Crystal Bases for Quantum Generalized Kac-Moody Algebras

Transcription

1 Crystal Bases for Quantum Generalized Kac-Moody Algebras Seok-Jin Kang Department of Mathematical Sciences Seoul National University 20 June 2006

2 Monstrous Moonshine 1.Monstrous Moonshine M: Monster simple group = largest sporadic simple group

3 Monstrous Moonshine 1.Monstrous Moonshine M: Monster simple group = largest sporadic simple group M =

4 Monstrous Moonshine 1.Monstrous Moonshine M: Monster simple group = largest sporadic simple group M = = M =

5 Monstrous Moonshine Moonshine Conjecture (Conway-Norton 1979)

6 Monstrous Moonshine Moonshine Conjecture (Conway-Norton 1979) 1 a Z-graded representation V = n= 1 V n of M s.t. dimv n = c(n), where j(τ) 744 = n= 1 c(n)qn, q = e 2πiτ, Imτ > 0

7 Monstrous Moonshine Moonshine Conjecture (Conway-Norton 1979) 1 a Z-graded representation V = n= 1 V n of M s.t. dimv n = c(n), where j(τ) 744 = n= 1 c(n)qn, q = e 2πiτ, Imτ > 0 2 T g (τ) = n= 1 tr(g V n)q n : Mckay-Thompson series = T g (τ) is the Hauptmodul for some discrete genus 0 subgroup Γ g of PSL 2 (R) That is, i) Ĥ/Γ g S 2 ii) Every modular function of weight 0 w.r.t Γ g is a rational function in T g (τ)

8 Monstrous Moonshine Proof of Moonshine Conjecture

9 Monstrous Moonshine Proof of Moonshine Conjecture 1 Borcherds, Frenkel-Lepowsky-Meurman (1985) : introduced vertex(operator) algebras

10 Monstrous Moonshine Proof of Moonshine Conjecture 1 Borcherds, Frenkel-Lepowsky-Meurman (1985) : introduced vertex(operator) algebras 2 Frenkel-Lepowsky-Meurman(1988) : constructed the Moonshine module V

11 Monstrous Moonshine Proof of Moonshine Conjecture 1 Borcherds, Frenkel-Lepowsky-Meurman (1985) : introduced vertex(operator) algebras 2 Frenkel-Lepowsky-Meurman(1988) : constructed the Moonshine module V 3 Borcherds(1992) : completed the proof key ingredient : Monster Lie algebra = an example of generalized Kac-Moody algebra

12 Quantum GKM Algebra 2.Quantum GKM Algebra Definition Borcherds-Cartan matrix Let I be a finite or countably infinite index set. A real matrix A = (a ij ) i,j I is called a Borcherds-Cartan matrix if it satisfies the following conditions: i) a ii = 2 or a ii 0 for all i I, ii) a ij 0 if i j, iii) a ij Z if a ii = 2, iv) a ij = 0 if and only if a ji = 0. Assume that A is even, integral and symmetrizable. That is, i) a ii is even, i I ii) a ij Z, i, j I iii) diagonal matrix D = diag(s i Z >0 i I ) s.t. DA :symmetric

13 Quantum GKM Algebra Borcherds-Cartan datum (A, P, P, Π, Π ) consists of i) A = (a ij ) i,j I, a Borcherds-Cartan matrix, ii) a free abelian group P, the weight lattice, iii) P = Hom(P, Z), the dual weight lattice, iv) Π = {α i P i I }, the set of simple roots, v) Π = {h i i I } P, the set of simple coroots, satisfying the properties: a) h i, α j = a ij for all i, j I, b) i I, there exists Λ i P s.t. h j, Λ i = δ ij j I, c) Π is linearly independent.

14 Quantum GKM Algebra Definition Quantum GKM Algebra Quantum GKM Algebra U q (g) associated with (A, P, P, Π, Π ) = the associative algebra over Q(g) with 1 generated by the elements e i, f i (i I ), q h (h P ) with i) q 0 = 1, q h q h = q h+h for h, h P ii) q h e i q h = q α i (h) e i, q h f i q h = q α i (h) f i, for h P i I K i K 1 i iii) e i f j f j e i = δ ij q i q 1 i for i, j I, where K i = q s i h i

15 Quantum GKM Algebra Definition Quantum GKM Algebra iv) 1 a ij k=0 ( 1)k v) 1 a ij k=0 ( 1)k [ [ 1 a ij k 1 a ij k vi) e i e j e j e i = f i f j f j f i = 0 if a ij = 0 ] ] i i e 1 a ij k i e j e k i = 0 if i I re and i j f 1 a ij k i f j fi k = 0 if i I re and i j

16 Crystal Bases 3.Crystal Bases Definition Category O int the abelian category O int of U g (g)-module M satisfying the following properties: i) M = λ P M λ, where M λ := {u } M ; q h u = q λ(h) u for any h P ii) dim U + q (g)u < for any u M iii) wt(m) := {λ P ; M λ 0 } { λ P ; h i, λ 0, i I im } iv) f i M λ = 0, i I im and λ P s.t. h i, λ = 0 v) e i M λ = 0, i I im and λ P s.t. h i, λ a ii

17 Crystal Bases Proposition (Jeong-Kashiwara-K 2005)

18 Crystal Bases Proposition (Jeong-Kashiwara-K 2005) 1 The abelian category O int is semisimple

19 Crystal Bases Proposition (Jeong-Kashiwara-K 2005) 1 The abelian category O int is semisimple 2 { irreducible objects in O int } = {V (λ) λ P + }, where P + = {λ P h i, λ 0 for all i I }

20 Crystal Bases Proposition (Jeong-Kashiwara-K 2005) 1 The abelian category O int is semisimple 2 { irreducible objects in O int } = {V (λ) λ P + }, where P + = {λ P h i, λ 0 for all i I } 3 V (λ) = U q (g)u λ, where i) u λ has weight λ ii) e i u λ = 0 for all i I iii) f h i,λ +1 i u λ = 0 for any i I re vi) f i u λ = 0 if i I im and h i, λ = 0

21 Crystal Bases Kashiwara operators Let M be a U q (g)-module in O int. The Kashiwara operators ẽ i, f i (i I ) are defined by ẽ i u = k 1 f (k 1) i u k f i u = f (k+1) i u k k 0 where u = k 0 f (k) i u k with u k M µ+nαi s.t. e i u k = 0, u M µ and f (k) i = fi k /[k] i! fi k if i I re if i I im

22 Crystal Bases Definition Crystal Bases Let A 0 = {f /g Q(q) ; f, g Q[q], g(0) 0 } and M O int. A Crystal Basis of M is a pair (L, B), where i) L is a free A 0 -submodule L of M s.t. M Q(q) A 0 L ii) B is a Q-basis of L/gL iii) f i L L and ẽ i L L i I iv) f i B B {0} and ẽ i B B {0} i I v) f i b = b b = ẽ i b, for b, b B and i I

23 Crystal Bases Theorem (Jeong-Kashiwara-K 2005) For λ P +, let L(λ) = A 0 -submodule of V (λ) generated by { f i1 f ir u λ ; r 0, i k I } B(λ) = { f i1 f ir u λ + ql(λ) ; r 0, i k I } \ {0}

24 Crystal Bases Theorem (Jeong-Kashiwara-K 2005) For λ P +, let L(λ) = A 0 -submodule of V (λ) generated by { f i1 f ir u λ ; r 0, i k I } B(λ) = { f i1 f ir u λ + ql(λ) ; r 0, i k I } \ {0} = (L(λ), B(λ)) is a unique crystal basis of V (λ)

25 Crystal Bases Crystal basis for U q (g) Fix i I. For any u U q (g), unique v, w U q (g) s.t. e i u ue i = K iv K 1 i w q i q 1 i We define the endomorphism e i : U q (g) Uq (g) by e i (u) = w. Then every u Uq (g) has a unique i-string decomposition u = k 0 f (k) i u k, where e i u k = 0 for all k 0, and the Kashiwara operators ẽ i, f i (i I ) are defined by ẽ i u = k 1 f (k 1) i u k, f i u = f (k+1) i u k. k 0

26 Crystal Bases Definition A Crystal Basis of U q (g) is a pair (L, B), where i) L is a free A 0 -submodule L of U q (g) s.t. U q (g) Q(q) A 0 L ii) B is a Q-basis of L/gL iii) f i L L and ẽ i L L i I iv) f i B B and ẽ i B B {0} i I v) f i b = b b = ẽ i b, for b, b B and i I

27 Crystal Bases Theorem (Jeong-Kashiwara-K 2005) L( ) = A 0 -submodule of U q (g) generated by { fi1 f ir 1 ; r 0, i k I } B( ) = { f i1 f ir 1 + ql( ) ; r 0, i k I } \ {0}

28 Crystal Bases Theorem (Jeong-Kashiwara-K 2005) L( ) = A 0 -submodule of U q (g) generated by { fi1 f ir 1 ; r 0, i k I } B( ) = { f i1 f ir 1 + ql( ) ; r 0, i k I } \ {0} = (L( ), B( )) is a unique crystal basis of U q (g)

29 Crystal Bases Theorem (Jeong-Kashiwara-K 2005) L( ) = A 0 -submodule of U q (g) generated by { fi1 f ir 1 ; r 0, i k I } B( ) = { f i1 f ir 1 + ql( ) ; r 0, i k I } \ {0} = (L( ), B( )) is a unique crystal basis of U q (g) Problem How to realize B(λ) and B( )?

30 Abstract Crystals 4.Abstract Crystals Definition Abstract Crystals A set B with the maps wt: B P, ẽ i, f i : B B {0} and ε i, ϕ i : B Z { } (i I ) where i) wt(ẽ i b) = wt b + α i if i I and ẽ i b 0, ii) wt( f i b) = wt b α i if i I and f i b 0, iii) i I and b B, ϕ i (b) = ε i (b) + h i, wt b, iv) i I and b, b B, f i b = b b = ẽ i b, Notation wt i (b) = h i, wt b for i I and b B

31 Abstract Crystals Definition Abstract Crystals v) i I and b B s.t. ẽ i b 0, we have ε i (ẽ i b) = ε i (b) 1, ϕ i (ẽ i b) = ϕ i (b) + 1 if i I re, ε i (ẽ i b) = ε i (b) and ϕ i (ẽ i b) = ϕ i (b) + a ii if i I im, vi) i I and b B s.t. f i b 0, we have ε i ( f i b) = ε i (b) + 1 and ϕ i ( f i b) = ϕ i (b) 1 if i I re, ε i ( f i b) = ε i (b) and ϕ i ( f i b) = ϕ i (b) a ii if i I im, vii) i I and b B s.t. ϕ i (b) =, we have ẽ i b = f i b = 0.

32 Abstract Crystals Definition Morphism of Crystals Let B 1 and B 2 be crystals. A morphism of crystals ψ : B 1 B 2 is a map ψ : B 1 B 2 s.t. i) for b B 1 we have wt(ψ(b)) = wt(b) and ε i (ψ(b)) = ε i (b), ϕ i (ψ(b)) = ϕ i (b) for all i I ii) if b B 1 and i I satisfy f i b B 1, then ψ( f i b) = f i ψ(b)

33 Abstract Crystals Definition Morphism of Crystals Let ψ : B 1 B 2 be a morphism of crystals i) ψ is called a strict morphism if ψ(ẽ i b) = ẽ i ψ(b), ψ( f i b) = f i ψ(b) for all i I and b B 1 where ψ(0) = 0 ii) ψ is called an embedding if the underlying map ψ : B 1 B 2 is injective. we say that B 1 is a subcrystal of B 2. If ψ is a strict embedding, we say that B 1 is a full subcrystal of B 2.

34 Abstract Crystals Example 1 B = B(λ), λ P +, b = f i1 f ir u λ + ql(λ) wt(b) def = λ (α i1 + α ir ) max { k 0 ; ẽi k ε i (b) = b 0 } for i I re, 0 for i I im, { max k 0 ; f } i k b 0 for i I re, ϕ i (b) = wt i (b) for i I im,

35 Abstract Crystals Example 2 B = B( ) and b = f i1 f ir 1 + ql( ) wt(b) = (α i1 + + α ir ) max { k 0 ; ẽi k ε i (b) = b 0 } for i I re, 0 for i I im, ϕ i (b) = ε i (b) + wt i (b) (i I ).

36 Abstract Crystals Definition Tensor Product of Crystals Let B 1 B2 = {b 1 b2 b 1 B 1, b 2 B 2 }, and define the maps wt, ε i, ϕ i as follows. wt(b b ) = wt(b) + wt(b ), ε i (b b ) = max(ε i (b), ε i (b ) wt i (b)), ϕ i (b b ) = max(ϕ i (b) + wt i (b ), ϕ i (b )). For i I, we define f i (b b ) = f i b b if ϕ i (b) > ε i (b ), b f i b if ϕ i (b) ε i (b ),

37 Abstract Crystals Definition Tensor Product of Crystals For i I re, we define ẽ i (b b ) = ẽ i b b if ϕ i (b) ε i (b ), b ẽ i b if ϕ i (b) < ε i (b ), and, for i I im, we define ẽ i b b if ϕ i (b) > ε i (b ) a ii, ẽ i (b b ) = 0 if ε i (b ) < ϕ i (b) ε i (b ) a ii, b ẽ i b if ϕ i (b) ε i (b ).

38 Abstract Crystals Proposition

39 Abstract Crystals Proposition 1 B 1 B2 is a crystal.

40 Abstract Crystals Proposition 1 B 1 B2 is a crystal. 2 (B 1 B2 ) B 3 B 1 (B2 B3 ) (b 1 b2 ) b 3 b 1 (b2 b3 )

41 Abstract Crystals Proposition 1 B 1 B2 is a crystal. 2 (B 1 B2 ) B 3 B 1 (B2 B3 ) (b 1 b2 ) b 3 b 1 (b2 b3 ) Remark Hence the category of crystals forms a tensor category.

42 Abstract Crystals Example

43 Abstract Crystals Example 1 For λ P, let T λ = {t λ } and define wt(t λ ) = λ, ẽ i t λ = f i t λ = 0 for all i I, ε i (t λ ) = ϕ i (t λ ) = for all i I. = T λ is a crystal.

44 Abstract Crystals Example 1 For λ P, let T λ = {t λ } and define wt(t λ ) = λ, ẽ i t λ = f i t λ = 0 for all i I, ε i (t λ ) = ϕ i (t λ ) = for all i I. = T λ is a crystal. 2 Let C = {c} with wt(c) = 0 and ε i (c) = ϕ i (c) = 0, f i c = ẽ i c = 0, i I = C is a crystal.

45 Abstract Crystals Example

46 Abstract Crystals Example 3 For each i I, let B i = {b i ( n) ; n 0 } with wt b i ( n) = nα i, ẽ i b i ( n) = b i ( n + 1), fi b i ( n) = b i ( n 1), ẽ j b i ( n) = f j b i ( n) = 0 if j i, ε i (b i ( n)) = n, ϕ i (b i ( n)) = n if i I re, ε i (b i ( n)) = 0, ϕ i (b i ( n)) = wt i (b i ( n)) = na ii if i I im, ε j (b i ( n)) = ϕ j (b i ( n)) = if j i. = B i is a crystal. The crystal B i is called an elementary crystal.

47 Abstract Crystals Example B 1, B 2 : crystals for U q (g)-modules in O int i) a ii = 2 ii) a ii 0

48 Abstract Crystals Example

49 Abstract Crystals Example 1 T λ Tµ T λ+µ

50 Abstract Crystals Example 1 T λ Tµ T λ+µ 2 B(λ) B( ) T λ

51 Abstract Crystals Example 1 T λ Tµ T λ+µ 2 B(λ) B( ) T λ 3 B(λ + µ) B(λ) B(µ)

52 Abstract Crystals Example 1 T λ Tµ T λ+µ 2 B(λ) B( ) T λ 3 B(λ + µ) B(λ) B(µ) 4 B C B

53 Abstract Crystals Example i = (i 1, i 2, ), i k I s.t. every i I appear infinitely many times in i B(i) def = { b ik ( x k ) b i1 ( x 1 ) B ik B i1 ; x k Z 0, and x k = 0 for k 0}.

54 Abstract Crystals Example i = (i 1, i 2, ), i k I s.t. every i I appear infinitely many times in i B(i) def = { b ik ( x k ) b i1 ( x 1 ) B ik B i1 ; x k Z 0, and x k = 0 for k 0}. = B(i) becomes a crystal; b = b ik ( x k ) b i1 ( x 1 ) B(i) wt(b) = k x kα ik for i I re, ε i (b) = max {x k + } l>k h i, α il x l ; 1 k, i = i k ϕ i (b) = max { x k } 1 l<k h i, α il x l ; 1 k, i = i k for i I im, ε i (b) = 0 and ϕ i (b) = wt i (b)

55 Abstract Crystals Example For i I re, we have b ine ( x ne + 1) b i1 ( x 1 ) if ε i (b) > 0, ẽ i b = 0 if ε i (b) 0, f i b = b inf ( x nf 1) b i1 ( x 1 ), where n e (resp. n f ) is the largest (resp. smallest) k 1 such that i k = i and x k + l>k h i, α il x l = ε i (b). Note such an n e exists if ε i (b) > 0.

56 Abstract Crystals Example When i I im, let n f = min{k i k = i and l>k h i, α il x l = 0} Then f i b = b inf ( x nf 1) b i1 ( x 1 ) and b inf ( x nf + 1) b i1 ( x 1 ) if x nf > 0 and k<l n ẽ i b = f h i, α il x l < a ii for any k such that 1 k < n f and i k = i, 0 otherwise.

57 Crystal Embedding Theorem 5.Crystal Embedding Theorem Theorem (Jeong-Kashiwara-K-Shin 2006) For all i I,! strict embedding Ψ i : B( ) B( ) B i 1 1 b i (0)

58 Crystal Embedding Theorem Let i = (i 1, i 2, ) I Observe : B( ) B( ) B i1 B( ) B i2 Bi1

59 Crystal Embedding Theorem Let i = (i 1, i 2, ) I Observe : B( ) B( ) B i1 B( ) B i2 Bi1 Proposition B( ) B(i) = { b ik ( x k ) b i1 ( x 1 )}

60 Crystal Embedding Theorem Let i = (i 1, i 2, ) I Observe : B( ) B( ) B i1 B( ) B i2 Bi1 Proposition B( ) B(i) = { b ik ( x k ) b i1 ( x 1 )} Corollary B( ) connected component of B(i) containing b ik (0) b i1 (0)

61 Crystal Embedding Theorem Theorem (Jeong-Kashiwara-K-Shin 2006)

62 Crystal Embedding Theorem Theorem (Jeong-Kashiwara-K-Shin 2006) 1 Let B be a crystal s.t. i) wt(b) Q +, ii) b 0 B s.t. wt(b 0 ) = 0, iii) for any b B s.t. b b 0, some i I s.t. ẽ i b 0, iv) for all i, a strict embedding Ψ i : B B B i. Then B B( )

63 Crystal Embedding Theorem Theorem (Jeong-Kashiwara-K-Shin 2006) 1 Let B be a crystal s.t. i) wt(b) Q +, ii) b 0 B s.t. wt(b 0 ) = 0, iii) for any b B s.t. b b 0, some i I s.t. ẽ i b 0, iv) for all i, a strict embedding Ψ i : B B B i. Then B B( ) 2 Let λ P +. Then B(λ) connected component of B( ) T λ C containing 1 t λ c

64 Crystal Embedding Theorem Example Let I ( = {1, 2} and ) i = (1, 2, 1, 2,... ) and 2 a A = for some a, b Z >0 and c 2Z 0. b c B def = { b 2 ( x 2k ) b 1 ( x 2k 1 ) b 2 ( x 2 ) b 1 ( x 1 ) i) ax 2k x 2k+1 0 for all k 1, ii) k 2 with x 2k > 0, we have x 2k 1 > 0 and ax 2k x 2k+1 > 0.}

65 Crystal Embedding Theorem Example Let I ( = {1, 2} and ) i = (1, 2, 1, 2,... ) and 2 a A = for some a, b Z >0 and c 2Z 0. b c B def = { b 2 ( x 2k ) b 1 ( x 2k 1 ) b 2 ( x 2 ) b 1 ( x 1 ) i) ax 2k x 2k+1 0 for all k 1, ii) k 2 with x 2k > 0, we have x 2k 1 > 0 and ax 2k x 2k+1 > 0.} = B B( )

66 Crystal Embedding Theorem Example B λ def = { b 2 ( x 2k ) b 1 ( x 2k 1 ) b 2 ( x 2 ) b 1 ( x 1 ) t λ c i) ax 2k x 2k+1 0 for all k 1, ii) k 2 with x 2k > 0, we have x 2k 1 > 0 and ax 2k x 2k+1 > 0, iii) 0 x 1 h 1, λ, iv) if x 2 > 0 and h 2, λ = 0, then x 1 > 0. }

67 Crystal Embedding Theorem Example B λ def = { b 2 ( x 2k ) b 1 ( x 2k 1 ) b 2 ( x 2 ) b 1 ( x 1 ) t λ c i) ax 2k x 2k+1 0 for all k 1, ii) k 2 with x 2k > 0, we have x 2k 1 > 0 and ax 2k x 2k+1 > 0, iii) 0 x 1 h 1, λ, iv) if x 2 > 0 and h 2, λ = 0, then x 1 > 0. } = B λ B(λ)

68 Crystal Embedding Theorem Example Quantum Monster Algebra I = {(i, t) i = 1, 1, 2, 1 t c(i)} A = ( (i + j)) (i,t),(j,s) I =

69 Crystal Embedding Theorem Example i = (i(k)) k=1 = (( 1, 1), (1, 1),..., (1, c(1)); ( 1, 1), (1, 1),..., (1, c(1)), (2, 1),..., (2, c(2)); ( 1, 1), (1, 1),..., (1, c(1)), (2, 1),..., (2, c(2)), (3, 1),..., (3, c(3)); ( 1, 1),... ) Note i) every (i, t) I appears infinitely times in i ii) ( 1, 1) appears at the b(n)-th position for n 0, where b(n) = nc(1) + (n 1)c(2) + + c(n) + n + 1.

70 Crystal Embedding Theorem Example For k Z >0, k ( ) = the largest integer l < k s.t. i(l) = i(k). B def = { b i(k) ( x k ) b i(1) ( x 1 ) B(i) i) x b(1) = 0, ii) n 1, b(n)<l<b(n+1) h ( 1,1), α i(l) x l x b(n+1), iii) if i(k) ( 1, 1), x k > 0 and k ( ) > 0, then h i(k), α i(l) x l < 0. k ( ) <l<k If x l = 0 for all k ( ) < l < k s.t. i(l) ( 1, 1), then h ( 1,1), α i(l) x l > x b(n+1) s.t. k ( ) < b(n) < k.} b(n)<l<b(n+1)

71 Crystal Embedding Theorem Example For k Z >0, k ( ) = the largest integer l < k s.t. i(l) = i(k). B def = { b i(k) ( x k ) b i(1) ( x 1 ) B(i) i) x b(1) = 0, ii) n 1, b(n)<l<b(n+1) h ( 1,1), α i(l) x l x b(n+1), iii) if i(k) ( 1, 1), x k > 0 and k ( ) > 0, then h i(k), α i(l) x l < 0. k ( ) <l<k If x l = 0 for all k ( ) < l < k s.t. i(l) ( 1, 1), then h ( 1,1), α i(l) x l > x b(n+1) s.t. k ( ) < b(n) < k.} b(n)<l<b(n+1) = B B( )

72 Crystal Embedding Theorem Example B λ def = { b i(k) ( x k ) b i(1) ( x 1 ) t λ c (i) (iii) in the previous example, iv) 0 x 1 h ( 1,1), λ, v) if i(k) ( 1, 1), h i(k), λ = 0, x k > 0 and k ( ) = 0, then l s.t. 1 l < k, h i(k), α i(l) < 0 and x l > 0. }

73 Crystal Embedding Theorem Example B λ def = { b i(k) ( x k ) b i(1) ( x 1 ) t λ c (i) (iii) in the previous example, iv) 0 x 1 h ( 1,1), λ, v) if i(k) ( 1, 1), h i(k), λ = 0, x k > 0 and k ( ) = 0, then l s.t. 1 l < k, h i(k), α i(l) < 0 and x l > 0. } = B λ B(λ)

74 Crystal Embedding Theorem THANK YOU