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THE CENTER OF (n ). q ω U HANS P. JAKOBSEN 5 Abstract. We determine the centerofa localizationofUq(nω)⊆Uq+(g)by the covariant 1 elements (non-mutable elements) by means of constructions and results from quantum 0 cluster algebras. In our set-up, g is any finite-dimensional complex Lie algebra and ω is 2 any element in the Weyl group W. The non-zero complex parameter q is mostly assumed l not to be a root of unity, but our method also gives many details in case q is a primitive u J root of unity. We point to a new and very useful direction of approachto a general set of problemswhichweexemplifyherebyobtainingtheresultthatthecenterisdeterminedby 7 2 thenullspaceof1+ω. Further,weusethistogiveageneralizationtodoubleSchubertCell algebras where the center is proved to be given by ωa+ωc. Another family of quadratic ] algebras is also considered and the centers determined. A Q . h t 1. Introduction a m The topics of quantum groups, quantized function algebras, quantized ma- [ trix algebras, and quantum cluster algebras have since long been seen to be 3 intrinsically interwoven. v 6 The groundbreaking research of Drinfeld ([13],[14]) and Jimbo ([29],[30]) was 3 followed by deep results of Lusztig ([41],[45])), Kashiwara ([31],[32]). Then 1 6 Levendorskii and Soibelman ([40], [38]) and later de Concini and Procesi ([11]) 0 added the quadratic algebra side to this distinguished family. With the advent . 1 of the cluster algebras of Fomin and Zelevinsky ([17]) and Berenstein-Zelevinsky 0 5 quantized cluster algebras ([4]) many new dimensions were added to the function 1 algebra side. : v Through many years, quantized function algebras have attracted a lot of at- i X tention ([11], [10], [12], [15], [19], [22], [24] [39], [40], [41], [46], [49], and r many others). Many special examples were considered in the beginning, but a also general families have more recently been considered ([18]). The current research has its focus on the quadratic algebra side. It utilizes fundamental results in ([4]) and ([18]). In certain families of examples ([25],[26]), it was seen that certain (signed) permutation matrices contained much information about the quantized matrix algebras. The topic of this article is to explain exactly the reason for that, while Date: July 28, 2015. 2010 Mathematics Subject Classification. MSC 17B37 (primary), MSC 17A45 (primary), MSC 13F60 (primary), MSC 20G42 (secondary), and MSC 16T20 (secondary). 1 at the same time giving the full description of the centers. That we thereby also obtain an insight into very algebraic properties of quantized function algebras, even specializing these to roots of unity, is clear, but will not be pursued in this article until the very last section. For now, and until then, we assume that q is a not roots of unity. An important tool in our investigation is a family of quantized minors introduced by Berenstein and Zelevinsky ([4]). Later C. Geiss, B. leclerc, and J. Schroër ([18]) have modified these in a way that turns out to be exactly suitable for our needs. Given an element w in the Weyl group W one may construct, using Lusztig ([42]), a family of elements Z ,...,Z +. It is a key result of Levendorskii 1 ℓ(w) q ⊂ U and Soibelman’s ([40], [38])) that, for 1 i < j ℓ(w), ≤ ≤ Z Z = q(γi,γj)Z Z + terms involving only elements Z with i k j. i j j i k ≤ ≤ These relations can be taken as the defining relations of (n ). q ω U Procesi and de Concini later reproved this result and introduced the associated quasi-polynomial algebra (n ) with generators z ,...,z , and relations q ω 1 ℓ(w) U z z = q(γi,γj)z z . i j j i They proved that eg. the P.I. degree of (n ) could be determined from this q ω U much simpler algebra in the case where q = ε is a primitive root of unity. We return briefly to this degree in the last section of this article. The success of the present endeavor rests on the choice of a good basis of the associated quasi Laurent algebra (n ). While looking at specific cases ([26])) q ω L such a basis was found essentially as the “diagonals” of the quantized minors of Berenstein-Zelevinsky. L Consider the symplectic form defined by the relations above, that is, the skew symmetric form defined by L i < j : = (γ ,γ ). i,j i j Recall that a symplectic form may be brought to a block diagonal form by using matrices with integer coefficients and determinant 1. The center of (n ) is given by the null space of L. q ω L The second major step forward comes with the construction, quite explicitly, B L and while again using ideas from ([25],[26]), of a partial inverse to . As an aside, we mention that we actually construct a quantum seed. With these steps taken, the center of (n ) is easily determined. q ω L 2 The final step towards determining the center of (n ) comes when one q ω L L L realizes that actually is also the symplectic form for a certain family of B q commuting quantized minors and likewise is expressible in terms of these L B minors and their inverses and hence we get a compatible pair ( , ). As mentioned in the abstract, the center is given by the null space of 1 + ω. More precisely, to each fundamental weight Λ there is a covariant element s C (ω) and the center is given by those Cns(ω) for which s s s (1 + ω)( Qn Λ ) = 0. s s s X We discuss some special examples of this in Subsection 5.3, Section 6, and Section 8. The results have previously been obtained by other methods by Ph. Caldero [7], [8] in the case where ω is the longest element. Recently, a full treatment was obtained by M. Yakimov [52] also by different methods. Besides the use of a number of deep results, our methods are completely elementary and focuses on quantum seeds. We also point to certain useful structures related to subalgebras of a fixed parabolic subalgebra p and this, we hope, will be seen as significant points of the article. One such structure is the assignment of a unique pair (c,d) N2 to each Schubert Cell. Another is, in ∈ the terminology of cluster algebras, a compatible pair given quite explicitly. With this available, we can even give a generalization to the centers of algebras connected to double Schubert Cells defined by minimal left coset representatives ωa,ωc in W W with, say, ωa < ωc. Here the center is given by the null space p of ωa + ωc \ Here is a more detailed account of the content: Section 2: Background; saturated sets of positive roots are discussed. Sec- tion 3: It is determined that (n ) has the structure of quadratic algebra. Sec- q ω U tion 4: Basics; a diagrammatic way of representing n and ∆+(n ) is introduced. ω ω Section 5: The case of the associated quasi-polynomial algebra is described and an example is given. Section 6 is a Diophantine interlude in which the centers are computed for some specific elements w W in type A . Then in Section 7 the n ∈ quantum minors of Berenstein-Zelevinsky are introduced, the twist by ([18]) is given, and two series of what we call Levendorskii-Soibelman quadratic algebras are introduced. After that, the way has been paved for Section 8 in which the previous results are extended to the general setting for these series of quadratic 3 algebras. A more general class of Double Schubert Cell algebras is also briefly discussed. Finally in the last section, the case where q is a primitive root of unity is discussed. 2. On Parabolics The origin of the following lies in A. Borel [5], and B. Kostant [35]. Other main contributors are [3] and [51]. See also [9]. We have also found ([50]) useful. We consider a simple Lie algebra g. A is the Cartan matrix of g and is assumed to be of finite type. Π denotes a fixed choice of simple roots, and E denotes the euclidean space spanned by the simple roots. The fundamental Π weights corresponding to the simple roots are denoted by Λ . i Definition 2.1. Let w W. Set ∈ Φ = α ∆+ w 1α ∆ = w(∆ ) ∆+. ω − − − { ∈ | ∈ } ∩ We have that ℓ(w) = ℓ(w 1) = Φ . − ω | | Definition 2.2. A subset S of ∆+ is saturated if whenever α,β S and ∈ α + β is a root, then α + β S. ∈ Theorem 2.3 ([35]). The map w Φ ω 7→ defines a bijection between W and the set of all subsets Φ ∆+ for which both Φ and ∆+ Φ are saturated. ⊆ \ In passing we observe that, trivially, for a saturated set Φ, both Φ and ∆+ Φ \ correspond to nilpotent subalgebras. We will from now on set Φ = ∆+(w). ω We will consider nilpotent quantized enveloping algebras of the form (n ), q ω U where ω is an arbitrary element in the Weyl group W, and n is the quantized ω nilpotent defined by the roots α Φ (This is n of ([18]). It is convenient ω ω−1 ∈ for us, also with an eye to forthcoming investigations, to assume that we are working with a fixed parabolic p with a Levi decomposition p = l + u, (1) where l is the Levi subalgebra, and such that, on the classical level, n u. ω ⊆ There is no loss of generality in that. Finally set 4 Definition 2.4. W = w W Φ ∆+(l) p ω { ∈ | ⊆ } Wp = w W Φ ∆+(u) . ω { ∈ | ⊆ } Wp is a set of distinguished representatives of the right coset space W W. p \ It is well known (see eg ([50])) that any w W can be written uniquely as w = w wp with w W and wp Wp. ∈ p p p ∈ ∈ 3. The quadratic algebras Let ω = s s ...s be an element of the Weyl group written in reduced α α α 1 2 t form. Using his braid operators, given in a special case as T (e ) = ( q ) be(a)e e(b), i j i − i j i − a+b=r X wherer = h ,α , Lusztig in ([42]) construct a sequence of elements Z ,...,Z i j 1 t −h i ⊆ +(g). Specifically, q U Z = T (E ), i = 2,...,t, and Z = E , i ωi−1 αi 1 α1 where, for each i = 1,...,t, ω = s s ...s . In particular, ω = ω . The i α α α t 1 2 i weight of Z is given by γ = ω (α ). Here, and throughout, we use the i i i 1 i notation of ([27]). − The following result is well known Theorem 3.1 ([39],[38]). Suppose that 1 i < j t. Then ≤ ≤ Z Z = q(γi,γj)Z Z + terms involving only elements Z with i k j. i j j i k ≤ ≤ Our statement follows [27],[28]. Other authors, eg. [39], [18] have used the other Lusztig braid operators. The result is just a difference between q and q 1. − Proofs of this theorem which are more accessible are available ([11],[28]). Proposition 3.2. (n ) is a quadratic algebra. q ω U It is known that this algebra is isomorphic to the algebra of functions on (n ) satisfying the usual finiteness condition. It is analogously equivalent to q ω U the algebra of functions on (n ) satisfying a similar finiteness condition. See q− ω U eg ([18]) and ([27]). We will not distinguish between these algebras. 5 4. basic structure Consider a fixed basis Π = α ,α ,...,α of Φ. Let us agree to write σ 1 2 R α { } i in W just as σ for i = 1,...,R. i We consider a fixed parabolic p with a Levi decomposition p = l + u (2) with u = 0 . 6 { } Adapting to the language of [17], [4], and others, we will often label structures derived from ωr by the reduced word r. The full structure with double words will not be required here. Let ωp be the maximal element in Wp. It is the one which maps all roots in ∆+(u) to ∆ . (Indeed: To ∆ (u).) Let ω be the longest element in W and − − 0 ω the longest in the Weyl group of l, Then L p ω ω = ω . (3) L 0 Let ωr = σ σ σ Wp be fixed and written in a fixed reduced form. i i i Then ℓ(ωr) = r1. 2 ··· r ∈ Set ∆+(ωr) = β ,...,β . (4) i i { 1 r} Consider α Π and set ω = ωr s . i 1 i r+1 ∈ ◦ r+1 Case 1: r(α ) = γ ∆+(u). Then ℓ(w ) = ℓ(ωr) + 1 and i 1 r+1 ∈ ∆+(w ) = β ,...,β γ and γ = β = ωr(α ). (5) 1 i i r+1 i { 1 r} ∪ { } r+1 Thus, ω Wp. 1 ∈ Case 2: ωr(α ) = γ ∆ (u). Then ℓ(w ) = r 1 and i − 1 r+1 − ∈ − ∆+(w ) = β ,...,β γ and γ = β for some β ∆+(ωr). (6) 1 i i i i { 1 r} \ { } s s ∈ We must always be in at least one of these cases since otherwise ωr would map all simple roots, hence ∆(g), to ∆(l). If ωr is maximal and α is a simple root i such that ωr(α ) ∆ (u) then α ∆+(u). It is easy to see that under the i − i ∈ ∈ same assumptions, ωr(α ) ∆ (l) is not possible. Furthermore, if α ∆+(l) i − i ∈ ∈ then ωr(α ) ∆+(g). In conclusion, a maximal ωr maps ∆+(l) to ∆+(l) and i ∈ ∆+(u) to ∆ (u). Thus, if ωr is maximal, ω ωr = ωrω = ω . Hence ωr = ωp. − L L 0 It follows easily that we have the following conclusion: Let ωr Wp with ℓ(ωr) = r. Then we may write ∈ ωr = s s s where for all j = 1...,r : w = s s Wp. i i i s i i 1◦···◦ s◦···◦ r 1◦···◦ j ∈ (7) 6 Furthermore, ∆+(ωr) = β ,...,β ,...,β (8) i i i { 1 s r} where for each i : s β = s s (α ) for s > 1, and β = α . (9) is i1 ◦ ··· ◦ is−1 is i1 i1 Moreover, ωr = σ σ σ = σ σ σ . (10) i i i β β β 1 2 ··· r r ··· 2 1 From now on, q is a fixed element of C which is not a root of unity, ωr Wp is given with a fixed decomposition as in (10), and ∆+(wr) is our univers∈e. Definition 4.1. Let b denote the map Π 1,2,...,R defined by b(α ) = i → { } i. Let π : 1,2,...,r Π be given by r { } → π (j) = α . (11) r i j If π (j) = α we say that α (or σ ) occurs at position j in ωr, and we say r α that π 1(α) are the positions at which α occurs in w. Set −r π = b π . (12) r r ◦ Let, for 1 n r, ω = σ σ σ . Thus, we have a 1-dimensional n α α α ≤ ≤ i1 i2 ··· in presentation of the situation given by the (ordered) set 1,2,...,r . { } The following 2-dimensional presentation is even more useful and informative: Definition 4.2. U(r) = (13) N N (s,t) n such that s = π (n) and ω = ω σ ω ...ω σ . r n 1 i 2 t i { ∈ × | ∃ n n} In the above, it is understood that each ω W e is reduced and does i ∈ \ { } not contain any σ . i n We also identify n (s,t) (and β β ) if n,s,t are connected as above. n s,t We denote the ident↔ifications between↔1,2,...,r and U(r) simply as n { } ↔ (s,t). (And β β ) n s,t ↔ We define a map π for such ω in analogy with that of π . ω n r n If ωr = ω ω˜ and ω = ω ωˆ with ω ,ω Wp and all Weyl group elements m m n n m ∈ reduced, we say that ω < ω if ωˆ = e. n m 6 Definition 4.3. If n (s,t) and m (c,d) we define ↔ ↔ (s,t) < (c,d) ω < ω . (14) s,t c,d ⇔ 7 For a fixed s 1,2,...,R we let s denote the maximal such t. This is the r number of tines∈σ{ occurs in ω}r. We then have s U(r) = (s,t) N N 1 s R and 1 t s . (15) r { ∈ × | ≤ ≤ ≤ ≤ } Finally, notice that if (s,t) U(r) then we may construct a subset U(s,t) of U by the above recipe, replac∈ing ωr by ω . In this subset t is maximal. s,t 5. The quasi-polynomial algebra 5.1. The first definitions and computations. Definition 5.1. (r) denotes the C-algebra generated by elements z ;j = q j M { 1,...,r indexed by the elements β defined as above and with relations j } z z = q(βi,βj)z z if i < j. (16) i j j i We let (r) denote the associated quasi-Laurent algebra, and we let (r) q q L X denote the center of (r). We will also label the generators by z as dis- q s,t L cussed in Section 4. Let s Im(π ). It is then straightforward to see that r ∈ ω(Λ ) + Λ = β + β + + β . (17) − s s s,1 s,2 ··· s,sr Definition 5.2. Let s Im(π ). We define the element C (r) in the quasi- r s ∈ polynomial algebra (r) by q M C (r) = z z z . (18) s s,1 s,2 · ··· · s,sr Proposition 5.3. The following holds for all s Im(π ) and all (a,b) r U(r): ∈ ∈ z C (r) = q (βa,b,(1+ωr)(Λs))C (r)z , (19) a,b s − s a,b Proof. We will be using repeatedly that β = ω (α ). Consider a decom- s,t s,t i − s position ωr = ω σ ω with π (ℓ) = a = s and suppose that β < β < A α B r s,t a,b iℓ 6 β for some t. It is here understood that ℓ = (a,b). The elements ω ,ω s,t+1 A B of course depend on ℓ, indeed, ω σ = ω . Then A i ℓ ℓ zℓzs,t+1...zs,sr = q(βiℓ,(−ωr+ωA)(Λs)))zs,t+1zs,t+2...zs,srzℓ (α ,( σ ω (Λ )+Λ )) = q iℓ − αiℓ B s s zs,t+1zs,t+2...zs,srzℓ (20) = q(αiℓ,ωB(Λs))zs,t+1...zs,srzℓ. (21) 8 Similarly, zℓzs,1...zs,t = q−(αiℓ,(ωA−1(Λs))zs,1...zs,tzℓ. (22) The statement then follows directly. To complete this part of the picture, we need to consider z < z and z > z ℓ s,1 ℓ s,sr The case z < z easily results in the exponent (β ,(1 ωr)(Λ )), but ℓ s,1 ℓ s − here (β ,Λ ) = 0. The case z > z gives an exponent (β ,(1 ωr)(Λ )), ℓ s ℓ s,sr − ℓ − s and here (β ,ωr(Λ ) = 0. ℓ s Next, we observe the following simple formulas, where ω and ω now are A B determined by (s,t): z z ...z = q 1+(αs,ωB(Λs))z ...z z , (23) s,t s,t+1 s,sr − s,t+1 s,sr s,t and, similarly, zs,tzs,1...zs,t 1 = q1−(αs,ωA−1(Λs))zs,1...zs,t 1zs,t. (24) − − These formulas also hold at the extreme positions of z and z , where s,1 s,sr either ω = 1 or ω = 1. A B So, indeed for any simple root α = π (ℓ) and decomposition ωr = ω ω = ω ℓ B ω s ω , we get, for the corresponding z , A α B ℓ zℓCs(r) = q(α,(ωB−ωA−1)(Λs))Cs(r)zℓ, (25) which, by the previousdefinitionsis equivalent to the statementin the propo- (cid:3) sition. If s / Im(π ), we set C (r) = 1. To any linear combination r s ∈ n Λ i i i X with integer coefficients we may consider the element in the quasi-Laurent algebra (C (r))n1 ...(C (r))nk. (26) s s 1 k If C (r) = 1 we set n = 0. s s Let S = Span α s Im(π ) . r s r { | ∈ } It is obviously invariant under ωr. We view tacitly the elements Λ as re- s stricted to this space. 9 Proposition 5.4. Let n ,...,n be integers and let s ,...,s Im(π ). 1 k 1 k r ∈ k (C (r))n1 ...(C (r))nk (r) (1 + ωr)( n Λ ) = 0. s s q j s 1 k ∈ Z ⇔ j j=1 X Proof. The commutation between this and any z is given by ℓ z (C (r))n1 ...(C (r))nk = ℓ s s 1 k q−(βℓ,(1+ωr))( iniΛi))(Cs (r))n1 ...(Cs (r))nkzℓ. (27) 1 k P (cid:3) This actually determines the center as will be proved below after some preparation. Remark 5.5. Since 1+ωr is an integer matrix, there is an R basis of the null space given by vectors with integer coordinates in the basis of fundamental weights. 5.2. More definitions and computations. The center of the quasi- polynomial algebra. We first make a very useful observation: Lemma 5.6. Let α Φ. Then i ∈ (s + 1)(Λ ) + a (Λ ) = 0. i i ji j j=i X6 (cid:3) Proof. This is actually equivalent to [17, (2.27)] by the definition of a . ki Definition 5.7. Let (s,t) U(r). Set ∈ M↓ = z ...z . (28) s,t s,1 s,t This element has weight p = w (Λ ) + Λ = β + + β . (29) s,t s,t s s s,1 s,t − ··· Proposition 5.8. Let (a,b),(s,t) U(r). Then ∈ z M↓ = qEa,bM↓ z , (30) a,b s,t s,t a,b where the exponent E is given as follows: CASE 1 : (a,b) (s,t): a,b ≤ E = (β ,(1 + w )(Λ )). (31) a,b s,t s − 10