Quantum cluster algebra structures on quantum nilpotent algebras K. R. Goodearl M. T. Yakimov Author address: Department of Mathematics, University of California, Santa Bar- bara, CA 93106, U.S.A. E-mail address: [email protected] Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. E-mail address: [email protected] To the memory of Andrei Zelevinsky Contents Chapter 1. Introduction 1 1.1. Quantum cluster algebras and a general formulation of the main theorem 1 1.2. Definition of quantum nilpotent algebras 3 1.3. Constructing an initial quantum seed and a precise statement of the main result 5 1.4. Additional clusters 8 1.5. Strategy of the proof 9 1.6. Organization of the paper and reading suggestions 11 1.7. Notation 12 Chapter 2. Quantum cluster algebras 15 2.1. General quantum tori 15 2.2. Based quantum tori 16 2.3. Compatible pairs 17 2.4. Mutation of compatible pairs 18 2.5. Quantum seeds and mutations 20 2.6. Quantum cluster algebras and the Laurent phenomenon 23 Chapter 3. Iterated skew polynomial algebras and noncommutative UFDs 29 3.1. Equivariant noncommutative unique factorization domains 29 3.2. CGL extensions 29 3.3. Symmetric CGL extensions 34 3.4. Further CGL details 36 Chapter 4. One-step mutations in CGL extensions 39 4.1. A general mutation formula 39 4.2. A base change and normalization of the elements y j from Theorem 3.6 43 4.3. Almost cluster mutations between CGL extension presentations 44 4.4. Scalars associated to mutation of prime elements 46 Chapter 5. Homogeneous prime elements for subalgebras of symmetric CGL extensions 49 5.1. The elements y 49 [i,sm(i)] 5.2. The elements of Ξ 50 N 5.3. A subset of Ξ 52 N 5.4. Sequences of homogeneous prime elements 53 5.5. An identity for normal elements 55 Chapter 6. Chains of mutations in symmetric CGL extensions 59 v vi CONTENTS 6.1. The leading coefficients of u 59 [i,sm(i)] 6.2. Rescaling of the generators of a symmetric CGL extension 63 6.3. Toric frames and mutations in symmetric CGL extensions 64 Chapter 7. Division properties of mutations between CGL extension presentations 69 7.1. Main result 69 7.2. Proof of the main result 71 Chapter 8. Symmetric CGL extensions and quantum cluster algebras 75 8.1. General setting 75 8.2. Statement of the main result 77 8.3. Cluster variables 79 8.4. Auxiliary results 81 8.5. An overview of the proof of Theorem 8.2 84 8.6. Recursive construction of quantum seeds for τ ∈Γ 85 N 8.7. Proofs of parts (a), (b) and (c) of Theorem 8.2 88 8.8. Intersections of localizations 90 8.9. Completion of the proof of Theorem 8.2 93 8.10. Some vectors f 94 [i,s(i)] Chapter 9. Quantum groups and quantum Schubert cell algebras 97 9.1. Quantized universal enveloping algebras 97 9.2. Quantum Schubert cell algebras 98 9.3. Quantum function algebras and homomorphisms 100 9.4. Quantum minors and sequences of prime elements 102 Chapter 10. Quantum cluster algebra structures on quantum Schubert cell algebras 105 10.1. Statement of the main result 105 10.2. Compatibility of the toric frame Mw and the matrix B(cid:101)w 109 10.3. Proof of Theorem 10.1 111 Bibliography 115 Index 117 Abstract All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild con- ditions. Furthermore, itisshownthatthesequantumclusteralgebrasalwaysequal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quan- tized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have abroadrangeofapplicationstotheseproblems,includingtheconstructionofquan- tum cluster algebra structures on quantum unipotent groups and quantum double Bruhatcells(theBerenstein–Zelevinskyconjecture),andtreattheseproblemsfrom aunifiedperspective. Allsuchapplicationsalsoestablishequalitybetweenthecon- structed quantum cluster algebras and their upper counterparts. The proofs rely onChatters’notionofnoncommutativeuniquefactorizationdomains. Toricframes are constructed by considering sequences of homogeneous prime elements of chains of noncommutative UFDs (a generalization of the construction of Gelfand–Tsetlin subalgebras) and mutations are obtained by altering chains of noncommutative UFDs. Along the way, an intricate (and unified) combinatorial model for the ho- mogeneousprimeelementsinchainsofnoncommutativeUFDsandtheiralterations is developed. When applied to special families, this recovers the combinatorics of Weyl groups and double Weyl groups previously used in the construction and cat- egorification of cluster algebras. It is expected that this combinatorial model of sequencesofhomogeneousprimeelementswillhaveapplicationstotheunifiedcat- egorification of quantum nilpotent algebras. Receivedbytheeditor23December2013. 2010MathematicsSubjectClassification. Primary16T20;Secondary13F60,17B37,14M15. Keywordsandphrases. Quantumclusteralgebras,quantumnilpotentalgebras,iteratedOre extensions,noncommutativeuniquefactorizationdomains. TheresearchofK.R.G.waspartiallysupportedbyNSFgrantDMS-0800948. The research of M.T.Y. was partially supported by NSF grants DMS-1001632 and DMS- 1303038. vii CHAPTER 1 Introduction 1.1. Quantum cluster algebras and a general formulation of the main theorem Cluster algebras were invented by Fomin and Zelevinsky in [11] based on a novel construction of producing infinite generating sets via a process of mutation. The initial goal was to set up a combinatorial framework for studying canonical bases and total positivity [10]. Remarkably, for the past twelve years cluster al- gebras and the procedure of mutation have played an important role in a large number of diverse areas of mathematics, including representation theory of finite dimensional algebras, combinatorial and geometric Lie theory, Poisson geometry, integrable systems, topology, commutative and noncommutative algebraic geome- try, and mathematical physics. The quantum counterparts of cluster algebras were introduced by Berenstein and Zelevinsky in [3]. We refer the reader to the recent surveys [16, 31, 36, 38] and the book [18] for more information on some of the abovementioned aspects of this theory. A major direction in the theory of cluster algebras is to prove that important (quantized) coordinate rings of algebraic varieties arising from Lie theory admit (quantum)clusteralgebrastructuresorupper(quantum)clusteralgebrastructures. For example, the Berenstein–Zelevinsky conjecture [3] states that the quantized coordinate rings of double Bruhat cells in all finite dimensional simple Lie groups admit explicit quantum cluster algebra structures. The motivation for this type of problem is that once (quantum) cluster algebra structures are constructed on families of (quantized) coordinate rings, they can then be used in the study of canonical bases of those rings. In the classical case, a cluster algebra structure on the coordinate ring of a variety can be used to investigate its totally positive part. Going back to the general problem, a second part asks if the constructed up- per(quantum)clusteralgebraequalsthecorresponding(quantum)clusteralgebra. For example, ten years ago Berenstein, Fomin and Zelevinsky proved [1] that the coordinate rings of double Bruhat cells in all simple algebraic groups admit upper clusteralgebrastructures. Yetitwasunknowniftheseupperclusteralgebrasequal the corresponding cluster algebras, i.e., if the coordinate rings of double Bruhat cells are actually cluster algebras. Previous approaches to the above problems relied on a construction of an ini- tialseedandsomeadjacentseedsintermsof(quantum)minorsandrelatedregular functions [1, 3, 14, 15]. After that point, two different approaches were followed. The first one, due to Berenstein, Fomin and Zelevinsky [1], used the methods of uniquefactorizationdomainstoprovethatthecoordinateringsunderconsideration areupperclusteralgebras. ItwasfirstappliedtocoordinateringsofdoubleBruhat cells [1]. This approach was developed further in [18, 19] and [17]. The second 1 2 1. INTRODUCTION approachwasviatheconstructionofacategorificationbasedonconcretecombina- torial data from Weyl groups and then to prove that the corresponding (quantum) cluster algebra equals the (quantized) coordinate ring under consideration. This approachisduetoGeiß–Leclerc–Schr¨oer[14, 15], whoappliedittothecoordinate rings of the unipotent groups U ∩w(U ) and the quantum Schubert cell algebras + − U (n ∩w(n ))(alsocalledquantumunipotentgroups)forsymmetricKac–Moody q + − groups G, where w is a Weyl group element. In both of the above approaches, one relied on specific data in terms of Weyl group combinatorics for the concrete family of coordinate rings. Moreover, the ini- tial(quantum)seedswerebuiltviaadirectconstructionbyconsidering(quantum) minors. Thegoalofthispaperistopresentanewalgebraicapproachtoquantumcluster algebrasbasedonnoncommutativeringtheory. Weproduceageneralconstruction of quantum cluster algebra structures on a broad class of algebras and construct initial clusters and mutations in a uniform and intrinsic way, without ad hoc con- structions with quantum minors. We first state the main theorem of the paper in a general form. The following sections contain a precise formulation of it. Main Theorem I: General Form. Each algebra in a very large, axiomati- cally defined class of quantum nilpotent algebras admits a quantum cluster algebra structure. Furthermore, for all such algebras, the latter equals the corresponding upper quantum cluster algebra. Thetheoremhasabroadrangeofapplicationsbecausemanyimportantfamilies of algebras fall within this axiomatic class. In particular, the previously mentioned families are special subfamilies of this class of algebras. Furthermore, the required axioms are easy to verify for additional families of algebras. The proof of the theoremisconstructive,sooneobtainsanexplicitquantumclusteralgebrastructure in each case. Initial clusters are constructed intrinsically as finite sequences of (homogeneous)elementsinchainsofnoncommutativeuniquefactorizationdomains. Another key feature of the result is that it holds for arbitrary base fields: there are no restrictions on their characteristic and they do not need to be algebraically closed. Theproofofthetheoremisbasedonpurelyringtheoreticargumentswhich are independent of the characteristic of the field and do not use specialization. Finally, when the methods are applied to algebras arising from quantum groups, thedeformationparameterqonlyneedstobeanon-rootofunitywhiletheprevious methods needed q to be transcendental over Q. In this paper, we apply the theorem to construct explicit quantum cluster algebra structures on the quantum Schubert cell algebras U (n ∩w(n )) for all q + − finite dimensional simple Lie algebras g. (The technique works for all Kac–Moody Liealgebrasg,butthegeneralcaserequirestechnicalitieswhichwouldonlyincrease the size of the paper and obstruct the main idea. Because of this, the minor additional details for infinite dimensional Lie algebras g will appear elsewhere.) If g is symmetric, this result is due to Geiß, Leclerc and Schr¨oer [15]. In this case we obtain the same quantum cluster algebra structure, but under milder assumptions on the base field and the deformation parameter. In a forthcoming publication [25], we will give a proof of the Berenstein– Zelevinskyconjecture[3]usingtheabovetheorem. Wewillshowthatthequantized coordinate rings of all double Bruhat cells are localizations of quantum nilpotent algebras and that applying the above theorem produces precisely the conjectured