imposing such a condition, the cluster algebra struc- and Definition 3 as follows. Consider the Weyl group elements tures can be completely unrelated. For example, every polynomial algebra d ... ; ≤ ≤ ; d : w≤k si1 sik 1 k N and w≤0 1 = K ; ...; R ½x1 xN In terms of these elements, the roots of the Lie algebra n+ ∩ wðn−Þ are over an infinite field K is a symmetric quantum nilpotent algebra with respect to the natural action of ðK*ÞN . The quantum cluster β = αi ; β = w≤1ðαi Þ; ...; β dw≤N−1ðαi Þ: algebra structure on R constructed in Theorem 5 has no ex- 1 1 2 2 N N ; ...; changeable indices, and its frozen variables are x1 xN . Each As in the works of Jantzen (19) and Lusztig (21), one associates polynomial algebra has many different presentations associated with those roots the Lusztig root vectors Eβ ; Fβ ; ...; Eβ ; 1 − 1 N with the elements of its automorphism group, and the corre- β ∈ F N UqðgÞ. The quantum Schubert cell algebra U ½w, defined sponding cluster algebra structures are not related in general. β ; ...; β as the subalgebra of UqðgÞ generated by F 1 F N , has an iterated skew polynomial extension presentation of the form Applications to Quantum Groups Quantized Universal Enveloping Algebra. Let g be a finite di- − U ½w = K Fβ Fβ ; σ2; δ2 ... Fβ ; σN ; δN ; [9] mensional complex simple Lie algebra of rank r with Cartan ma- 1 2 N K ∈ K* trix ðcijÞ. For an arbitrary field and a nonroot of unity q , for which conditions (a)–(c) of Definition 1 are satisfied with following the notation of Jantzen (19), one defines the quantized respect to the action of Eq. 8. Moreover, this presentation sat- universal enveloping algebra UqðgÞ with generators isfies the condition for a symmetric quantum nilpotent algebra in Definition 3 because of the Levendorskii–Soibelman straighten- ± 1; ; ; ≤ ≤ + Ki Ei Fi 1 i r ing law (19) in UqðgÞ. The opposite algebra U ½w, generated by Eβ ; ...; Eβ , has analogous properties and is actually isomorphic 1 − N and relations (19) to U ½w. The algebra Rq½Mm × n is isomorphic to one of the algebras −1 = −1 = ; = ; − Ki Ki KiKi 1 KiKj KjKi U ½w for g = slm+n and a certain choice of w ∈ Sm+n.

− −1 = cij ; −1 = cij ; Quantized Function Algebras. The irreducible finite dimensional KiEjKi qi Ej KiFjKi qi Fj modules of UqðgÞ on which the elements Ki act diagonally via powers of the scalars q are parametrized by the set P+ of − −1 i Ki Ki E F − F E = δ ; ; dominant integral weights of g. The module corresponding to i j j i i j − −1 qi qi such a weight λ will be denoted by VðλÞ. Let G be the connected, simply connected algebraic group − * 1Xcij with Lie algebra g. The Hopf subalgebra of UqðgÞ spanned by − − − − n 1 cij n 1 cij n = ; ≠ ; λ λ ∈ λ * ð 1Þ ðEiÞ EjðEiÞ 0 i j the matrix coefficients cf;y of all modules Vð Þ [where f Vð Þ = n q n 0 i and y ∈ VðλÞ] is denoted by Rq½G and called the quantum group λ corresponding to G. The weight spaces Vð Þwλ are one-dimensional together with the analogous relation for the generators Fi. Here, for all Weyl group elements w. Considering the fundamental fd ; ...; d g is the collection of relatively prime positive integers 1 r representations Vðϖ Þ; ...; Vðϖ Þ, and a normalized covector such that the matrix ðd c Þ is symmetric and q dqdi . The algebra 1 r i ij i and vector in each of those weight spaces, one defines, following UqðgÞ is a Hopf algebra with coproduct Berenstein and Zelevinsky (11), the quantum minors

ΔðKiÞ = Ki ⊗ Ki; ΔðEiÞ = Ei ⊗ 1 + Ki ⊗ Ei; − i Δ = ⊗ 1 + ⊗ : Δ ; = Δwϖ ;vϖ ∈ Rq½G; 1 ≤ i ≤ r; w; v ∈ W: ðFiÞ Fi Ki 1 Fi w v i i ± The quantum Schubert cell algebras U ½w, parametrized by The subalgebras of Rq½G spanned by the elements of the form λ λ the elements w of the Weyl group W of g, were introduced by De cf;y, where y is a highest or lowest weight vector of Vð Þ and * ± Concini et al. (20) and Lusztig (21). In the work of Geiß et al. f ∈ VðλÞ , are denoted by R . They are quantum analogs of the (12), the term quantum unipotent groups was used. These base affine space of G. With the help of the Demazure modules + λ = + λ + algebras are quantum analogs of the universal enveloping alge- Vw ð Þ Uq ðgÞVð Þwλ [where Uq ðgÞ is the unital subalgebra of ± ∓ ± bras Uðn ∩ wðn ÞÞ, where n are the nilradicals of a pair of UqðgÞ generated by E1; ...; Er], one defines the ideal

9700 | www.pnas.org/cgi/doi/10.1073/pnas.1313071111 Goodearl and Yakimov Downloaded by guest on September 24, 2021 n o + λ closest to the combinatorial setting of (10, 11). It goes with the − SPECIAL FEATURE I = Span c ; jλ ∈ P+; fj + λ = 0; y ∈ VðλÞλ w f y Vw ð Þ reverse presentation of U ½w, − p p p p + − − = K β β ; σ ; δ ... β ; σ ; δ : U ½w F N F N − 1 N−1 N−1 F 1 1 1 of R . Analogously one defines an ideal Iw of R . For all pairs of Weyl group elements ðw; vÞ, the quantized coordinate ring of the 9 double Bruhat cell (9) The transition from the original presentation in Eq. to the above one amounts to interchanging the roles of the predecessor Gw;vdB+wB+\ B−vB−; and successor functions. As a result, the set of exchangeable indices for the latter presentation is ± where B are opposite Borel subgroups of G, is defined by − exwdfk ∈ ½1; Njk ≠−∞g: [10] h i w;v d + − + + − Δi −1; Δi −1 ; Rq½G Iw R R Iv w;1 vw ;w 0 0 Theorem 6. Consider an arbitrary finite dimensional complex simple Lie algebra g, a Weyl group element w ∈ W, a reduced expression where the localization is taken over all 1 ≤ i ≤ r and w denotes 0 of w, an arbitrary base field K, and a nonroot of unity q ∈ K* such the longest element of the Weyl group W. pffiffiffi − that q ∈ K. The quantum Schubert cell algebra U ½w possesses a To connect cluster algebra structures on the two kinds of canonical quantum cluster algebra structure for which no frozen algebras (quantum Schubert cells and quantum double Bruhat + + + i −1 cluster variables are inverted and the set of exchangeable indices is cells), we use certain subalgebras S of ðR =I Þ½ðΔ ; Þ ; 1 ≤ i ≤ r, w w w 1 exw. Its initial seed consists of the cluster variables which were defined by Joseph (22). They are the subalgebras ffiffiffi 2 generated by the elements p ðw − w≤ − Þϖ 2 i i −1 jj k 1 ik jj φ Δ k Δ k ; q w ; ð ; Þ w≤ k−1 1 w 1 − ϖ + ðΔi Þ 1 c i + I ; w;1 f; y w 1 ≤ k ≤ N. The exchange matrix B~ of this seed has entries given by 8 for 1 ≤ i ≤ r, f ∈ Vðϖ Þ*, and y a highest weight vector of Vðϖ Þ. > ; = i i > 1 if k pðlÞ These algebras played a major role in the study of the spectra <> −1; if k = sðlÞ

of quantum groups (22, 23) and the quantum Schubert cell MATHEMATICS bkl = ci i ; if pðkÞ < pðlÞ < k < l algebras (24). In ref. 23, Yakimov constructed an algebra > k l > −ci i ; if pðlÞ < pðkÞ < l < k antiisomorphism : k l 0; otherwise φ : + → − : w Sw U ½w for all 1 ≤ k ≤ N and l ∈ exw. Furthermore, this quantum cluster algebra equals the corresponding upper quantum cluster algebra. An earlier variant of it for UqðgÞ equipped with a different co- m For all k ∈ ½1; N, m ∈ Z≥ such that s ðkÞ ∈ ½1; N, the elements product appeared in the work of Yakimov (24). 0 ffiffiffi 2 p w≤ m − w≤ − ϖ 2 Cluster Structures on Quantum Schubert Cell Algebras. Denote by jjð s ðkÞ k 1Þ ik jj q h:; :i the Weyl group invariant bilinear form on the vector space −1 × φ Δik Δik w≤ m w≤ − ;1 w≤ m ;1 Rα1⊕⋯⊕Rαr normalized by hαi; αii = 2 for short roots αi. Let s ðkÞ k 1 s ðkÞ γ 2d γ; γ jj jj h i. − Fix a Weyl group element w, and consider the quantum are cluster variables of U ½w. − For symmetric Kac–Moody algebras g, the theorem is due to Schubert cell algebra U ½w. A reduced expression w = si ...si − 1 N – gives rise to the presentation in Eq. 9 of U ½w as a symmetric Geiß et al. (12). Our proof also works for all Kac Moody algebras g, but, here, we restrict to the finite dimensional case for quantum nilpotent algebra. The result of the application of simplicity of the exposition. Theorem 6 is proved in section 10 of Theorem 1 to it is as follows. The function η can be chosen as ref. 17. Example 3 and Example 4 can be recovered as special cases of ηðkÞ = ik for all 1 ≤ k ≤ N: Theorem 6 for g = slm+n and a particular choice of the Weyl − group element w ∈ S + . In this case, the torus action can be The predecessor function p is the function k↦k , which plays m n pffiffiffi a key role in the works of Fomin and Zelevinsky (1, 9), used to kill the power of q. Remark 2: It follows from the definition of the antiisomorphism φ : + → − < = ; ; w Sw U ½w in the work of Yakimov (23, 24) that the element −d maxfl kjil ikg if such l exists k −∞; : otherwise −1 φ Δik Δik [11] w w≤ k−1;1 w;1 The successor function s is the function k↦k+ in the works of Fomin and Zelevinsky (1, 9). The sequence of prime elements is obtained (up to a minor term) by evaluating y1; ...; yN consists of scalar multiples of the elements Δik ⊗ w ; − ð w≤ − ;w idÞðR Þ φ Δik Δik 1 ; ≤ ≤ ; k 1 w n ; ð ; Þ 1 k N w≤ p k ðkÞ−1 1 w≤ k 1 where Rw, calledP the R-matrix for the Weyl group element w, equals + − + − + where nk denotes the maximal nonnegative integer n such that the infinite sum u ⊗ u for dual bases fu g and fu g of U ½w − − j j j j j pnðkÞ ≠−∞. The presentation in Eq. 9 of U ½w as a symmetric and U ½w. Because of this, the element in Eq. 11 can be identified i quantum nilpotent algebra satisfies the conditions (A) and (B) if with Δ k ; , and thus can be thought of as a quantum minor. Such pffiffiffi w≤k−1 w − q ∈ K, in which case Theorem 5 produces a canonical cluster a construction of cluster variables of U ½w via quantum minors is − algebra structure on U ½w. Among all of the clusters in Theorem due to the work of Geiß et al. (12), who used linear maps that are 5, indexed by the elements of the subset ΞN of the symmetric not algebra (anti)isomorphisms. group SN, the one corresponding to the longest element of SN is More generally,

Goodearl and Yakimov PNAS | July 8, 2014 | vol. 111 | no. 27 | 9701 Downloaded by guest on September 24, 2021 − ⊆ − ≤ ≤ products are defined (22, 23) from the Drinfeld R-matrix com- U w≤ j U ½w for 1 j N φ mutation relations of Rq½G. Using the antiisomorphism w and its negative counterpart [which turns out to be an isomorphism and the cluster variables in Theorem 6, (23)], one converts −1 + − − op + ik ik − S ⋈ S ≅ U ½w ⋈ U ½v; φ Δ ; Δ ; ∈ U w≤ m ; w v w≤ smðkÞ w≤ k−1 1 w≤ smðkÞ 1 s ðkÞ op i where R stands for the algebra with opposite product. We then can be identified with the quantum minors Δ k ; . w≤ k−1 w≤ smðkÞ establish that the right-hand algebra above is a symmetric quantum nilpotent algebra satisfying the conditions (A) and (B). The Berenstein–Zelevinsky Conjecture. Consider a pair of Weyl group ; proof is completed by applying Theorem 5, and showing that the elements ðw vÞ with reduced expressions localization that we started with is, in fact, a localization by all w = si ...si and v = s 0 ...s 0 : frozen cluster variables. 1 N i1 iM

Let η : ½1; r + M + N → ½1; r be the function given by Poisson Nilpotent Algebras and Cluster Algebras. In this subsection, 8 we will assume that the base field K has characteristic 0. A prime < k; for 1 ≤ k ≤ r; element p of a Poisson algebra R with Poisson bracket f:; :g will η = 0 ; + ≤ ≤ + ; be called Poisson prime if ðkÞ : ik−r for r 1 k r M ik−r−M ; for r + M + 1 ≤ k ≤ r + M + N: fR; pg = Rp: The following set will be used as the set of exchangeable indices ; In other words, this requires that the principal ideal Rp be a for a quantum cluster algebra structure on R ½Gw v: q Poisson ideal as well as a prime ideal.

ex w;vd½1; k⊔fk ∈ ½r + 1; r + M + NjsðkÞ ≠ +∞g; For a commutative algebra R equipped with a rational action of a torus H by algebra automorphisms, we will denote by ∂h the where s is the successor function for the level sets of η. Set derivation of R corresponding to an element h of the Lie algebra eðkÞd1 for k ≤ r + M and eðkÞd − 1 for k > r + M. Following of H. Berenstein and Zelevinsky (11), define the ðr + M + NÞ × ex ma- Definition 4: A nilpotent semiquadratic Poisson algebra is a ~ K ; ...; :; : trix Bw;v with entries polynomial algebra ½x1 xN with a Poisson structure f g and r 8 a rational action of a torus H = ðK*Þ by Poisson algebra auto- > −e ; = ; ; ...; > ðlÞ if k pðlÞ morphisms for which x1 xN are H-eigenvectors and there exist > −e ; < < < ; e = e ; ...; > ðlÞcηðkÞ;ηðlÞ if k l sðkÞ sðlÞ ðlÞ ðsðkÞÞ elements h1 hN in the Lie algebra of H such that the following <> or k < l ≤ r + M < sðlÞ < sðkÞ; two conditions are satisfied for 1 ≤ k ≤ N: d e ; < < < ; e = e bkl ðkÞcηðkÞ;ηðlÞ if l k sðlÞ sðkÞ ðkÞ ðsðlÞÞ ∈ dK ; ...; > (a) For all b Rk−1 ½x1 xk−1 > or l < k ≤ r + M < sðkÞ < sðlÞ; > > eðkÞ; if k = sðlÞ; ; = ∂ + δ : fxk bg hk ðbÞxk kðbÞ 0; otherwise: for some δkðbÞ ∈ Rk−1 and the map δk : Rk−1 → Rk−1 is locally The following theorem proves the Berenstein-Zelevinsky con- nilpotent. jecture (11). (b) The hk-eigenvalue of xk is nonzero.

Theorem 7. Let G be an arbitrary complex simple Lie group and Such an algebra will be called symmetric if the above condition is ; ...; ðw; vÞ a pair of elements of the corresponding Weyl group. For any satisfied for the reverse order of generators xN x1 (with different pffiffiffi base field K and a nonroot of unity q ∈ K* such that q ∈ K*, the choices of elements h•). quantum double Bruhat cell algebra R Gw;v possesses a canonical The adjective semiquadratic refers to the leading term in the q½ ; λ structure of quantum cluster algebra for which all frozen cluster Poisson bracket fxk xlg, which is forced to have the form klxkxl λ ∈ K* variables are inverted and the set of exchangeable indices is ex ; . for some kl . ~ w v The initial seed has exchange matrix Bw;v,asdefined above, and w;v Theorem 8. Every symmetric nilpotent semiquadratic Poisson algebra cluster variables y ; ...; y + + ∈ R ½G , given by 8 1 r M N q as above satisfying the Poisson analog of condition (B) has a canon- k ical structure of cluster algebra for which no frozen variables are < Δ − ; for 1 ≤ k ≤ r; 1;v 1 inverted and the compatible Poisson bracket in the sense of Gekhtman i′ = ξ Δ k − r ; + ≤ ≤ + ; :; : yk k ; −1 for r 1 k r M et al. (25) is f g. Its initial cluster consists, up to scalar multiples, of : 1 v v≤k−r i − − those Poisson prime elements of the chain of Poisson subalgebras ξ Δ k r M ; for r + M + 1 ≤ k ≤ r + M + N k w≤k−r−M ;1 K½x ⊂ K½x ; x ⊂ ... ⊂ K½x ; ...; x ξ ∈ K* 1 1 2 1 N for some scalars k . Furthermore, this quantum cluster algebra coincides with the that are H-eigenvectors. Each generator xk,1≤ k ≤ N, is a cluster corresponding upper quantum cluster algebra. variable of this cluster algebra. We briefly sketch the relationship of the quantum double Furthermore, this cluster algebra coincides with the corresponding w;v Bruhat cell algebras Rq½G to quantum nilpotent algebras and upper cluster algebra. the proof of the theorem. We first show, using results of Joseph Applying this theorem in a similar fashion to Theorem 7,we w;v (22), that Rq½G is a localization of h i obtain the following result. ± ± + ⋈ − #K Δ1 1; ...; Δr 1 ; Sw Sv 1;v−1 1;v−1 Theorem 9. Let G be an arbitrary complex simple Lie group. For all pairs of elements ðw; vÞ of the Weyl group of G, the Berenstein– − − – where Sv is the Joseph subalgebra of R defined in a similar way Fomin Zelevinsky upper cluster algebra (10) on the coordinate ring + + “ ” w;v (22) to the subalgebra Sw of R . The bicrossed and smash of the double Bruhat cell G coincides with the corresponding

9702 | www.pnas.org/cgi/doi/10.1073/pnas.1313071111 Goodearl and Yakimov Downloaded by guest on September 24, 2021 cluster algebra. In other words, the coordinate rings of all double A. Zelevinsky for helpful discussions and comments. We also thank the referee, Bruhat cells C½Gw;v are cluster algebras with initial seeds con- whose suggestions helped improve the exposition. Moreover, we would like to SPECIAL FEATURE thank the Mathematical Sciences Research Institute for its hospitality during the structed by Berenstein et al. (10). programs in “Cluster Algebras” and “Noncommutative Algebraic Geometry and Representation Theory” whenpartsofthisprojectwerecompleted.This ACKNOWLEDGMENTS. We thank S. Fomin, A. Berenstein, Ph. Di Francesco, work was partially supported by the National Science Foundation Grants DMS- R. Kedem, B. Keller, A. Knutson, B. Leclerc, N. Reshetikhin, D. Rupel and 0800948 (to K.R.G.), and DMS-1001632 and DMS-1303038 (to M.T.Y.).

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