Diagrammatic categorification of quantum groups I
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1 Diagrammatic categoriicatio o quatum groups I Aaro Lauda Columbia Uiversity Jauary 28th, 2010 Available at lauda/talks/msri Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
2 Reshetikhi-Turaev ivariat For g a simple Lie algebra the quatum deormatio U q (g) o the evelopig algebra o g gives lik/tagle ivariats. Colour the strads o a tagle by a represetatio V λ o U q (g) λ λ V λ U q (g)-module ϕ(t) RT-ivariat U λ λ q (g)-module The ivariat ϕ(t) is a map o U q (g)-represetatios. Example g = sl 2 g = sl V m λ Joes polyomial, coloured Joes polyomial specializatios o the HOMFLYPT polyomial Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
3 Reshetikhi-Turaev ivariat For g a simple Lie algebra the quatum deormatio U q (g) o the evelopig algebra o g gives lik/tagle ivariats. Colour the strads o a tagle by a represetatio V λ o U q (g) λ λ V λ U q (g)-module ϕ(t) RT-ivariat U λ λ q (g)-module The ivariat ϕ(t) is a map o U q (g)-represetatios. Example g = sl 2 g = sl V m λ Joes polyomial, coloured Joes polyomial specializatios o the HOMFLYPT polyomial Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
4 Quatum sl(2) Deiitio The quatum group U q (sl 2 ) is the associative algebra (with uit) over É(q) with geerators E, F, K, K 1 ad relatios KK 1 = 1 = K 1 K, KE = q 2 EK, EF FE = K K 1 q q 1 KF = q 2 FK, Cosider a iite-dimesioal represetatio V with a weight decompositio V( + 2) E E V() F F V( 2) V = V() Kv = q v, v V() Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
5 Why categoriy quatum groups? Cojectured applicatios to low-dimesioal topology Represetatio theoretic explaatio o Khovaov homology Categoriicatio o the Reshetikhi-Turaev quatum kot ivariats. Crae-Frekel cojectured categoriied quatum groups would give 4-dimesioal TQFTs Categoriied represetatio theory should provide ew isights or ordiary represetatio theory. Geometric represetatio theory (algebraic/combiatorial aalog o perverse sheaves) Positivity ad itegrality properties or quatum groups Represetatio theory o the symmetric group! Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
6 represetatio category Quatum Group Braided mooidal category with duals Categoriied Quatum Group represetatio 2-category Braided mooidal 2-category with duals Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
7 Road map to categoriicatio There are several hits that suggest that quatum groups are really just shadows o a richer algebraic structure Lusztig s discovery o caoical basis or idempotet versio U o quatum groups. Geometric costructios o categorical quatum group actios The existece o semiliear orms, : U U É(q) with ice properties. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
8 Beiliso, Lusztig, ad MacPherso added orthogoal idempotets 1 or projectio oto V() to produce U := U q (sl 2 ), a É(q)-algebra without uit U q (sl 2 ) 1 U collectio o orthogoal idempotets 1 or K 1 = q 1 o more K E1 = 1 +2 E = 1 +2 E1 F 1 = 1 2 F = 1 2 F 1 EF 1 FE1 = []1 U has a basis {E a F b 1 } or, a, b 0 [] = q q q q 1 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
9 Beiliso, Lusztig, ad MacPherso added orthogoal idempotets 1 or projectio oto V() to produce U := U q (sl 2 ), a É(q)-algebra without uit U q (sl 2 ) 1 U collectio o orthogoal idempotets 1 or K 1 = q 1 o more K E1 = 1 +2 E = 1 +2 E1 F 1 = 1 2 F = 1 2 F 1 EF 1 FE1 = []1 U has a basis {E a F b 1 } or, a, b 0 [] = q q q q 1 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
10 Beiliso, Lusztig, ad MacPherso added orthogoal idempotets 1 or projectio oto V() to produce U := U q (sl 2 ), a É(q)-algebra without uit U q (sl 2 ) 1 U collectio o orthogoal idempotets 1 or K 1 = q 1 o more K E1 = 1 +2 E = 1 +2 E1 F 1 = 1 2 F = 1 2 F 1 EF 1 FE1 = []1 U has a basis {E a F b 1 } or, a, b 0 [] = q q q q 1 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
11 Beiliso, Lusztig, ad MacPherso added orthogoal idempotets 1 or projectio oto V() to produce U := U q (sl 2 ), a É(q)-algebra without uit U q (sl 2 ) 1 U collectio o orthogoal idempotets 1 or K 1 = q 1 o more K E1 = 1 +2 E = 1 +2 E1 F 1 = 1 2 F = 1 2 F 1 EF 1 FE1 = []1 U has a basis {E a F b 1 } or, a, b 0 [] = q q q q 1 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
12 Beiliso, Lusztig, ad MacPherso added orthogoal idempotets 1 or projectio oto V() to produce U := U q (sl 2 ), a É(q)-algebra without uit U q (sl 2 ) 1 U collectio o orthogoal idempotets 1 or K 1 = q 1 o more K E1 = 1 +2 E = 1 +2 E1 F 1 = 1 2 F = 1 2 F 1 EF 1 FE1 = []1 U has a basis {E a F b 1 } or, a, b 0 [] = q q q q 1 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
13 Itegral orms The itegral orm U o U q (sl 2 ) is spaed by products o divided diereces E (a) 1 := E a [a]! 1 F (b) 1 := F b [b]! 1 Crae ad Frekel cojectured (1994) that U could be categoriied usig Lusztig s caoical basis E (a) F (b) 1, F (b) E (a) 1, b a b a Structure costats are i Æ[q, q 1 ] U is the Grothedieck rig o some higher structure U. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
14 Itegral orms The itegral orm U o U q (sl 2 ) is spaed by products o divided diereces E (a) 1 := E a [a]! 1 F (b) 1 := F b [b]! 1 Crae ad Frekel cojectured (1994) that U could be categoriied usig Lusztig s caoical basis E (a) F (b) 1, F (b) E (a) 1, b a b a Structure costats are i Æ[q, q 1 ] U is the Grothedieck rig o some higher structure U. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
15 Why 2-categories? U is a ouital algebra with a collectio o mutually-orthogoal idempotets (small) pre-additive categories U is a pre-additive category objects: morphisms m: abelia group 1 m U1 idetities: 1 idempoteted rigs compositio: 1m U1m 1 U1 δ,m1 m U1 give by multiplicatio the categoriicatio U o U should be a 2-category (small) pre-additive 2-categories Grothedieck category/rig (small) pre-additive categories idempoteted additive mooidal categories idempoteted rigs Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
16 Why 2-categories? U is a ouital algebra with a collectio o mutually-orthogoal idempotets (small) pre-additive categories U is a pre-additive category objects: morphisms m: abelia group 1 m U1 idetities: 1 idempoteted rigs compositio: 1m U1m 1 U1 δ,m1 m U1 give by multiplicatio the categoriicatio U o U should be a 2-category (small) pre-additive 2-categories Grothedieck category/rig (small) pre-additive categories idempoteted additive mooidal categories idempoteted rigs Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
17 Why 2-categories? U is a ouital algebra with a collectio o mutually-orthogoal idempotets (small) pre-additive categories U is a pre-additive category objects: morphisms m: abelia group 1 m U1 idetities: 1 idempoteted rigs compositio: 1m U1m 1 U1 δ,m1 m U1 give by multiplicatio the categoriicatio U o U should be a 2-category (small) pre-additive 2-categories Grothedieck category/rig (small) pre-additive categories idempoteted additive mooidal categories idempoteted rigs Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
18 2-categories ad strig diagrams A 2-category is give by objects: represeted by regios i the plae x or y morphisms: represeted by lies separatig regios o the plae y x y x 2-morphisms: y g α x g y α x together with compositio operatios ad idetity 1 ad 2-morphisms. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
19 g vertical compositio y g β α x β y x α horizotal compositio z g β y g α x g g β α z y x By covetio we do ot draw idetity morphisms or 2-morphisms: 1 x x = x x y 1 x = y x 1 x Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
20 g vertical compositio y g β α x β y x α horizotal compositio z g β y g α x g g β α z y x By covetio we do ot draw idetity morphisms or 2-morphisms: 1 x x = x x y 1 x = y x 1 x Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
21 g vertical compositio y g β α x β y x α horizotal compositio z g β y g α x g g β α z y x By covetio we do ot draw idetity morphisms or 2-morphisms: 1 x x = x x y 1 x = y x 1 x Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
22 g vertical compositio y g β α x β y x α horizotal compositio z g β y g α x g g β α z y x By covetio we do ot draw idetity morphisms or 2-morphisms: 1 x x = x x y 1 x = y x 1 x Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
23 Examples 1 Cat: 1 Bim: 1 Π(X): objects: categories morphisms: uctors 2-morphisms: atural trasormatios objects: commutative rigs R, S, T,... morphisms: (S, R)-bimodules T N S compositio: T S SM R R := T T N S S S M R R 2-morphisms: bimodule homomorphisms objects: poits o a topological space X morphisms: paths i X 2-morphisms: homotopies betwee paths The last two examples are really weak 2-categories or bicategories. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
24 Examples 1 Cat: 1 Bim: 1 Π(X): objects: categories morphisms: uctors 2-morphisms: atural trasormatios objects: commutative rigs R, S, T,... morphisms: (S, R)-bimodules T N S compositio: T S SM R R := T T N S S S M R R 2-morphisms: bimodule homomorphisms objects: poits o a topological space X morphisms: paths i X 2-morphisms: homotopies betwee paths The last two examples are really weak 2-categories or bicategories. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
25 Examples 1 Cat: objects: categories morphisms: uctors 2-morphisms: atural trasormatios 1 Bim: 1 Π(X): objects: commutative rigs R, S, T,... morphisms: (S, R)-bimodules T N S compositio: T S SM R R := T T N S S S M R R 2-morphisms: bimodule homomorphisms objects: poits o a topological space X morphisms: paths i X 2-morphisms: homotopies betwee paths The last two examples are really weak 2-categories or bicategories. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
26 Examples 1 Cat: objects: categories morphisms: uctors 2-morphisms: atural trasormatios 1 Bim: 1 Π(X): objects: commutative rigs R, S, T,... morphisms: (S, R)-bimodules T N S compositio: T S SM R R := T T N S S S M R R 2-morphisms: bimodule homomorphisms objects: poits o a topological space X morphisms: paths i X 2-morphisms: homotopies betwee paths The last two examples are really weak 2-categories or bicategories. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
27 Examples 1 Cat: objects: categories morphisms: uctors 2-morphisms: atural trasormatios 1 Bim: 1 Π(X): objects: commutative rigs R, S, T,... morphisms: (S, R)-bimodules T N S compositio: T S SM R R := T T N S S S M R R 2-morphisms: bimodule homomorphisms objects: poits o a topological space X morphisms: paths i X 2-morphisms: homotopies betwee paths The last two examples are really weak 2-categories or bicategories. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
28 I α is a 2-morphism y g 3 g 2 z 3 z 2 z 1 g 4 α g 1 x 3 w 2 w the α becomes the strig diagram: g 4 g 3 g 2 g 1 y α x Now let s apply strig diagrams to adjoit uctors! Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
29 I α is a 2-morphism y g 3 g 2 z 3 z 2 z 1 g 4 α g 1 x 3 w 2 w the α becomes the strig diagram: g 4 g 3 g 2 g 1 y α x Now let s apply strig diagrams to adjoit uctors! Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
30 Deiitio A adjuctio i a 2-category cosists o objects x ad y morphisms y x ad x y 2-morphisms 1 x u: ε ad u 1 y : η x x := y such that the equalities u 1 x ε y u u u x x y u x := η y y 1 y hold. x y = x y y u x u u = y x u Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
31 U q (sl 2 ) Categoriicatio Grothedieck rig U q (sl 2 ) weight object o U q (sl 2 ) b basis elemet 1-morphism o U q (sl 2 ) I.e., 1, E1, F 1 1, E1, F1 q a b b{a} (1-morphisms should be graded) x y = z mz xyz m z xy structure costats i Æ[q, q 1 ] x y = z mz xyz 2-morphisms??? Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
32 U q (sl 2 ) Categoriicatio Grothedieck rig U q (sl 2 ) weight object o U q (sl 2 ) b basis elemet 1-morphism o U q (sl 2 ) I.e., 1, E1, F 1 1, E1, F1 q a b b{a} (1-morphisms should be graded) x y = z mz xyz m z xy structure costats i Æ[q, q 1 ] x y = z mz xyz 2-morphisms??? Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
33 U q (sl 2 ) Categoriicatio Grothedieck rig U q (sl 2 ) weight object o U q (sl 2 ) b basis elemet 1-morphism o U q (sl 2 ) I.e., 1, E1, F 1 1, E1, F1 q a b b{a} (1-morphisms should be graded) x y = z mz xyz m z xy structure costats i Æ[q, q 1 ] x y = z mz xyz 2-morphisms??? Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
34 U q (sl 2 ) Categoriicatio Grothedieck rig U q (sl 2 ) weight object o U q (sl 2 ) b basis elemet 1-morphism o U q (sl 2 ) I.e., 1, E1, F 1 1, E1, F1 q a b b{a} (1-morphisms should be graded) x y = z mz xyz m z xy structure costats i Æ[q, q 1 ] x y = z mz xyz 2-morphisms??? Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
35 U q (sl 2 ) Categoriicatio Grothedieck rig U q (sl 2 ) weight object o U q (sl 2 ) b basis elemet 1-morphism o U q (sl 2 ) I.e., 1, E1, F 1 1, E1, F1 q a b b{a} (1-morphisms should be graded) x y = z mz xyz m z xy structure costats i Æ[q, q 1 ] x y = z mz xyz 2-morphisms??? Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
36 U q (sl 2 ) Categoriicatio Grothedieck rig U q (sl 2 ) weight object o U q (sl 2 ) b basis elemet 1-morphism o U q (sl 2 ) I.e., 1, E1, F 1 1, E1, F1 q a b b{a} (1-morphisms should be graded) x y = z mz xyz m z xy structure costats i Æ[q, q 1 ] x y = z mz xyz 2-morphisms??? Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
37 Sice the morphisms are graded, there is a iteral 2-HOM give by takig homomorphisms o all degrees ( ) ( ) HOM U (,): 1morph U 1morph U GrVect k Decategoriicatio K 0, : K 0 gdim U U [q, q 1 ] Choice o 2-morphisms is cotrolled by such a orm o U Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
38 Sice the morphisms are graded, there is a iteral 2-HOM give by takig homomorphisms o all degrees ( ) ( ) HOM U (,): 1morph U 1morph U GrVect k Decategoriicatio K 0, : K 0 gdim U U [q, q 1 ] Choice o 2-morphisms is cotrolled by such a orm o U Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
39 This orm will be semiliear (or sesquiliear) sice x y deg = α y y{s } x{s} α y } s s α x x x{s} y deg = α s x y{s } deg = α + s q s x, y = gdim (HOM(x{s}, y)) = q s gdim (HOM(x, y)) = q s x, y x, q s y = gdim ( HOM(x, y{s }) ) = q s gdim (HOM(x, y)) = q s x, y Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
40 This orm will be semiliear (or sesquiliear) sice x y deg = α y y{s } x{s} α y } s s α x x x{s} y deg = α s x y{s } deg = α + s q s x, y = gdim (HOM(x{s}, y)) = q s gdim (HOM(x, y)) = q s x, y x, q s y = gdim ( HOM(x, y{s }) ) = q s gdim (HOM(x, y)) = q s x, y Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
41 This orm will be semiliear (or sesquiliear) sice x y deg = α y y{s } x{s} α y } s s α x x x{s} y deg = α s x y{s } deg = α + s q s x, y = gdim (HOM(x{s}, y)) = q s gdim (HOM(x, y)) = q s x, y x, q s y = gdim ( HOM(x, y{s }) ) = q s gdim (HOM(x, y)) = q s x, y Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
42 This orm will be semiliear (or sesquiliear) sice x y deg = α y y{s } x{s} α y } s s α x x x{s} y deg = α s x y{s } deg = α + s q s x, y = gdim (HOM(x{s}, y)) = q s gdim (HOM(x, y)) = q s x, y x, q s y = gdim ( HOM(x, y{s }) ) = q s gdim (HOM(x, y)) = q s x, y Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
43 From geometric cosideratios we expect that E (a) 1, E (a) 1 = grdimh (Gr(a, )) = a j=1 1 1 q 2j Example E1, E1 = 1 1 q 2 = 1 + q 2 + q 4 + q E 2 1, E 2 1 = [2][2] 1 1 q q 4 = (1 q 2 )( 1 1 q 2 ) Idea: use the semiliear orm to guess geeratig 2-morphisms Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
44 Example gdim ( HOM U (E1, E1 ) ) = E1, E1 = 1 1 q 2 = 1+q2 +q 4 +q The idetity 2-morphism E1 E1 must be degree zero ( ) deg = 0 cotributes q 0 = 1 to graded dimesio The q 2 ew 2-morphism E1 E1 ( with deg ) = 2 cotributes q 2 to graded dimesio Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
45 Example gdim ( HOM U (E1, E1 ) ) = E1, E1 = 1 1 q 2 = 1+q2 +q 4 +q The idetity 2-morphism E1 E1 must be degree zero ( ) deg = 0 cotributes q 0 = 1 to graded dimesio The q 2 ew 2-morphism E1 E1 ( with deg ) = 2 cotributes q 2 to graded dimesio Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
46 Example (cot.) Vertically composig the dot with itsel we get ( ) ( ( deg α := deg + 2 ) α ) = 2α, gdim ( HOM U (E1, E1 ) ) = deg ( ) + deg ( ) ( + deg 2 ) + = 1 + q 2 + q 4 + = 1 1 q 2 = E1, E1 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
47 Example (cot.) Vertically composig the dot with itsel we get ( ) ( ( deg α := deg + 2 ) α ) = 2α, gdim ( HOM U (E1, E1 ) ) = deg ( ) + deg ( ) ( + deg 2 ) + = 1 + q 2 + q 4 + = 1 1 q 2 = E1, E1 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
48 Example ( HOM U (E2 1, E 2 1 ) ) 0 α 1,0 α 2 deg ( ) + 4 α 2 α 1 = (1 + q 2 + q )(1 + q ) = ( 1 ) 2 1 q 2 The semiliear orm gives gdim U(EE1, EE1 ) ( ) 1 2 = EE1, EE1 = [2][2] E (2) 1, E (2) 1 = (1 + q 2 ) 1 q 2 a additioal geeratig 2-morphism o degree -2: + 4 := Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
49 Example (cot.) gdim U(EE1, EE1 ) = (1 q 2 )(1 + q 2 + q )(1 + q 2 + q ) ( ) deg = 4 = 0 The space o degree zero 2-morphisms is 3-dimesioal are ot liearly idepedet. Add relatios o these 2-morphisms Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
50 Example (cot.) gdim U(EE1, EE1 ) = (1 q 2 )(1 + q 2 + q )(1 + q 2 + q ) ( ) deg = 4 = 0 The space o degree zero 2-morphisms is 3-dimesioal are ot liearly idepedet. Add relatios o these 2-morphisms Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
51 Example (cot.) gdim U(EE1, EE1 ) = (1 q 2 )(1 + q 2 + q )(1 + q 2 + q ) ( ) deg = 4 = 0 The space o degree zero 2-morphisms is 3-dimesioal are ot liearly idepedet. Add relatios o these 2-morphisms Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
52 Cosideratios o adjoits i the geometric settig suggest that ux, y = x,τ(u)y or u U ad x, y U, where τ : U U is such that τ(xy) = τ(y)τ(x) or all x U (atihomomorphism) τ(1 ) = 1 τ(1 E1 2 ) = q 1 1 F 1 +2 τ(1 F 1 +2 ) = q E1 or all. Example gdim HOM U (FE1, 1 ) = FE1, 1 = E1,τ(F)1 = q1+ 1 q 2 gdim HOM U (EF1, 1 ) = EF 1, 1 = F 1,τ(E)1 = q1 1 q 2 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
53 Add geerators geerator ad dots so that degree gdim HOM U (FE1, 1 ) = gdim α=0 α ad = q 1 (1 + q 2 + q ) = q1 1 q 2 = FE1, 1 gdim HOM U (EF1, 1 ) = gdim ( α=0 α ) = q 1 (1 + q 2 + q ) = q1 1 q 2 = EF 1, 1 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
54 Similar calculatios or HOM U (1, FE1 ) ad HOM U (1, EF1 ) suggest geerators geerator degree Usig these 2-morphisms we ca costruct ew 2-morphisms E1 E1 o the orm Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
55 Similar calculatios or HOM U (1, FE1 ) ad HOM U (1, EF1 ) suggest geerators geerator degree Usig these 2-morphisms we ca costruct ew 2-morphisms E1 E1 o the orm Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
56 Recall that gdim ( HOM U (E1, E1 ) ) = E1, E1 = 1 = 1 + q 2 + q 4 + q q 2 But + 2 ( ) ( ) deg + 2 = deg + deg similarly deg = ( + 2) = = 0 Hece these two 2-morphisms must be a multiple o the idetity 2-morphism E1 E1 + 2 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
57 Recall that gdim ( HOM U (E1, E1 ) ) = E1, E1 = 1 = 1 + q 2 + q 4 + q q 2 But + 2 ( ) ( ) deg + 2 = deg + deg similarly deg = ( + 2) = = 0 Hece these two 2-morphisms must be a multiple o the idetity 2-morphism E1 E1 + 2 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
58 Deiitio U is a additive k-liear 2-category. The 2-category U cosists o objects: or. The homs U(, ) betwee two objects, are additive k-liear categories cosistig o: objects o U(, ): a 1-morphism i U rom to is a ormal iite direct sum o 1-morphisms E α 1 F β1 E αm F βm 1 {s} = 1 E α 1 F β1 E αm F βm 1 {s} or ay s ad = + 2 α i 2 β i. morphisms o U(, ): the k-vector space o 2-morphisms E α 1F β 1 E αm F βm 1 {s} E α 1F β 1 E α m F β m 1 {s } give by liear combiatios o degree s s diagrams, modulo certai relatios, built rom composites o: Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
59 Geeratig 2-morphisms: idetities Degree zero idetity 2-morphisms 1 x or each 1-morphism x i U; we write 1 E1{s} 1 F1λ {s} ad more geerally, the idetity 2-morphism 1 E α 1 F β 1 E αm F β m 1 {s}1 λ {s} is represeted as }{{}}{{} }{{}}{{} α 1 β 1 α m β m Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
60 2-morphisms II + 2 F For example + 2 deg 2 deg 2 deg -2 deg -2 E E F F E E F deg +1 deg 1- deg +1 deg 1- E 3 F 3 1 {s } E 3 F 3 1 {s} take degree s s diagrams this makes the total degree =0 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
61 2-morphisms II + 2 F For example + 2 deg 2 deg 2 deg -2 deg -2 E E F F E E F deg +1 deg 1- deg +1 deg 1- E 3 F 3 1 {s } E 3 F 3 1 {s} take degree s s diagrams this makes the total degree =0 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
62 Local relatios E1 ad F1 are biadjoit up to gradig shit = = + 2 = = + 2 NilHecke relatios = 0, = = = Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
63 Local relatios E1 ad F1 are biadjoit up to gradig shit = = + 2 = = + 2 NilHecke relatios = 0, = = = Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
64 Topological ivariace + 2 = + 2 = + 2 = = We ca deie := = := = Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
65 Topological ivariace + 2 = + 2 = + 2 = = We ca deie := = := = Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
66 Topological ivariace + 2 = + 2 = + 2 = = We ca deie := = := = Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
67 Positivity o bubbles All dotted bubbles o egative degree are zero. That is, deg = 2(1 ) + 2β deg = 2(1 + ) + 2β β β β = 0 i β < 1 β = 0 i β < 1 It is coveiet to emphasize the degree o a bubble o its label α 0 deg = 2α deg = 2α ( 1)+α ( 1)+α Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
68 Positivity o bubbles All dotted bubbles o egative degree are zero. That is, deg = 2(1 ) + 2β deg = 2(1 + ) + 2β β β β = 0 i β < 1 β = 0 i β < 1 It is coveiet to emphasize the degree o a bubble o its label α 0 deg = 2α deg = 2α ( 1)+α ( 1)+α Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
69 Positivity o bubbles All dotted bubbles o egative degree are zero. That is, deg = 2(1 ) + 2β deg = 2(1 + ) + 2β β β β = 0 i β < 1 β = 0 i β < 1 It is coveiet to emphasize the degree o a bubble o its label α 0 deg = 2α deg = 2α ( 1)+α ( 1)+α Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
70 Positivity o bubbles All dotted bubbles o egative degree are zero. That is, deg = 2(1 ) + 2β deg = 2(1 + ) + 2β β β β = 0 i β < 1 β = 0 i β < 1 It is coveiet to emphasize the degree o a bubble o its label α 0 deg = 2α deg = 2α ( 1)+α ( 1)+α sometimes this is a egative umber! Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
71 Fake bubbles α α deg deg α α 0 α < 0 0 α < 0 These ormal symbols are iductively deied Iiite ( Grassmaia equatio: ) + t + + t α α ( ) + + t α + = α We set the degree zero bubbles equal to 1 or coveiece. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
72 Fake bubbles α α deg deg α α 0 α < 0 0 α < 0 These ormal symbols are iductively deied Iiite ( Grassmaia equatio: ) + t + + t α α ( ) + + t α + = α We set the degree zero bubbles equal to 1 or coveiece. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
73 Fake bubbles α α deg deg α α 0 α < 0 0 α < 0 These ormal symbols are iductively deied Iiite ( Grassmaia equatio: ) + t + + t α α ( ) + + t α + = α We set the degree zero bubbles equal to 1 or coveiece. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
74 Reductio to bubbles = = 2 ( 1)+ 1 = g 1 +g 2 = ( 1)+g 1 g 2 EF decompositio = = = 1 g 1 +g 2 +g 3 = g 1 g 3 ( 1)+ 2 ( 1)+g 2 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
75 Reductio to bubbles = = 2 ( 1)+ 1 = g 1 +g 2 = ( 1)+g 1 g 2 EF decompositio = = = 1 g 1 +g 2 +g 3 = g 1 g 3 ( 1)+ 2 ( 1)+g 2 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
76 For [] = q 1 + q q 1 write [] 1 := 1 { 1} 1 { 3} 1 {1 } Litig sl 2 relatios EF1 = FE1 [] 1 or 0 FE1 = EF1 [ ] 1 or 0 EF l FE1 1 { 1} 1 { 1 2l} 1 {1 } 1+j 2 j 1 +j 2 =l 1 j 1 1+j 2 j 1 +j 2 = 1 j 1 EF1 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
77 For [] = q 1 + q q 1 write [] 1 := 1 { 1} 1 { 3} 1 {1 } Litig sl 2 relatios EF1 = FE1 [] 1 or 0 FE1 = EF1 [ ] 1 or 0 EF l FE1 1 { 1} 1 { 1 2l} 1 {1 } 1+j 2 j 1 +j 2 =l 1 j 1 1+j 2 j 1 +j 2 = 1 j 1 EF1 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
78 Example ( = 0, EF1 0 = FE10 ) FE1 0 EF1 0 These maps are isomorphisms sice or = = 0 0 = 0 Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
79 Spaig sets Usig relatios oe ca id spaig sets or the space o 2-morphisms: Example U(FE 2 F1, EF1 ) a 2 E a 1 F a 3 F E E F 1+α 1 1+α 2 1+α 3 1+α 4 a 3 E a 1 F a 3 F E E F 1+α 1 1+α 2 1+α 3 o strad itersects itsel all closes diagrams are reduced to o-ested bubbles with the same orietatio ad are moved to the ar right o each diagram all dots are coied to a small iterval o each strad Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
80 Iterated lag varieties Fl(k, k + 1, N) Gr(k, N) {0 k k+1 N } Gr(k + 1, N) {0 k N } {0 k+1 N } H k,k+1 := H (Fl(k, k + 1, N) H k := H (Gr(k, N)) H k+1 := H (Gr(k + 1, N)) Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
81 Iterated lag varieties Fl(k, k + 1, N) Gr(k, N) {0 k k+1 N } Gr(k + 1, N) {0 k N } {0 k+1 N } H k,k+1 := H (Fl(k, k + 1, N) H k := H (Gr(k, N)) H k+1 := H (Gr(k + 1, N)) Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
82 H k+1,k H k,k 1 H N =. H k+1 H k H k 1. H 0 = H k+1,k H k,k 1 Γ N : U Flag { N Hk = 2k N 0 2k N E1 H k+1,k F1 H k 1,k E1 +2 E1 H k+2,k+1 Hk+1 H k+1,k H k+2,k+1 Hk+1 H k+1,k bimodule map H k+2,k+1 Hk+1 H k+1,k Theorem Γ N is a 2-uctor, all relatios i U hold i the iterated lag category Flag N. Γ N categoriies the irreducible N + 1-dimesioal represetatio. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
83 H k+1,k H k,k 1 H N =. H k+1 H k H k 1. H 0 = H k+1,k H k,k 1 Γ N : U Flag { N Hk = 2k N 0 2k N E1 H k+1,k F1 H k 1,k E1 +2 E1 H k+2,k+1 Hk+1 H k+1,k H k+2,k+1 Hk+1 H k+1,k bimodule map H k+2,k+1 Hk+1 H k+1,k Theorem Γ N is a 2-uctor, all relatios i U hold i the iterated lag category Flag N. Γ N categoriies the irreducible N + 1-dimesioal represetatio. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
84 H k+1,k H k,k 1 H N =. H k+1 H k H k 1. H 0 = H k+1,k H k,k 1 Γ N : U Flag { N Hk = 2k N 0 2k N E1 H k+1,k F1 H k 1,k E1 +2 E1 H k+2,k+1 Hk+1 H k+1,k H k+2,k+1 Hk+1 H k+1,k bimodule map H k+2,k+1 Hk+1 H k+1,k Theorem Γ N is a 2-uctor, all relatios i U hold i the iterated lag category Flag N. Γ N categoriies the irreducible N + 1-dimesioal represetatio. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
85 H k+1,k H k,k 1 H N =. H k+1 H k H k 1. H 0 = H k+1,k H k,k 1 Γ N : U Flag { N Hk = 2k N 0 2k N E1 H k+1,k F1 H k 1,k E1 +2 E1 H k+2,k+1 Hk+1 H k+1,k H k+2,k+1 Hk+1 H k+1,k bimodule map H k+2,k+1 Hk+1 H k+1,k Theorem Γ N is a 2-uctor, all relatios i U hold i the iterated lag category Flag N. Γ N categoriies the irreducible N + 1-dimesioal represetatio. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
86 Theorem (arxiv: ) This graphical calculus is cosistet ad categoriies U U = K0 ( U) the Grothedieck rig/category o this 2-category x y U [x] + [y] K 0 ( U) x{s} q s [x] K 0 ( U) Idecomposable 1-morphisms Lusztig caoical basis elemet Graded 2Hom HOM U (x, y) categoriies the semiliear orm x, y The 2-category U acts o cohomology o iterated lag varieties, categoriyig the irreducible N-dimesioal rep o U q (sl 2 ) Various kow (ati) liear (ati)automorphism o U q (sl 2 ) have categoriicatios that are give o 2-morphisms by symmetries o the graphical calculus. ω: U U E1 F 1 F 1 E1 Ivert orietatio Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
87 Joit with Mikhail Khovaov 2-category U has a extesio to a categoriicatio o U(sl ). There is a categoriicatio o U + (g) or ay Kac-Moody algebra g usig a similar diagrammatic calculus (arxiv: , arxiv: ). Cojectural categoriicatio o the itegral orm o U(g) or ay Kac-Moody algebra (arxiv: ). arxiv , arxiv: Closely related 2-categories were studied by Chuag ad Rouquier. arxiv: Geometric otios o categorical sl 2 -actios ad their coectio with the 2-categories above have bee studied by Cautis-Kamitzer-Licata. Aaro Lauda (Columbia Uiversity) Diagrammatic categoriicatio Ja 28th, / 41
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