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[ x ,δ σ, o all for ) x ; omtswt h elements the with commutes ,δ σ, ⊲ sioopi to isomorphic is ] n ] 1 n , ⊲ fdimension of n a o.111 vol. 0 σ ∈ Let = of A. { A 0 K } n skew- a and ea arbi- an be o og,LA Rouge, ton o 27 no. K A m -algebra A [ x [ struc- sa as ] x there pect .It ]. sing ses 1 he so v- – t- d s 9 - f 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∗, de- fine the algebra of quantum matrices R [M ] as the∈ algebra max ll, min l>k η(l)= η(k) , if such l exists, −1 s(k)= { | } tij tkl tkltij =(q q )tiltkj , for i k. −2 ∈ hij tij = q tij . · In particular, yk is a homogeneous prime element of Rk, for ∗ Thus, for all q K which are not roots of unity, the algebras all k [1, N]. The sequence y1,...,yN and the level sets of a ∈ ∈ Rq[Mm×n] are examples of quantum nilpotent algebras. function η with these properties are both unique.

2 www.pnas.org/cgi/doi/10.1073/pnas.1313071111 Footline Author Example 2. Given two subsets I = i1 < < ik [1,m] Def inition 3. A quantum nilpotent algebra R as in Definition { · · · } ⊂ and J = j1 < < jk [1, n], define the quantum minor 1 will be called symmetric if it can be presented as an iterated { · · · } ⊂ ∆I,J Rq[Mm×n] by skew polynomial extension for the reverse order of its genera- ∈ tors, ℓ(σ) ∆I,J = ( q) ti1jσ(1) ...tikjσ(k) K ∗ ∗ ∗ ∗ − R = [xN ][xN−1; σN−1, δN−1] [x1; σ1 , δ1 ], σX∈Sk · · · in such a way that conditions (a)–(c) in Definition 1 are sat- where Sk denotes the symmetric group and ℓ : Sk Z≥0 the ∗ ∗ → isfied for some choice of hN ,...,h1 . standard length function. A quantum nilpotent algebra R ∈is H symmetric if and only For the algebra of quantum matrices Rq [Mm×n], the se- if it satisfies the Levendorskii–Soibelman type straightening quence of prime elements from Theorem 1 consists of solid law quantum minors; more precisely, nl+1 nk−1 xkxl λklxlxk = ξnl+1,...,nk−1 xl+1 ...xk−1 y =∆ − Z (i−1)n+j [i−min(i,j)+1,i],[j−min(i,j)+1,j] nl+1,...,nXk−1∈ ≥0 ∗ for all 1 i m and 1 j n. Furthermore, the function for all lk. The defining commuta- → k · lk tion relations for the algebras of quantum matrices Rq[Mm×n] η((i 1)n + j) := j i. − − imply that they are symmetric quantum nilpotent algebras. Denote by ΞN the subset of the symmetric group SN con- Definition 2. The cardinality of the range of the function η sisting of all permutations that have the property that from Theorem 1 is called the rank of the quantum nilpotent algebra R and is denoted by rk(R). τ(k) = max τ([1,k 1]) + 1 or [4] − For example, the algebra of quantum matrices Rq [Mm×n] has τ(k) = min τ([1,k 1]) 1 [5] rank m + n 1. − − − for all 2 k N. In other words, ΞN consists of those τ S such≤ that≤ τ([1,k]) is an interval for all 2 k N. Embedded quantum tori. An N N q := (q ) with N kl For∈ each τ Ξ , a symmetric quantum nilpotent≤ algebra≤ R entries in K is called multiplicatively× skewsymmetric if N has the presentation∈

qklqlk = qkk = 1 for 1 l,k N. R = K[x ][x ; σ′′ , δ′′ ] [x ; σ′′ , δ′′ ] [6] ≤ ≤ τ(1) τ(2) τ(2) τ(2) · · · τ(N) τ(N) τ(N) Such a matrix gives rise to the quantum torus q which is the where σ′′ := σ and δ′′ := δ if Eq. [4] is satisfied, K ±1 ±1 T τ(k) τ(k) τ(k) τ(k) -algebra with generators Y1 ,...,YN and relations ′′ ∗ ′′ ∗ while στ(k) := στ(k) and δτ(k) := δτ(k) if Eq. [5] holds. This presentation satisfies the conditions (a)–(c) in Definition 1 for YkYl = qklYlYk for 1 l,k N. ′′ ′′ ≤ ≤ the elements hτ(k) , given by hτ(k) := hτ(k) in case of N ′′ ∈ H∗ Let e1,...,eN be the standard basis of the lattice Z . Eq. [4] and hτ(k) := hτ(k) in case of Eq. [5]. The use of the For a quantum{ nilpotent} algebra R of dimension N, define the term symmetric in Definition 3 is motivated by the presenta- eigenvalues λkl K: tions in Eq. [6] parametrized by the elements of the subset ∈ ΞN of the symmetric group SN . hk xl = λklxl for 1 ll, s(l) =+ . They are related to the eigenvalues λl by  −1 6 ∞ λlk , if k

∗ dη(l) ∗ dη(k) Cluster structures on quantum nilpotent algebras (λk) =(λl ) Symmetric quantum nilpotent algebras. for all k,l ex. ∈

Footline Author PNAS July 8, 2014 vol. 111 no. 27 3 Remark 1. With the exception of some two-cocycle twists, for Similarly, applying Theorem 4 to the presentation from all of the quantum nilpotent algebras R coming from the Eq. [6] we obtain the theory of quantum groups, the subgroup of K∗ generated by Z λ 1 l

4 www.pnas.org/cgi/doi/10.1073/pnas.1313071111 Footline Author For the first question we expect that the two quantum algebras ±[w] are symmetric quantum nilpotent algebras for cluster algebra structures on R are the same (i.e., the corre- all base fieldsU K and non-roots of unity q K∗. In fact, to sponding quantum seeds are mutation equivalent) if the max- each reduced expression ∈ imal tori for the two presentations are the same and act in the same way on R. However, proving such a fact appears to w = si1 ...siN , be difficult due to the generality of the setting. The condition one associates a presentation of ±[w] that satisfies Defini- on the tori is natural in light of Theorem 5.5 in [18] which U proves the existence of a canonical maximal torus for a quan- tions 1 and 3 as follows. Consider the Weyl group elements tum nilpotent algebra. Without imposing such a condition, w≤k := si ...si , 1 k N, and w≤0 := 1. the cluster algebra structures can be completely unrelated. 1 k ≤ ≤ For example, every polynomial algebra In terms of these elements the roots of the Lie algebra n+ w(n−) are R = K[x1,...,xN ] ∩

β1 = αi1 , β2 = w≤1(αi2 ), ..., βN := w≤N−1(αiN ). over an infinite field K is a symmetric quantum nilpotent algebra with respect to the natural action of (K∗)N . The As in [19, 21], one associates with those roots the Lusztig root quantum cluster algebra structure on R constructed in The- vectors Eβ1 , Fβ1 ,...,EβN , FβN q(g). The quantum Schu- − ∈ U orem 5 has no exchangeable indices and its frozen variables bert cell algebra [w], defined as the subalgebra of q (g) U U are x1,...,xN . Each polynomial algebra has many different generated by Fβ1 ,...,FβN , has an iterated skew polynomial presentations associated to the elements of its automorphism extension presentation of the form group and the corresponding cluster algebra structures are not − [w]= K[Fβ ][Fβ ; σ2, δ2] . . . [Fβ ; σN , δN ] [9] related in general. U 1 2 N for which conditions (a)–(c) of Definition 1 are satisfied with respect to the action Eq. [8]. Moreover, this presentation Applications to quantum groups satisfies the condition for a symmetric quantum nilpotent al- Quantized universal enveloping algebra. Let g be a finite di- gebra in Definition 3 because of the Levendorskii–Soibelman + mensional complex simple Lie algebra of rank r with Cartan straightening law [19] in q(g). The opposite algebra [w], matrix (cij ). For an arbitrary field K and a non-root of unity U U ∗ generated by Eβ1 ,...,EβN , has analogous properties and is q K , following the notation of [19], one defines the quan- − ∈ actually isomorphic [19] to [w]. tized universal enveloping algebra q (g) with generators U The algebra Rq[Mm×n] is isomorphic to one of the alge- U − ±1 bras [w] for g = slm+n and a certain choice of w Sm+n. K ,Ei, Fi, 1 i r U ∈ i ≤ ≤ Quantized function algebras. The irreducible finite dimen- and relations sional modules of q(g) on which the elements Ki act diag- U −1 −1 onally via powers of the scalars qi are parametrized by the Ki Ki = KiKi = 1, KiKj = Kj Ki, set P+ of dominant integral weights of g. The module corre- −1 cij −1 −cij KiEj Ki = qi Ej , KiFj Ki = qi Fj , sponding to such a weight λ will be denoted by V (λ). Let G be the connected, simply connected algebraic group K K−1 i i g g ∗ EiFj Fj Ei = δi,j − −1 , with Lie algebra . The Hopf subalgebra of q( ) spanned − qi qi λ U − by the matrix coefficients cf,y of all modules V (λ) (where f 1−c ∗ ∈ ij V (λ) and y V (λ)) is denoted by Rq[G] and called the quan- n 1 cij n 1−cij −n ∈ ( 1) − (Ei) Ej (Ei) = 0, i = j, tum group corresponding to G. The weight spaces V (λ)wλ − n 6 n=0 qi are one dimensional for all Weyl group elements w. Consider- X   ing the fundamental representations V (̟1),...,V (̟r), and a together with the analogous relation for the generators Fi. normalized covector and vector in each of those weight spaces, Here d1,...,dr is the collection of relatively prime posi- { } one defines, following [11], the quantum minors tive integers such that the matrix (dicij ) is symmetric, and di i qi := q . The algebra q(g) is a Hopf algebra with coproduct ∆ =∆w̟ ,v̟ Rq[G], 1 i r, w, v W. U w,v i i ∈ ≤ ≤ ∈

∆(Ki)= Ki Ki, ∆(Ei)= Ei 1+ Ki Ei, The subalgebras of Rq [G] spanned by the elements of the form ⊗ ⊗ ⊗ λ −1 cf,y where y is a highest or a lowest weight vector of V (λ) and ∆(Fi)= Fi K + 1 Fi. ⊗ i ⊗ f V (λ)∗ are denoted by R±. They are quantum analogs of ∈ ± the base affine space of G. With the help of the Demazure The quantum Schubert cell algebras [w], parametrized + + + U modules V (λ) = (g)V (λ)wλ (where (g) is the unital by the elements w of the Weyl group W of g, were intro- w Uq Uq subalgebra of q(g) generated by E1,...,Er) one defines the duced by De Concini–Kac–Procesi [20] and Lusztig [21]. In U [12] the term quantum unipotent groups was used. These ideal algebras are quantum analogs of the universal enveloping al- + λ Iw = Span cf,y λ P+, f + = 0, y V (λ)λ gebras (n± w(n∓)) where n± are the nilradicals of a pair { | ∈ |Vw (λ) ∈ } U ∩ ∗ r of opposite Borel subalgebras of g. The torus := (K ) acts + − − H of R . Analogously one defines an ideal Iw of R . For all on q(g) by U pairs of Weyl group elements (w, v), the quantized coordinate ±1 ±1 −1 ring of the double Bruhat cell [9] h K = K , h Ei = ξiEi, h Fi = ξ Fi, [8] · i i · · i Gw,v := B+wB+ B−vB−, for all h = (ξ1,...,ξr) and 1 i r. The subalgebras ∩ ± ∈ H ≤ ≤ [w] are preserved by this action. where B± are opposite Borel subgroups of G, is defined by U Denote by α1,...,αr the set of simple roots of g and by w,v + − + − i −1 i −1 s1,...,sr W the corresponding set of simple reflections. All Rq [G ]:=(I R + R I )[(∆ ) , (∆ ) ] ∈ w v w,1 vw0,w0

Footline Author PNAS July 8, 2014 vol. 111 no. 27 5 ∗ where the localization is taken over all 1 i r and w0 unity q K such that √q K. The quantum Schubert cell ≤ ≤ ∈ ∈ denotes the longest element of the Weyl group W . algebra −[w] possesses a canonical quantum cluster algebra To connect cluster algebra structures on the two kinds structureU for which no frozen cluster variables are inverted and of algebras (quantum Schubert cells and quantum dou- the set of exchangeable indices is exw. Its initial seed consists + ble Bruhat cells), we use certain subalgebras Sw of of the cluster variables + + i −1 (R /Iw )[(∆w,1) , 1 i r], which were defined by Joseph ≤ ≤ 2 [22]. They are the subalgebras generated by the elements k(w−w≤k−1)̟i k /2 ik ik −1 √q k ϕw ∆ (∆ ) , w≤k−1,1 w,1 i −1 ̟i +   (∆w,1) (cf,y + Iw ), 1 k N. The exchange matrix B of this seed has entries ∗ ≤ ≤ for 1 i r, f V (̟i) , and y a highest weight vector given by ≤ ≤ ∈ of V (̟i). These algebras played a major role in the study e of the spectra of quantum groups [22, 23] and the quantum 1, if k = p(l) Schubert cell algebras [24]. In [23], Yakimov constructed an 1, if k = s(l) algebra antiisomorphism − bkl = cikil , if p(k)

6 www.pnas.org/cgi/doi/10.1073/pnas.1313071111 Footline Author The Berenstein–Zelevinsky conjecture. Consider a pair of Poisson nilpotent algebras and cluster algebras Weyl group elements (w, v) with reduced expressions In this subsection, we will assume that the base field K has characteristic 0. A prime element p of a Poisson algebra R w = si ...si and v = si′ ...si′ . 1 N 1 M with Poisson bracket ., . will be called Poisson prime if Let η : [1,r + M + N] [1,r] be the function given by { } → R,p = Rp. k, for 1 k r, { } ≤ ≤ η(k)= i′ , for r + 1 k r + M,  k−r ≤ ≤ In other words, this requires that the principal ideal Rp be a ik−r−M , for r + M + 1 k r + M + N. Poisson ideal as well as a prime ideal. ≤ ≤ For a commutative algebra R equipped with a rational ac- The following set will be used as the set of exchangeable in-  w,v tion of a torus by algebra automorphisms, we will denote dices for a quantum cluster algebra structure on Rq[G ]: H by ∂h the derivation of R corresponding to an element h of exw,v := [1,k] k [r + 1,r + M + N] s(k) =+ the Lie algebra of . ⊔ { ∈ | 6 ∞} H where s is the successor function for the level sets of η. Set Def inition 4. A nilpotent semi-quadratic Poisson algebra is a ǫ(k) := 1 for k r + M and ǫ(k) := 1 for k >r + M. polynomial algebra K[x1,...,xN ] with a Poisson structure ≤ − ., . and a rational action of a torus =(K∗)r by Poisson al- Following [11], define the (r + M + N) ex matrix Bw,v with { } H entries × gebra automorphisms for which x1,...,xN are -eigenvectors and there exist elements h ,...,h in theH Lie algebra of e 1 N ǫ(l), if k = p(l), such that the following two conditions are satisfied for − H ǫ(l)cη(k),η(l), if k

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8 www.pnas.org/cgi/doi/10.1073/pnas.1313071111 Footline Author