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.