Quantum Cluster Algebras and Quantum Nilpotent Algebras

Quantum cluster algebras and quantum nilpotent algebras K. R. Goodearl ∗ and M. T. Yakimov † ∗Department of Mathematics, University of California, Santa Barbara, CA 93106, U.S.A., and †Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. Proceedings of the National Academy of Sciences of the United States of America 111 (2014) 9696–9703 A major direction in the theory of cluster algebras is to construct construction does not rely on any initial combinatorics of the (quantum) cluster algebra structures on the (quantized) coordinate algebras. On the contrary, the construction itself produces rings of various families of varieties arising in Lie theory. We prove intricate combinatorial data for prime elements in chains of that all algebras in a very large axiomatically defined class of noncom- subalgebras. When this is applied to special cases, we re- mutative algebras possess canonical quantum cluster algebra struc- cover the Weyl group combinatorics which played a key role tures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results in categorification earlier [7, 9, 10]. Because of this, we expect for a large class of Poisson nilpotent algebras. Many important fam- that our construction will be helpful in building a unified cat- ilies of coordinate rings are subsumed in the class we are covering, egorification of quantum nilpotent algebras. Finally, we also which leads to a broad range of application of the general results prove similar results for (commutative) cluster algebras using to the above mentioned types of problems. As a consequence, we Poisson prime elements. prove the Berenstein–Zelevinsky conjecture for the quantized coor- The results of the paper have many applications since a dinate rings of double Bruhat cells and construct quantum cluster number of important families of algebras arise as special cases algebra structures on all quantum unipotent groups, extending the in our axiomatics. Berenstein, Fomin, and Zelevinsky proved theorem of Geiß, Leclerc and Schröer for the case of symmetric Kac– [10] that the coordinate rings of all double Bruhat cells in ev- Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein, Fomin and Zelevinsky associated to double Bruhat ery complex simple Lie group are upper cluster algebras. It cells coincide with the corresponding cluster algebras. was an important problem to decide whether the latter coincide with the corresponding cluster algebras. We resolve quantum cluster algebras noncommutative unique factorization domains this problem positively. On the quantum side, we prove the quantum groups quantum| Schubert cell algebras | Berenstein–Zelevinsky conjecture [11] on cluster algebra struc- | tures for all quantum double Bruhat cells. Finally, we establish that the quantum Schubert cell algebras for all complex he theory of cluster algebras provides a unified frame- simple Lie groups have quantum cluster algebra structures. work for treating a number of problems in diverse areas T Previously this was known for symmetric Kac–Moody groups of mathematics such as combinatorics, representation theory, due to Geiß, Leclerc and Schröer [12]. topology, mathematical physics, algebraic and Poisson geom- etry, and dynamical systems [1, 2, 3, 4, 5, 6, 7]. The construction of cluster algebras was invented by Fomin and Zelevin- Prime elements of quantum nilpotent algebras sky [1] who also obtained a number of fundamental results Definition of quantum nilpotent algebras. Let K be an arbi- on them. This construction builds algebras in a novel way K by producing infinite generating sets via a process of muta- trary base field. A skew polynomial extension of a -algebra tion rather than the classical approach using generators and A, A A[x; σ, δ], relations. 7→ The main algebraic approach to cluster algebras relies on is a generalization of the classical polynomial algebra A[x]. It representations of finite dimensional algebras and derived cat- is defined using an algebra automorphism σ of A and a skew- egories [5, 8]. In this paper we describe a different algebraic derivation δ. The algebra A[x; σ, δ] is isomorphic to A[x] as a approach based on noncommutative ring theory. vector space and the variable x commutes with the elements arXiv:1508.03086v1 [math.QA] 12 Aug 2015 An important range of problems in the theory of clus- of A as follows: ter algebras is to prove that the coordinate rings of certain algebraic varieties coming from Lie theory admit cluster alge- xa = σ(a)x + δ(a) for all a A. bra structures. The idea is that, once this is done, one can ∈ use cluster algebras to study canonical bases in such coor- For every nilpotent Lie algebra n of dimension m there dinate rings. Analogous problems deal with the correspond- exists a chain of ideals of n ing quantizations. The approach via representation theory to this type of problem needs combinatorial data for quivers as n = nm ⊲ nm−1 ⊲ . ⊲ n1 ⊲ n0 = 0 a starting point. Such might not be available a priori. This { } approach also differs from one family of varieties to another, which means that in each case one needs to design an appro- Reserved for Publication Footnotes priate categorification process. We prove that all algebras in a very general, axiomatically defined class of quantum nilpotent algebras are quantum cluster algebras. The proof is based on constructing quantum clusters by considering sequences of prime elements in chains of subalgebras which are noncommutative unique factorization domains. These clusters are canonical relative to the mentioned chains of subalgebras, which are determined by the presentation of the quantum nilpotent algebra. The www.pnas.org/cgi/doi/10.1073/pnas.1313071111 PNAS July 8, 2014 vol. 111 no. 27 1–9 such that dim(nk/nk−1)=1 and [n, nk] nk−1, for 1 k Unique factorization domains. The notion of unique factoriza- ⊆ ≤ ≤ m. Choosing an element xk in the complement of nk−1 in tion domain plays an important role in algebra and number nk for each 1 k m leads to the following presentation theory. Its noncommutative analog was introduced by Chat- of the universal≤ enveloping≤ algebra (n) as an iterated skew ters in [15]. A nonzero, non-invertible element p of a domain polynomial extension: U R (a ring without zero divisors) is called prime if pR = Rp and the factor R/Rp is a domain. A noetherian domain R is (n) = K[x1][x2; id, δ2] . [xm; id, δm] U ∼ called a unique factorization domain (UFD) if every nonzero prime ideal of R contains a prime element. Such rings pos- where all the derivations δ2, . , δm are locally nilpotent. sess the unique factorization property for all of their nonzero Definition 1. An iterated skew polynomial extension normal elements – the elements u R with the property that ∈ R = K[x1][x2; σ2, δ2] [xN ; σN , δN ] [1] Ru = uR. If the ring R is acted upon by a group G, then one · · · can introduce an equivariant version of this property: Such is called a quantum nilpotent algebra if it is equipped with a an R is called a G-UFD if every nonzero G-invariant prime rational action of a K-torus by K-algebra automorphisms H ideal of R contains a prime element which is a G-eigenvector. satisfying the following conditions: It was shown in [16] that every quantum nilpotent algebra (a) The elements x1,...,xN are -eigenvectors. H R is a noetherian -UFD. An -eigenvector of such a ring R (b) For every 2 k N, δk is a locally nilpotent σk-der- H H ≤ ≤ will be called a homogeneous element since this corresponds ivation of K[x1] [xk−1; σk−1, δk−1]. · · · to the homogeneity property with respect to the canonical (c) For every 1 k N, there exists hk such that ≤ ≤ ∈ H induced grading of R by the character lattice of . In par- σk = (hk ) and the hk-eigenvalue of xk, to be denoted by λk, H · ticular, we will use the more compact term of homogeneous is not a root of unity. prime element of R instead of a prime element of R which is The universal enveloping algebras of finite dimensional an -eigenvector. nilpotent Lie algebras satisfy all of the conditions in the defi- H nition except for the last part of the third one. More precisely, in that case one can take = 1 , conditions (a)–(b) are sat- Sequences of prime elements. Next, we classify the set of all H { } homogeneous prime elements of the chain of subalgebras isfied, and in condition (c) we have λk = 1. Thus, condition (c) is the main feature that separates the class of quantum 0 R1 R2 . RN = R nilpotent algebras from the class of universal enveloping alge- { }⊂ ⊂ ⊂ ⊂ bras of nilpotent Lie algebras. The torus is needed in order H of a quantum nilpotent algebra R, where R is the subalgebra to define the eigenvalues λk. k The algebras in Definition 1 are also known as Cauchon– of R generated by the first k variables x1,...,xk. Z Z Goodearl–Letzter (CGL) extensions. The axiomatics came We will denote by and ≥0 the sets of all integers and nonnegative integers, respectively. Given two integers l k, from the works [13, 14] which investigated in this generality ≤ the stratification of the prime spectrum of an algebra into set [l,k] := l,l + 1,...,k . For a function{ η : [1}, N] Z, define the predecessor strata associated to its -prime ideals. → The Gelfand–KirillovH dimension of the algebra in Eq. [1] function p : [1, N] [1, N] and successor function s : [1, N] [1, N] →+ for⊔ its{−∞} level sets by equals N. → ⊔ { ∞} Example 1. For two positive integers m and n, and q K∗, define the algebra of quantum matrices R [M ] as the∈ algebra max l<k η(l)= η(k) , if such l exists, q m×n p(k)= { | } with generators tij , 1 i m and 1 j n, and relations , otherwise, ≤ ≤ ≤ ≤ (−∞ tij tkj = qtkj tij , for i<k, and tij til = qtiltij , for j <l, tij tkl = tkltij , for i<k, j>l, min l>k η(l)= η(k) , if such l exists, −1 s(k)= { | } tij tkl tkltij =(q q )tiltkj , for i<k, j<l.

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