Lecture Notes Introduction to Cluster Algebra

Transcription

1 Lecture Notes Introducton to Cluster Algebra Ivan C.H. Ip Updated: Ma 7, Defnton and Examples of Cluster algebra 3.1 Quvers We frst revst the noton of a quver. Defnton 3.1. A quver s a fnte orented graph. We allow multple arrows, but no 1-ccles and 2-ccles. 1-ccle 2-ccle We wll let some vertces be frozen, whle others be mutable. We assume there are no arrows between frozen vertces. Defnton 3.2. Let k be mutable vertex n a quver Q. A quver mutaton µ k transforms Q nto a new quver Q := µ k (Q) b the three steps: (1) For each par of drected edges k j, ntroduce a new edge j (unless both, j are frozen) (2) Reverse drecton of all edges ncdent to k (3) Remove all orented 2-ccles. Example 3.3. Consder the example: (Let u, v be frozen) u x z v u µ z Center for the Promoton of Interdscplnar Educaton and Research/ Department of Mathematcs, Graduate School of Scence, Koto Unverst, Japan Emal: van.p@math.koto-u.ac.jp x z v 1

2 Proposton 3.4. Mutaton s an nvoluton µ k (µ k (Q)) = Q. If k and l are two mutable vertces wth no arrows between them, then the mutatons at k and l commute µ l (µ k (Q)) = µ k (µ l (Q)). Defnton 3.5. Two quvers Q and Q are called mutaton equvalent f Q can be transformed nto Q b a sequence of mutatons. The mutaton equvalence class [Q] s the set of all quvers whch are mutaton equvalent to Q. A quver Q s sad to have fnte mutaton tpe f [Q] s fnte. Example 3.6. other. All orentatons of a tree are mutaton equvalent to each The mutaton equvalence class [Q] of the Markov quver Q conssts of a sngle element. Defnton 3.7. Let Q be a quver wth m vertces, and n of them mutable. The extended exchange matrx of Q s the m n matrx B(Q) = (b j ) defned b r f there are r arrows from to j n Q b j = r f there are r arrows from j to n Q otherwse The exchange matrx B(Q) s the n n skew-smmetrc submatrx of B(Q) occupng the frst n rows: B(Q) = (b j ),j [1,n] Lemma 3.8. The extended exchange matrx B = (b j ) of the mutated quver µ k(q) s gven b b j k {, j} b b j = j + b k b kj b k > 0 and b kj > 0 (3.1) b j b k b kj b k < 0 and b kj < 0 b j otherwse or more compactl: or or b j = { bj k {, j} b j + [b k ] + b kj + b k [ b kj ] + otherwse. { b bj k {, j} j = b j + b k b kj +b k b kj 2 otherwse. { b bj k {, j} j = b j + sgn(b k )[b k b kj ] + otherwse. 2

3 Defnton 3.9. An n n matrx B s skew-smmetrzable, f there exsts ntegers d 1,..., d n such that d b j = d j b j. An m n nteger matrx wth top n n submatrx skew-smmetrzable s called extended skew-smmetrzable matrx. Defnton The dagram of a skew-smmetrzable n n matrx B s the weghted drected graph Γ(B) such that there s a drected edge from to j ff b j > 0, and ths edge s assgned the weght b j b j. 3.2 Cluster algebra of geometrc tpe Now we can defne algebracall the noton of cluster algebra. We frst defne cluster algebra of geometrc tpe (wthout coeffcents). Let m n be two postve ntegers. Let the ambent feld F be the feld of ratonal functons over C n m ndependent varables. Defnton A labeled seed of geometrc tpe n F s a par ( x, B) where x = (x 1,..., x m ) s an m-tuple of elements of F formng a free generatng set,.e. x 1,..., x m are algebracall ndependent, and F = C(x 1,..., x m ) B s an m n extended skew-smmetrzable nteger matrx. We have the termnolog: x s the (labeled) extended cluster of the labeled seed ( x, B) x = (x 1,..., x n ) s the (labeled) cluster of ths seed; x 1,..., x n are ts cluster varables; The remanng x n+1,..., x m of x are the frozen varables; B s called the extended exchange matrx of the seed The top n n submatrx B of B s the exchange matrx Labeled means we also care about the order (ndex) of the seeds. Defnton A seed mutaton µ k n drecton k transform the labeled seed ( x, B) nto a new labeled seed ( x, B ) := µ k ( x, B) where B s defned n (3.1), and x = (x 1,..., x m) s gven b x j = x j, j k, and x k F defned b the exchange relaton x k x k = + or equvalentl x k x k = b k >0 m =1 x b k x [b k] b k <0 m =1 x b k x [ b k] +

4 We sa two skew-smmetrzable matrces B and B are mutaton equvalent f one can get from B to B b a sequence of mutatons, possbl followed b smultaneous renumberng of rows and columns. Defnton Let T n denote the n-regular tree. A seed pattern s defned b assgnng a labeled seed ( x(t), B(t)) to ever vertex t T n, so that the seeds assgned k to the end ponts of an edge t t are obtaned from each other b the seed mutaton n drecton k. To a seed pattern, we can assocate an exchange graph whch s n-regular, whose vertex are seeds and edges are mutatons (the exchange graph s T n onl when no seeds repeat.) Fgure 1: Exchange graph Defnton Let ( x(t), B(t)) be a seed pattern, and let X := t T n x(t) be the set of all cluster varables appearng n ts seeds. Let the ground rng R = C[x n+1,..., x m ] be the polnomal rng generated b the frozen varables. The cluster algebra of geometrc tpe A of rank n s the R-subalgebra of F generated b all cluster varables A = R[X ] Usuall, we pck an ntal seed ( x 0, B 0 ), and buld a seed pattern out of t. Then the correspondng cluster algebra A( x 0, B 0 ) s generated over R b all cluster varables appearng n the seeds mutaton equvalent to ( x 0, B 0 ). Hence f we let S denote the set of all seeds, then we can wrte A = A(S). 3.3 Examples Rank 1 Case. T 1 s ver smple 4

5 1 We have two seeds and two clusters (x 1 ) and (x 1). B0 can be an m 1 matrx wth top entr 0. A F = C(x 1, x 2,..., x m ) s generated b x 1, x 1, x 2,..., x m subject to relaton of the form x 1 x 1 = M 1 + M 2 where M are monomals n the frozen varables x 2,..., x m whch do not share a common factor. ( ) a b Example C[SL 2 ] = C[a, b, c, d] s a cluster algebra, wth ad = c d 1 + bc. We have two extended clusters {a, b, c} and {b, c, d} and clusters {a} and {d}. Example C[SL 3 /N]: Recall we have the Plücker relaton 2 13 = Then C[SL 3 /N] has frozen varables { 1, 12, 23, 3 } and clusters { 2 }, { 13 }. Rank 2 Case. An 2 2 skew-smmetrzable matrx look lke ths: ( ) 0 b ± c 0 for some postve ntegers b, c, or both zero. µ 1 or µ 2 smpl changes ts sgn. Example b = c = 0. Ths reduce to the rank 1 case. Example( Let ) A = A(b, c) denote cluster algebra of rank 2 wth exchange 0 b matrx ± and no frozen varables. Then we have c 0 { x c x k+1 x k 1 = k + 1 k s even x b k + 1 k s odd Ths s the same as the Conwa-Coxeter freze pattern for (d 1, d 2 ) = (c, b). The exchange graph s fnte onl when (d 1, d 2 ) = (1, 1), (1, 2), (1, 3), such that the graph s pentagon, hexagon and octagon respectvel. In all other cases, the exchange graph s T 2, whch s an nfnte lne. Example Let us ntroduce frozen varable. ntal seed {z 1, z 2, } and exchange matrx B 0 = Consder a seed pattern wth we get p q z 1, z 2, z 3 = z 2 + p z 1, z 4 = p+q z 1 + z 2 + p z 1 z 2, z 5 = q z z 2, z 6 = z 1, z 7 = z 2 5

6 Agan there are 5 dstnct cluster varables. The cluster algebra s then defned to be A = R[X ] = C[ ±1 ] [z 1, z 2, z 2 + p, p+q z 1 + z 2 + p ], q z z 1 z 1 z 2 z 2 Example Grassmannan Comparng the propertes, we see that the coordnate rng of the Grassmannan C[Gr(2, n + 3)] s a cluster algebra, where cluster = {2 2 mnors} = trangulaton cluster varables =,j = dagonals frozen varables =,+1 = sdes mutaton = Plücker relaton = flppng of dagonals Smlarl, C[SL n /N] s a cluster algebra. Example Markov trples. Trples of ntegers satsfng Consder t as equaton n x 1 : x x x 2 3 = 3x 1 x 2 x 3. 2 (3x 2 x 3 ) + (x x 2 3) = 0. Then t has two roots: = x 1 and x 1 = x2 2 +x2 3 x 1. Startng wth x 1 = x 2 = x 3 = 1, replacng wth another root: Veta jumpng. B = ± Conjecture 3.22 (Unqueness). Maxmal elements of Markov trples are all dstnct. 6

7 Example Somos-4 sequence. Fgure 2: Markov trples x n x n+4 = x n+1 x n+3 + x 2 n+2 Mutate at 1 rotate the graph b 90 degree Somos-5 sequence. x n x n+5 = x n+1 x n+4 + x n+2 x n+3 Mutate at 1 rotate the graph b 72 degree. 7

8 In both Somos-r sequence, x n wll be Laurent polnomals n the ntal varables x 1,..., x r. In partcular the wll be ntegers f x 1 =... = x r = 1. = Laurent phenomenon! 3.4 Semfelds and coeffcents The mutaton does not reall use the frozen varables. So let us treat them as coeffcents, whch leads to a more general noton of cluster algebra wth semfelds as coeffcents. Let us denote m j := x bj, j = 1,..., n. =n+1 Then 1,..., n encodes the same nformaton as the lower (n m) n submatrx of B. Hence a labeled seed can equvalentl be presented as trples (x,, B) where x = (x 1,..., x n ), = ( 1,..., n ). Now the mutaton of x k becomes: x k x k = m =n+1 = k k 1 x [b k] + n =1 n =1 x [b k] + + x [b k] k 1 m =n+1 n =1 x [ b k] + x [ b k] + n =1 x [ b k] + where the semfeld addton s defned b x a x b := x mn(a,b) n partcular, 1 x b := x [ b]+ The mutaton of the frozen varables x n+1,..., x m also nduces the mutaton of the coeffcent -varables: ( 1,..., n) := µ k ( 1,..., n ) 8

9 j := 1 k = k j ( k 1) b kj f j k and b kj 0 j ( 1 k 1) b kj f j k and b kj 0 Ths s called tropcal Y -seed mutaton rule Ths s general, we can use an semfeld! Defnton A semfeld (P,, ) s an abelan group (P, ) (wrtten multplcatvel) together wth a bnar operator such that : P P P (p, q) p q s commutatve, assocatve, and dstrbutve: Note: ma not be nvertble! p (q r) = p q p r Example Examples of semfeld (P,, ): (R >0,, +) (R, +, mn) (Q sf (u 1,..., u m ),, +) subtracton-free ratonal functons (T rop( 1,..., m ),, ) Laurent monomals wth usual multplcaton and x a x b := x mn(a,b) P s called the coeffcent group of our cluster algbera. Proposton If (P,, ) s a semfeld, then (P, ) s torson free (f there exsts p, m such that p m = 1, then p = 1). Let ZP be the group rng of (P, ). Then t s a doman (p q = 0 = p = 0 or q = 0). Can defne feld of fractons QP of ZP Proof. (1) If p m = 1, then note that 1 p... p m 1 P but 0 / P, we can wrte 1 p... pm 1 p = p 1 p... p m 1 = p p2... p m 1 p... p m 1 = 1 (2) Let p, q ZP wth p q = 0. Then p and q are contaned n ZH for some fntel generated subgroup H of P. Snce H P s abelan, H Z n, hence ZH Z(x 1,..., x n ) conssts of all Laurent polnomals n x. In partcular ZH s an ntegral doman, and hence p = 0 or q = 0. 9

10 Then we can set our ambent feld to be F := QP(u 1,..., u n ). rewrte prevous defntons and results: Now we can Defnton A labeled seed n F s (x,, B) wth x = {x 1,..., x n } free generatng set of F = { 1,..., n } P an elements B = n n skew-smmetrzable Z-matrx We have mutatons for all x, and B, together wth the exchange patterns. A cluster algebra wth coeffcents P s then [ ] A(x,, B) := ZP x(t) t T n Example For rank n = 2 we have n the most general case: t ( B t ) t x t x 1 x 2 ( ) x ( 2 1) 2 x x 2( 2 1) ( ) x x ( 2 1) 2 ( )x 1x 2 x 2( 2 1) ( ) x x 2 1+x ( )x 1x 2 x 1( 1 1) ( ) x 1+x 2 2 x 1( 1 1) ( ) x 2 x 1 Remark The prevous cluster algebra of geometrc tpe = cluster algebra wth coeffcents P = T rop(x n+1,..., x m ). We onl need the n n matrx B There are no frozen varables Mutaton of onl nvolve two varables But we need to mutate all varables, there are usuall more varables than cluster varables Y -pattern do not n general exhbt Laurent phenomenon 10

11 Defnton Two cluster algebra A(S) and A(S ) are called strongl somorphc f there exsts a ZP-algebra somorphsm F F sendng some seed n S nto a seed n S, thus nducng a bjecton S S of seeds and an algebra somorphsm A(S) A(S ) An cluster algebra A s unquel determned b an sngle seed (x,, B). Hence A s determned b B and up to strong somoprhsm, and we can wrte A = A(B, ). 11