DOCSLIB.ORG

Quantum Cluster Algebras and Quantum Nilpotent Algebras

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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 ca- nonical 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 À Á = ⊳ ⊳ ... ⊳ ⊳ = −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.
Load more

Recommended publications
  • An Introduction to Lie Algebra
    California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 12-2017 An Introduction to Lie Algebra Amanda Renee Talley California State University – San Bernardino Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Algebra Commons Recommended Citation Talley, Amanda Renee, "An Introduction to Lie Algebra" (2017). Electronic Theses, Projects, and Dissertations. 591. https://scholarworks.lib.csusb.edu/etd/591 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. An Introduction to Lie Algebra A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Amanda Renee Talley December 2017 An Introduction to Lie Algebra A Thesis Presented to the Faculty of California State University, San Bernardino by Amanda Renee Talley December 2017 Approved by: Dr. Giovanna Llosent, Committee Chair Date Dr. Gary Griffing, Committee Member Dr. Hajrudin Fejzic, Committee Member Dr. Charles Stanton, Chair, Dr. Corey Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract An (associative) algebra is a vector space over a field equipped with an associa- tive, bilinear multiplication. By use of a new bilinear operation, any associative algebra morphs into a nonassociative abstract Lie algebra, where the new product in terms of the given associative product, is the commutator.
    [Show full text]
  • Nonassociative Algebras∗
    Nonassociative Algebras∗ Murray R. Bremner† Department of Mathematics and Statistics University of Saskatchewan, Saskatoon, Canada Lucia´ I. Murakami‡ Instituto de Matem´atica e Estat´ıstica Universidade de S˜aoPaulo, S˜ao Paulo, Brasil Ivan P. Shestakov§ Instituto de Matem´atica e Estat´ıstica Universidade de S˜aoPaulo, S˜ao Paulo, Brasil and Sobolev Institute of Mathematics, Novosibirsk, Russia One of the earliest surveys on nonassociative algebras is the article by Shirshov [Shi58] which introduced the phrase “rings that are nearly associative”. The first book in the English language devoted to a systematic study of nonassociative algebras is Schafer [Sch66]. A comprehensive exposition of the work of the Russian school is Zhevlakov, Slinko, Shestakov and Shirshov [ZSS82]. A collection of open research problems in algebra, including many problems on nonassociative algebra, is the Dniester Notebook [FKS93]; the survey article by Kuzmin and Shestakov [KS95] is from the same period. Three books on Jordan algebras which contain substantial material on general nonassociative algebras are Braun and Koecher [BK66], Jacobson [Jac68] and McCrimmon [McC04]. Recent research appears in the Proceedings of the International Con- ferences on Nonassociative Algebra and its Applications [Gon94], [CGG00], [SSS05]. The present section provides very limited information on Lie algebras, since they are the subject of Section §16.4. The last part (§9) of the present section presents three applications of computational linear algebra to the study of polynomial identities for nonassociative algebras: pseudorandom vectors in a nonassociative algebra, the expansion matrix for a nonassociative operation, and the representation theory of the symmetric group. 1 Introduction Definitions: An algebra is a vector space A over a field F together with a bilinear multiplication (x, y) 7→ xy from ∗The authors thank Irvin Hentzel for helpful comments on an earlier version of this article.
    [Show full text]
  • Quantum Groups
    Lie Groups and Quantum Groups Prof. Vera Serganova notes by Theo Johnson-Freyd UC-Berkeley Mathematics Department Spring Semester 2010 Contents Contents 1 Introduction 4 0.1 Conventions and numbering................................ 5 Lecture 1 Jan 20, 20105 1.1 ................................................ 5 1.2 Root and Weight lattices.................................. 7 Lecture 2 Jan 22, 20109 2.1 Weight Modules ...................................... 9 2.2 Weyl Character Formula.................................. 10 2.3 Verma module ....................................... 11 Lecture 3 Jan 25, 2010 12 3.1 Casimir element ...................................... 13 3.2 Category O (BGG category) ............................... 14 Lecture 4 Jan 27, 2010 16 4.1 Kostant partition function................................. 16 4.2 BGG resolution....................................... 17 4.3 Dimension formula..................................... 17 4.4 Application to tensor product............................... 18 Lecture 5 Jan 29, 2010 19 5.1 Center of universal enveloping algebra .......................... 20 5.2 Harish-Chandra homomorphism.............................. 22 1 Lecture 6 Feb 1, 2010 22 Lecture 7 Feb 3, 2010 25 7.1 Nilpotent cone ....................................... 27 Lecture 8 Feb 5, 2010 28 Lecture 9 Feb 8, 2010 31 9.1 G as an algebraic group .................................. 31 Lecture 10 Feb 10, 2010 34 10.1 General facts about algebraic groups........................... 35 10.2 Compact Groups.....................................
    [Show full text]
  • Constants of Lie Algebra Actions
    JOURNAL OF ALGEBRA 114, 452465 (1988) Constants of Lie Algebra Actions JEFFREY BERGEN Deparrment of Mathematics, DePaul University, Chicago, Illinois 60614 Communicated by I. N. Herstein Received May 5, 1985 Let R be an associative algebra over a field k and let L be a Lie algebra over k acting on R as derivations. We will be interested in studying the relationship between R and the ring of constants RL. Our situation is a special case of the study of rings of invariants of Hopf algebra actions and is also analogous to another special case of Hopf algebra actions, namely, the study of fixed rings of finite group actions. Many of the results in this paper are analogs of results on group actions and give rise to more general questions regarding Hopf algebra actions. In Section 1, we study actions on non-nilpotent algebras. We first show that any algebraic derivation has non-zero constants and then prove that any finite dimensional nilpotent Lie algebra of algebraic derivations must act with non-zero constants. It then follows, as a corollary, that any finite dimensional nilpotent restricted Lie algebra must act with non-zero constants. These are analogs of results on groups proved in [ 12, 5). In Section 2, we prove a necessary condition for the existence of non- zero invariants of certain Hopf algebra actions. We then use this result to prove the converse of our theorem on nilpotent restricted Lie algebras. We conclude the section with several questions on general Hopf algebra actions. 1. NON-NILPOTENT ALGEBRAS In this section we will prove the existence of non-zero constants of cer- tain Lie algebra actions.
    [Show full text]