NOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules

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1 NOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules EVAN WILSON Quantum groups Consder the Le algebra sl(n), whch s the Le algebra over C of n n trace matrces together wth the commutator bracket: [A, B] = AB BA It s easy to verfy that ths s generated by the elements e := E,+, f := E +,, h := E, E +,+, =,, n where E,j s the matrx wth n the th row and jth column and everywhere else and satsfes the followng relatons: () [h, h j ] =,, j =,,, n, () [e, f j ] = δ j h,, j =,,, n, (3) [h, e j ] = a j e j,, j =,,, n (4) [h, f j ] = a j f j,, j =,,, n (5) (ad e ) a j (e j ) =, (ad f ) a j (f j ) =, for j, where A = (a j ) n,j= s the matrx: and ad x (y) := [x, y] s the adjont operator In fact, sl(n) s somorphc to the Le algebra generated by e, f, h, =,,, n subject to relatons ()-(5) (see for example [H]) Ths s our motvatng example for the noton of Kac-Moody algebras, whch we defne next A standard reference s [K] Every Kac-Moody algebra s determned by a generalzed Cartan matrx (GCM), whch s a matrx A = (a j ),j I, a j Z, where I s a fnte ndex set, satsfyng the followng condtons: () a =, () a j f j, (3) a j < f and only f a j < Date: June,

2 A GCM s called symmetrzable f there exsts a dagonal matrx D = dag(s ) I such that s Z >, I and DA s a symmetrc matrx We wll consder only symmetrzable GCMs It may happen that A s sngular Fx a subset S of I of cardnalty I rank(a) Defnton Let A = (a j ),j I be a (symmetrzable) GCM The Kac-Moody algebra g(a) s the Le algebra over C generated by the elements e, f, I, and d j, j S satsfyng the followng relatons: () [h, h j ] =,, j I, [h, d j ] =, I, j S, [d, d j ] =,, j S () [e, f j ] = δ j h,, j I (3) [h, e j ] = a j e j,, j I, [d, e j ] = δ j e j, I, j S (4) [h, f j ] = a j f j,, j I, [d, f j ] = δ j f j, I, j S (5) (ad e ) a j (e j ) =, (ad f ) a j (f j ) =, for j I, Where there s no confuson about A, we wrte g for g(a) We remark that the Kac-Moody algebra g(a) s fnte dmensonal f and only f A s a postve defnte matrx In ths case, t s a fnte dmensonal semsmple Le algebra (see [H]) We recall also the noton of the unversal envelopng algebra of a Le algebra g whch s the unque assocatve algebra U(g) equpped wth a map ι : g U(g) such that ι([x, y]) = ι(x)ι(y) ι(y)ι(x) whch satsfes the followng unversal property: f A s any assocatve algebra and κ : g A s a map satsfyng κ([x, y]) = κ(x)κ(y) κ(y)κ(x) then there exsts a unque homomorphsm of algebras φ : U(g) A such that φ ι = κ, alternately, such that the followng dagram commutes: U(g) ι g κ Recall also (by the Poncaré-Brkhoff-Wtt theorem) that g embeds va ι somorphcally n U(g), whch can be realzed as a sutable quotent of the tensor algebra of g In order to quantze the unversal envelopng algebra of a Kac-Moody algebra, t s helpful to gve an explct realzaton n terms of generators and relatons: Proposton (see [HK]) Let g be a Kac-Moody algebra wth GCM A Then U(g) s somorphc to the assocatve algebra over C wth generated by e, f, h, I, d j, j S satsfyng relatons ()-(4) of defnton (wth Le bracket replaced by commutator bracket) and the followng: (5) a j k= ( )k ( aj k (6) a j k= ( )k ( aj k ) A!φ e a j k e j e k = for j, ) f a j k f j f k = for j

3 Now we are ready to gve the defnton of a quantum group Let C(q) be the feld of ratonal functons n the ndetermnate q and defne [m] q = qm q m C(q) to be q q the q nteger m [ Defne ] the q-factoral nductvely by [] q! =, [m] q! = [m] q [m] q! m [m] Fnally, defne = q! n [n] q![m n] q! to be the q bnomal coeffcent Notce that q the lmt as q of each of these functons s m, m! and ( m n) respectvely Then we make the followng defnton: Defnton Let g be a Kac-Moody algebra wth GCM A The quantum group or quantzed unversal envelopng algebra U q (g) s the assocatve algebra over C(q) wth generated by the elements e, f, K, K I and D j, D j, j S wth the followng defnng relatons: () K K = K K = D j D j = D j D j =, I, j S () [K, K j ] =,, j I, [K, D j ] =, I, j S, [D, D j ] =,, j S, (3) K e j K (4) K f j K K (5) [e, f j ] = δ K j q q (6) a j k= ( )k [ aj k (7) a j k= ( )k [ aj k Where q = q s = q a j e j,, j I, D e j D = q a j f j,, j I, D f j D for, j I, ] ] q e aj k = q δ j e j, I, j S = q δ j e j, I, j S e j e k = for j, f aj k f j f k = for j q It s possble to vew U(g) as the formal lmt as q of U q (g), hence the termnology (see [HK]) There s also an analogous theory of U q (g)-modules wth parallels to the theory of U(g)-modules (equvalently representatons of g) We outlne the man ponts next Let λ P := {λ h λ(h ), λ(d j ) Z, I, j S}, where h = span C {h, I, d j, j S} s the Cartan subalgebra of g Then there exsts a U q (g)-module V q (λ) whch s an rreducble hghest weght module wth hghest weght λ e, there exsts v λ V q (λ) such that V q (λ) = U q (g)v λ and () e v λ = {}, I () K v λ = q λ(h ) v λ, I, D v λ = q λ(d j) v λ, j S For µ P we defne the set V q (λ) µ := {v V q K v = q µ(h ) v, I, D j v = q µ(d j), j S} to be the weght space of V q (λ) of weght µ One of the man results s the followng, due to Lusztg: Theorem Let λ P + := {λ P λ(h ), λ(d ), I, j S} Then dm C(q) (V q (Λ) µ ) = dm C (V (λ) µ ), µ P where V (λ) s the rreducble hghest-weght U(g)-module wth hghest weght λ 3

4 Therefore, the characters of ntegrable U q (g)-modules are the same as the correspondng U(g)-modules Crystal Bases Crystal bases ([Ka],[Lu]) are essentally bases of U q (g) modules n the formal lmt as q Before we defne crystal bases, we need the noton of the Kashwara operators ẽ, f, I These are certan modfed root vectors for the quantum group U q (g) But frst, we need a prelmnary result: Lemma ([Ka]) Let λ P + and V q (λ) be the hghest weght U q (g)-module of hghest weght λ For each I, every weght vector u V q (λ) µ (µ P ) may be wrtten n the form u = u + f u + + f (N) u N, where N Z and u k V q (λ) µ+kα ker e for all k =,,, N, α P s defned by α (h j ) = a j,, j I, α (d j ) = δ,j, I, j S, and f (n) := f n [n] q! Here, each u k n the expresson s unquely determned by u and u k only f µ(h ) + k We now have the followng: Defnton 3 Let λ P + The Kashwara operators ẽ and f ( I) on V q (λ) are defned by N N ẽ u = f (k ) u k, f u = f (k+) u k k= We also need an auxlary defnton of a crystal lattce, whch wll make t possble to formally take the lmt as q Let A := {f C(q) f s regular at } Defnton 4 Let λ P + and V q (λ) be the hghest weght U q (g)-module of hghest weght λ A free A -submodule L of V q (λ) s called a crystal lattce f () L generates V q (λ) as a vector space over C(q), () L = µ P L µ, where L µ = L V q (λ) µ, (3) ẽ L L, f L L for all I Fnally, we have the followng: Defnton 5 A crystal base of the rreducble hghest weght U q (g)-module V q (λ), λ P + s a par (L, B) such that () L s a crystal lattce of V q (λ), () B s a C-bass of L/qL, (3) B = µ P B µ, where B µ = B (L µ /ql µ ), (4) ẽ B B {}, f B B {} for all I, (5) for any b, b B and I, we have f b = b f and only f b = ẽ b 4 k=

5 The set B s called the crystal or crystal graph of (L, B) Ths s because B can be regarded as a colored, orented graph by defnng b b f b = b Example: Let V q (m), m be the hghest weght U q (sl())-module span C(q) {v, v,, v m } wth the followng actons of the generators: K v j = q m j v j f v j = [j + ] q v j+ e v j = [m j + ] q v j where v j s understood to be f j < or j > m Then L = span A {v j j =,,, m} s a crystal lattce of V q (m) and (L, B) s a crystal base, where B = { v, v,, v m } (we wrte v for v + ql) The acton of the Kashwara operators on B s ẽ( v j ) = v j, f( v j ) = v j+, agan v j s understood to be f j < or j > m Therefore the crystal graph s: v v v v m In ths way, theory of crystal bases enables us to study combnatoral skeleton of U q (g)-modules In partcular, the crystal base of a U q (g)-module satsfes the condtons for an (abstract) crystal Defnton 6 A crystal assocated wth U q (g) s a set B together wth maps wt : B P, ẽ, f : B B {}, and ε, ϕ : B Z { }, for I satsfyng the followng propertes: () ϕ (b) = ε (b) + h, wt(b) for all I, () wt(ẽ b) = wt(b) + α f ẽ b B, (3) wt( f b) = wt(b) α f f b B, (4) ε (ẽ b) = ε (b), ϕ (ẽ b) = ϕ (b) + f ẽ b B, (5) ε ( f b) = ε (b) +, ϕ ( f b) = ϕ (b) f f b B, (6) f b = b f and only f b = ẽ b for b, b B, I, (7) f ϕ (b) = for b B, then ẽ b = f b = Then one may easly prove the followng: Proposton ([Ka]) Let (L, B) be the crystal bass of a U q (g)-module V q (λ) Then B s a crystal f we defne n addton to ẽ, f : wt(b) = µ f b B µ, ε (b) = max{k ẽ k (b) }, ϕ (b) = max{k f k (b) } 5

6 + + + Fgure Color pattern for an extended Young dagram of charge All labels are reduced modulo n 3 Realzaton of ŝl(n)-modules We probably should say combnatoral realzaton of U q (ŝl(n))-modules, but n lght of Theorem the termnology s justfed The Kac-Moody algebra ŝl(n) has GCM A = (a,j ),j I where I = {,,, n} and The Kac-Moody algebra ŝl(n) s the nfnte-dmensonal analogue of sl(n), hence the termnology Let I := {,, n } denote the ndex set for ŝl(n) An extended Young dagram s a collecton of I-colored boxes arranged n rows and columns, such that the number of boxes n each row s greater than or equal to the number of boxes n the row below To every extended Young dagram we assocate a charge, I In each box, we put a color j I gven by j a b + (mod n) where a s the number of columns from the rght and b s the number of rows from the top (see fgure ) For example, s an extended Young dagram of charge for n = 3 The null dagram wth no boxes denoted by s also consdered as an extended Young dagram A column n an extended Young dagram s -removable f the bottom box contans and can be removed leavng another extended Young dagram A column s -admssble f a box contanng could be added to gve another extended Young dagram An extended Young dagram s called n-regular f there are at most (n ) rows wth the same number of boxes Let Y(), I denote the collecton of all n-regular extended Young dagrams of charge j Then Y() can be gven the structure of a crystal wth the followng actons of ẽ j, fj, ε j, ϕ j, and wt( ) For each j I and b Y() we defne the j-sgnature of b to be the strng of + s, 6

7 and s n whch each j-admssble column receves a + and each j-removable column receves a readng from rght to left The reduced j-sgnature s the result of recursvely cancelng all + pars n the -sgnature leavng a strng of the form (,,, +, +) The Kashwara operator ẽ j acts on b by removng the box correspondng to the rghtmost, or maps b to f there are no mnus sgns Smlarly, fj adds a box to the bottom of the column correspondng to the leftmost +, or maps b to f there are no plus sgns The functon ϕ j (b) s the number of + sgns n the reduced j-sgnature of b and ε j (b) s the number of sgns We defne wt: Y() P by b Λ n j= # {j-colored boxes n b}α j where Λ P s the th fundamental weght gven by Λ (h j ) = δ j, Λ (d) = Then we have the followng: Theorem (MM) Let B(Λ ) be the crystal graph of the U q (ŝl(n))-module V q (Λ ) Then Y() = B(Λ ) as crystals We gve the top part of Y() for ŝl(n) below: References [H] Humphreys, James Introducton to Le Algebras and Representaton Theory, Sprnger- Verlag 97 [HK] Hong, J Kang, S-J Introducton to Quantum Groups and Crystal Bases, Amercan Mathematcal Socety [K] Kac, V Infnte Dmensonal Le Algebras, 3 rd edton, Cambrdge Unversty Press 99 [Ka] Kashwara, M Crystalzng the q-analogue of unversal envelopng algebras, Commun Math Phys 33 (99),

8 [Lu] Lusztg, G Canoncal bases arsng from quantzed envelopng algebras, J Amer Math Soc 3 (99), [MM] Msra, K C, Mwa, T Crystal bases for the basc representaton of U q (ŝl(n)), Commun Math Phys 34 (99),