Quantum groups and quantized q-difference birational Weyl group actions

Transcription

1 q Weyl Quantum groups and quantzed q-dfference bratonal Weyl group actons ( ) Gen KUROKI (Tohoku Unversty, Japan) 24 September (24 September 2010, Verson 1.7)

2 Quantum bratonal Weyl group acton GCM [a j ] d a j = d j a j, h, α = a j, α = d 1 α Ore doman q-serre: k ( 1)k[ 1 a j e k ]q 1 a j k e j e k = 0 ( j) A q = e, a I Parameters a = q α : a e j = e j a Feld of fractons K q = Q(A q ) = { ab 1 a, b A q, b 0 } e n e je n s an n-ndependent ratonal functon of q n q n a = q α (q = q d ) Weyl group acton on K q s (e j ) = e α e j e α s (a j ) = a a j a j ( s (α j ) = α j a j α ) Both q-dfference analogue and canoncal quantzaton of the bratonal Weyl group acton gven by Noum-Yamada math/

3 Explct formulae for s (e j ) = e α e j e α [x, y] q := xy qyx, q(k) := q 2k+a j ( j) Defne (ad q e ) k (e j ) for k = 0, 1, 2,... by (ad q e ) k (e j ) = [e, [, [e, [e, e j ] q(0) ] q(1) ] q(k 2) ] q(k 1). Then (ad q e ) k (e j ) = k=0 k [ ] ( 1) ν q ν(k 1+a j) k e k ν e j e ν. ν q ν=0 q-serre relatons (ad q e ) k (e j ) = 0 f j and k > a j. e ( = j), a s (e j ) = j [ ] q (k+a j)(α k) α (ad q e ) k (e j )e k ( j). k q 2

4 Quantum geometrc crystal structure on K q GCM [a j ] d a j = d j a j, h, α = a j, α = d 1 α Ore doman q-serre: k ( 1)k[ 1 a j e k ]q 1 a j k e j e k = 0 ( j) A q = e, a I Parameters a = q α : a e j = e j a Feld of fractons K q = Q(A q ) = { ab 1 a, b A q, b 0 } e n e je n s an n-ndependent ratonal functon of q n q n t Quantum geometrc crystal str. on K q e t (e j) = e n e je n e t (a j) = t a ja j q n t The Verma relatons for e t = Weyl group acton s = e a 3

5 Generalzaton q-serre relatons Verma relatons Assumpton k ( 1)k[ 1 a j e k ]q 1 a j k e j e k e k ek+l j = 0 ( j) e l = el j ek+l e k j f a ja j = 1, etc. e n xe n s a ratonal fucnton of q n. q n t Quantum e t (x) = en xe n q n t geomtrc crystal e t (a j) = t a ja j (acton on parameters) Weyl group acton s = e α (a = q α ) Actons of the lattce parts of affne Weyl groups q-dfference quantum Panlevé systems 4

6 Quantum Schubert cell Reduced expresson of w W : w = s 1 s N, = ( 1,..., N ). A = x ν, a, K = Q(A q ) = { ab 1 a, b A, b 0 } Defnng relatons: x ν x µ = q µν x µ x ν (µ < ν), a center. q µν := q b µν, b j := d a j. (x 1,..., x N ) e q1 (x 1 F 1 ) e qn (x N F N ) s quantzaton of a postve structure of a Schubert cell. e := { q-serre relatons for ν = x e, ν e n x νe n s a ratonal functon of q n. Quantum geometrc crystal structure on K. 5

7 Explct formulae for e t (x ν) X := µ =, µ<ν x µ, Y := µ =, µ>ν x µ.. Then e = X + δ ν x ν + Y. If ν =, then e t (x ν ) = t 2 x ν If ν, then a ν 0 and 1 + q 2(x ν + q 2Y )X q 2Y (X + x ν) q 2t 2 (x ν + q 2Y )X q 2t 2 Y (X + x ν ) 1, a ν e t (x ν ) = t a νx ν k=1 1 + q 2(k 1) t 2 Y X q 2(k 1) Y X. 1 6

8 Remarks All the expressons for e t (x ν) are subtracton-free. K depends only on w W up to canoncal postve somorphsms. generalzaton Varous quantum postve geometrc crystals classcal lmt Varous postve geometrc crystals ultra-dscretzaton Varous crystals φ and ε. e t (e ) = e, e t (qα ) = t 2 q α. ε := const.q α e, φ := const.q α e. (ε = q 2α φ ) Then e t (φ ) = t 2 φ, e t (ε ) = t 2 ε. quantum t 2, q 2α classcal c, α 7

9 Fles Old verson of ths fle kurok/latex/ Nagoya.pdf Quantum M-matrx for A case 1.6 of kurok/latex/ Osaka.pdf Quantzaton of the bratonal acton of W (A (1) m 1 ) W (A(1) n 1 ) gven by Kajwara-Noum-Yamada nln/ for mutually prme m, n kurok/latex/ WxW.pdf Theory of quantum geometrc crystals n preparaton For more detals see the followng pages. 8

10 Symmetrzable GCM and root datum Let A = [a j ],j I be a symmetrzable GCM: a = 2, a j 0 ( j), a j = 0 a j = 0; d a j = d j a j, d Z >0. Let (, : Q P Z, {h } I Q, {α } I P ) be a root datum: fntely generated free Z-modules Q, P and perfect blnear parng, : Q P Z. {h } I Q s called a set of smple coroots. Q s called a coroot lattce. {α } I P s called a set of smple roots. P s called a weght lattce. h, α j = a j. 9

11 The group algebra F[q P ] of the weght lattce P Base feld F := Q(q). F[q P ] := λ P Fqλ, q λ q µ = q λ+µ (λ, µ P ). [x] q := qx q x q q, [k] q! := [1] 1 q [2] q [k] q (k Z 0 ). [ ] x := [x] q[x 1] q [x k + 1] q (q-bnomal coeffcents). k q [k] q! q := q d, α := d 1 α (= a smple coroot). [ ] Remark. q ±d α = q ±α α F[q P ] = F[q P ]. k q 10

12 Quantum algebra A q = q λ, e λ P, I Assumptons. (1) A q,0 s an assocatve algebra over F generated by e 0 ( I). 1 a j [ ] 1 (2) q-serre relatons: ( 1) k aj e 1 aj k e j e k = 0 ( j). k k=0 q (3) A q := F[q P ] F A q,0 s an Ore doman. Identfcaton. q λ = q λ 1 A q, e = 1 e A q. Remark. q λ e = e q λ n A q. Q(A q ) := (the quotent skew feld of A q ) = { as 1 a, s A q, s 0 }. Example. The root datum s of fnte or affne type = A q,0 = U q (n + ) satsfes all the assumptons above. 11

13 Iterated adjont by e Assume j. [x, y] q := xy qyx, q(k) := q 2k+a j Defne (ad q e ) k (e j ) for k = 0, 1, 2,... by (ad q e ) 0 (e j ) = e j, (ad q e ) 1 (e j ) = [e, e j ] q(0), (ad q e ) 2 (e j ) = [e, [e, e j ] q(0) ] q(1),..., (ad q e ) k (e j ) = [e, [, [e, [e, e j ] q(0) ] q(1) ] q(k 2) ] q(k 1). Then (ad q e ) k (e j ) = k [ ] ( 1) ν q ν(k 1+a j) k e k ν e j e ν. ν q ν=0 q-serre relatons (ad q e ) k (e j ) = 0 f j and k > a j. 12

14 Conjugaton by powers of e For n = 0, 1, 2,..., e n e j e n = e a j k=0 q (k+a j)(n k) [ ] n (ad q e ) k (e j )e k k q ( = j), ( j). Defne e α e α e j e α = e j e α e a j k=0 Q(A q ) by [ q (k+a j)(α k) α k ] (ad q e ) k (e j )e k q ( = j), ( j). x e n xe n s an algebra automorphsm of Q(A q ) = e j e α e j e α s unquely extended to an alg. autom. of Q(A q ). 13

15 Qauntzed bratonal Weyl group acton Theorem 1. The algebra automorphm s of Q(A q ) can be defned by s (e j ) = e α e j e α ( I), s (q λ ) = q λ h,λ α (λ P ). Then {s } I satsfes the defnng relatons of the Weyl group W : s s j = s j s (a j a j = 0), s s j s = s j s s j (a j a j = 1), s s j s s j = s j s s j s (a j a j = 2), s s j s s j s s j = s j s s j s s j s (a j a j = 3), s 2 = 1. Thus we obtan the acton of the Weyl group W on Q(A q ). Remark. Ths s a q-dfference verson of quantzaton of the bratonal Weyl group acton gven by Noum-Yamada math/

16 The Verma relatons of {e } I (Lusztg s book (1993)) q-serre relatons of {e } I mples (a j, a j ) = (0, 0) = e k el j = el j ek, (a j, a j ) = ( 1, 1) = e k ek+l j e l = el j ek+l e k j, (a j, a j ) = ( 1, 2) = e k e2k+l j (a j, a j ) = ( 1, 3) = e k e3k+l j e 2k+l e 3k+2l j e k+l e k+l e l j = el j ek+l These relatons are called the Verma relatons. e l j = el j ek+l e 2k+l j e k, e 3k+2l j e 2k+l e 3k+l j e k. The Verma relatons = {s } I satsfes the defnng relaton of the Weyl group. For detals see arxv:

17 Quantum geometrc crystal structure on A q The algebra homomorphsm e t : Q(A q) Q(A q )(t) s defned by e t (e j ) = e n e j e n (j I), q n t e t (q λ ) = t h,λ q λ (λ P ). Then e 1 = d Q(A q ), e t 1 e t 2 = e t 1t 2 : Q(A q ) Q(A q )(t 1, t 2 ). Furthermore {e t } I satsfes the Verma relatons: (a j, a j ) = (0, 0) = e t 1 e t 2 j = e t 2 j e t 1, (a j, a j ) = ( 1, 1) = e t 1 e t 1t 2 j e t 2 = e t 2 j e t 1t 2 e t 1 (a j, a j ) = ( 1, 2) = e t 1 e t 1t 2 j e t 1t 2 2 e t 2 j = e t 2 j e t 1t 2 2 e t 1t 2 j e t 1, (a j, a j ) = ( 1, 3) = e t 1 e t 1t 2 j e t2 1 t3 2 e t 1t 2 2 j e t 1t 3 2 e t 2 j = e t 2 j e t 1t 3 2 e t 1t 2 2 j e t2 1 t3 2 e t 1t 2 j e t 1. j, 16

18 Defnton of quantum geometrc crystal Defnton. (K, {e t } I) s called a quantum geometrc crystal f t satsfes the followng condtons: K s a skew feld. e t s an algebra homomorphsm K K(t). e t s regular at t = 1. e1 = d K, e t 1 e t 2 = e t 1t 2. {e t } I satsfes the Verma relatons. F[q P ] s a subalgebra of the center of K. e t s regular at t = qλ for any λ P. e t (qλ ) = t h,λ q λ for λ P. Remark. For the classcal case, see Berensten-Kazhdan math/ Proposton 2. (Q(A q ), {e t } I) s a quantum geometrc crystal. 17

19 Weyl group acton on a quantum geometrc crystal Proposton 3. Let (K, {e t } I) be a quantum geometrc crystal. Put a = q α = q α and s (x) = e a (x) for I, x K. Then s s an algebra automorphsm of K wth s (q λ ) = q λ h,λ α = q s (λ) for λ P. Moreover {s } I satsfes the defnng relatons of the Weyl group W and hence generates the acton of W on K. Propostons 2 and 3 = Theorem 1. 18

20 Quantum Schubert cell b j := d a j. := ( 1, 2,..., N ) I N. Then b j = b j and q b j = q a j. A,0 := the assocatve algebra over F = Q(q) generated by {x ν } N ν=1 wth defnng relatons: x ν x µ = q b µνx µ x ν (µ < ν). A := F[q P ] F A,0 = q λ, x ν λ P, 1 ν N. (Identfcaton. q λ 1 = q λ, 1 x ν = x ν ) Then A s an Ore doman. If w = s 1 s 2 s N s a reduced expresson of w W, then Q(A 0, ) depends only on w (Berensten q-alg/ ) and s the ratonal functon feld of a quantum Schubert cell. 19

21 Quantum geometrc crystal structure on A e := ν = x ν. Then {e } I satsfes the q-serre relatons. Assume { ν ν = 1...., N } = I. Then e 0 for all I. ( nessental assumton) Theorem 4. (quant. geom. crys. str. on A ) The algebra hom. e t : Q(A ) Q(A )(t) can be defned by e t (x ν ) = e n x ν e n q n, e t (q λ ) = t h,λ q λ. Then (Q(A ), {e } I ) s a quantum geometrc crystal. Remark. An nducton on n = 0, 1, 2,... proves that e n x νe n s an n-ndependent ratonal functon of q n. Explct formulae Next page 20

22 Explct formulae for e t (x ν) and ther postvty X := µ =, µ<ν x µ, Y := µ =, µ>ν x µ.. Then e = X + δ ν x ν + Y. If ν =, then e t (x ν ) = t 2 x ν If ν, then a ν 0 and a ν e t (x ν ) = t a νx ν 1 + q 2(x ν + q 2Y )X q 2Y (X + x ν) q 2t 2 (x ν + q 2Y )X q 2t 2 Y (X + x ν ) 1, k=1 1 + q 2(k 1) t 2 Y X q 2(k 1) Y X. 1 Postvty. All the formulae for e t (x ν) are subtracton-free. 21

23 Commentares φ and ε for A q and A cases. e t (e ) = e, e t (qα ) = t 2 q α. ε := const.q α e, φ := const.q α e. (ε = q 2α φ ) Then e t (φ ) = t 2 φ, e t (ε ) = t 2 ε. quantum t 2, q 2α classcal c, α Classcal lmt of A q,0 : Posson subvarety X U. (U = exp n, the lower maxmal unpotent subgroup pf G) Classcal lmt of A,0 : X = {(x 1,..., x N )} {y 1 (x 1 ) y N (x N )} U. (y (x) = exp(xf ), F = (the lower Chevalley generator of g)) 22

24 Fles Ths fle kurok/latex/ Nagoya.pdf Quantum M-matrx for A case 1.6 of kurok/latex/ Osaka.pdf Quantzaton of the bratonal acton of W (A (1) m 1 ) W (A(1) n 1 ) gven by Kajwara-Noum-Yamada nln/ for mutually prme m, n kurok/latex/ WxW.pdf Theory of quantum geometrc crystals n preparaton 23