Quantum supergroups and canonical bases

Transcription

1 Quantum supergroups and canonical bases Sean Clark University of Virginia Dissertation Defense April 4, 2014

2 WHAT IS A QUANTUM GROUP? A quantum group is a deformed universal enveloping algebra.

3 WHAT IS A QUANTUM GROUP? A quantum group is a deformed universal enveloping algebra. Let g be a semisimple Lie algebra (e.g. sl(n), so(2n + 1)). Π = {α i : i I} the simple roots.

4 WHAT IS A QUANTUM GROUP? A quantum group is a deformed universal enveloping algebra. Let g be a semisimple Lie algebra (e.g. sl(n), so(2n + 1)). Π = {α i : i I} the simple roots. U q (g) is the Q(q) algebra with generators E i, F i, K ±1 i for i I, Various relations; for example, K i q h i, e.g. K i E j K 1 i = q h i,α j Ej quantum Serre, e.g. F 2 i F j [2]F i F j F i + F j F 2 i = 0 (here [2] = q + q 1 is a quantum integer)

5 WHAT IS A QUANTUM GROUP? A quantum group is a deformed universal enveloping algebra. Let g be a semisimple Lie algebra (e.g. sl(n), so(2n + 1)). Π = {α i : i I} the simple roots. U q (g) is the Q(q) algebra with generators E i, F i, K ±1 i for i I, Various relations; for example, K i q h i, e.g. K i E j K 1 i = q h i,α j Ej quantum Serre, e.g. F 2 i F j [2]F i F j F i + F j F 2 i = 0 (here [2] = q + q 1 is a quantum integer) Some important features are: an involution q = q 1, K i = K 1 i, E i = E i, F i = F i ; a bar invariant integral Z[q, q 1 ]-form of U q (g).

6 CANONICAL BASIS AND CATEGORIFICATION U q (n ), the subalgebra generated by F i.

7 CANONICAL BASIS AND CATEGORIFICATION U q (n ), the subalgebra generated by F i. [Lusztig, Kashiwara]: U q (n ) has a canonical basis, which is bar-invariant, descends to a basis for each h. wt. integrable module, has structure constants in N[q, q 1 ] (symmetric type).

8 CANONICAL BASIS AND CATEGORIFICATION U q (n ), the subalgebra generated by F i. [Lusztig, Kashiwara]: U q (n ) has a canonical basis, which is bar-invariant, descends to a basis for each h. wt. integrable module, has structure constants in N[q, q 1 ] (symmetric type). Relation to categorification: U q (n ) categorified by quiver Hecke algebras [Khovanov-Lauda, Rouquier] canonical basis indecomp. projectives (symmetric type) [Varagnolo-Vasserot].

9 LIE SUPERALGEBRAS g: a Lie superalgebra (everything is Z/2Z-graded). e.g. gl(m n), osp(m 2n)

10 LIE SUPERALGEBRAS g: a Lie superalgebra (everything is Z/2Z-graded). e.g. gl(m n), osp(m 2n) Example: osp(1 2) is the set of 3 3 matrices of the form A = c d 0 e c } {{ } A a b b 0 0 a 0 0 } {{ } A 1 with the super bracket; i.e. the usual bracket, except [A 1, B 1 ] = A 1 B 1 +B 1 A 1. (Note: The subalgebra of the A 0 is = to sl(2).)

11 OUR QUESTION Quantized Lie superalgebras have been well studied (Benkart, Jeong, Kang, Kashiwara, Kwon, Melville, Yamane,...)

12 OUR QUESTION Quantized Lie superalgebras have been well studied (Benkart, Jeong, Kang, Kashiwara, Kwon, Melville, Yamane,...) U q (n ): algebra generated by F i satisfying super Serre relations. Is there a canonical basis à la Lusztig, Kashiwara?

13 OUR QUESTION Quantized Lie superalgebras have been well studied (Benkart, Jeong, Kang, Kashiwara, Kwon, Melville, Yamane,...) U q (n ): algebra generated by F i satisfying super Serre relations. Is there a canonical basis à la Lusztig, Kashiwara? Some potential obstructions are: Existence of isotropic simple roots: (α i, α i ) = 0 No integral form, bar involution (e.g. quantum osp(1 2)) Lack of positivity due to super signs Experts did not expect canonical bases to exist!

14 INFLUENCE OF CATEGORIFICATION [KL,R] ( 08): quiver Hecke categorify quantum groups

15 INFLUENCE OF CATEGORIFICATION [KL,R] ( 08): quiver Hecke categorify quantum groups [KKT11]: introduce quiver Hecke superalgebras (QHSA) (Generalizes a construction of Wang ( 06))

16 INFLUENCE OF CATEGORIFICATION [KL,R] ( 08): quiver Hecke categorify quantum groups [KKT11]: introduce quiver Hecke superalgebras (QHSA) (Generalizes a construction of Wang ( 06)) [KKO12]: QHSA s categorify quantum groups (Generalizes a rank 1 construction of [EKL11])

17 INFLUENCE OF CATEGORIFICATION [KL,R] ( 08): quiver Hecke categorify quantum groups [KKT11]: introduce quiver Hecke superalgebras (QHSA) (Generalizes a construction of Wang ( 06)) [KKO12]: QHSA s categorify quantum groups (Generalizes a rank 1 construction of [EKL11]) [HW12]: QHSA s categorify quantum supergroups (assuming no isotropic roots)

18 INSIGHT FROM [HW] Key Insight [HW]: use a parameter π 2 = 1 for super signs e.g. a super commutator AB + BA becomes AB πba

19 INSIGHT FROM [HW] Key Insight [HW]: use a parameter π 2 = 1 for super signs e.g. a super commutator AB + BA becomes AB πba π = 1 non-super case. π = 1 super case.

20 INSIGHT FROM [HW] Key Insight [HW]: use a parameter π 2 = 1 for super signs e.g. a super commutator AB + BA becomes AB πba π = 1 non-super case. π = 1 super case. There is a bar involution on Q(q)[π] given by q πq 1.

21 INSIGHT FROM [HW] Key Insight [HW]: use a parameter π 2 = 1 for super signs e.g. a super commutator AB + BA becomes AB πba π = 1 non-super case. π = 1 super case. There is a bar involution on Q(q)[π] given by q πq 1. [n] = (πq)n q n πq q 1, e.g. [2] = πq + q 1. Note πq + q 1 has positive coefficients. (vs. q + q 1 ) (Important for categorification: e.g. F 2 i = (πq + q 1 )F (2) i.)

22 ANISOTROPIC KM I = I 0 I1 (simple roots), parity p(i) with i I p(i). Symmetrizable generalized Cartan matrix (a ij ) i,j I : a ij Z, a ii = 2, a ij 0; positive symmetrizing coefficients d i (d i a ij = d j a ji ); (anisotropy) a ij 2Z for i I 1 ; (bar-compatibility) d i = p(i) mod 2, where i I p(i)

23 EXAMPLES (FINITE AND AFFINE) ( =odd root) < (osp(1 2n)) < < < > < > < < <>

24 FINITE TYPE The only finite type covering algebras have Dynkin diagrams <

25 FINITE TYPE The only finite type covering algebras have Dynkin diagrams < This diagram corresponds to the Lie superalgebra osp(1 2n)

26 FINITE TYPE The only finite type covering algebras have Dynkin diagrams < This diagram corresponds to the Lie superalgebra osp(1 2n) the Lie algebra so(1 + 2n)

27 FINITE TYPE The only finite type covering algebras have Dynkin diagrams < This diagram corresponds to the Lie superalgebra osp(1 2n) the Lie algebra so(1 + 2n) These algebras have similar representation theories. osp(1 2n) irreps half of so(2n + 1) irreps.

28 FINITE TYPE The only finite type covering algebras have Dynkin diagrams < This diagram corresponds to the Lie superalgebra osp(1 2n) the Lie algebra so(1 + 2n) These algebras have similar representation theories. osp(1 2n) irreps half of so(2n + 1) irreps. U q (osp(1 2n))/C(q) all of U q (so(2n + 1)) irreps. [Zou98]

29 RANK 1 [CW]: U q (osp(1 2))/Q(q) can be tweaked to get all reps. EF πfe = 1K K 1 πq q 1 }{{} even h.w. or πk K 1 πq q 1 }{{} odd h.w.

30 RANK 1 [CW]: U q (osp(1 2))/Q(q) can be tweaked to get all reps. EF πfe = πh K K 1 πq q 1 (h the Cartan generator) ( )

31 RANK 1 [CW]: U q (osp(1 2))/Q(q) can be tweaked to get all reps. EF πfe = πh K K 1 πq q 1 (h the Cartan generator) ( ) New definition: generators E, F, K ±1, J, relations J 2 = 1, JK = KJ, JEJ 1 = E, KEK 1 = q 2 E, JFJ 1 = F, KFK 1 = q 2 F, EF πf j E i = (If h is the Cartan element, K = q h and J = π h.) JK K 1 πq q 1 ; ( )

32 DEFINITION OF QUANTUM COVERING GROUPS Let A be a symmetrizable GCM. U is the Q(q)[π]-algebra with generators E i, F i, K ±1 i, J i and relations J i 2 = 1, J i K i = K i J i, J i J j = J j J i J i E j J 1 i = π a ij E j, J i F j J 1 i = π a ij F j. E i F j π p(i)p(j) J d i i K d i i K d i i F j E i = δ ij (πq) d i q d i and others (super quantum Serre, usual K relations). ;

33 DEFINITION OF QUANTUM COVERING GROUPS Let A be a symmetrizable GCM. U is the Q(q)[π]-algebra with generators E i, F i, K ±1 i, J i and relations J i 2 = 1, J i K i = K i J i, J i J j = J j J i J i E j J 1 i = π a ij E j, J i F j J 1 i = π a ij F j. E i F j π p(i)p(j) J d i i K d i i K d i i F j E i = δ ij (πq) d i q d i and others (super quantum Serre, usual K relations). Bar involution: q = πq 1, K i = J i K 1 i, E i = E i, F i = F i Can also define a bar-invariant integral Z[q, q 1, π]-form! ;

34 RELATION TO QUANTUM (SUPER)GROUPS By specifying a value of π, we have maps U π= 1 π=1 U π= 1 U π=1 U π=1 is a quantum group (forgets Z/2Z grading). U π= 1 is a quantum supergroup.

35 REPRESENTATIONS X: integral weights, X + : dominant integral weights. A weight module is a U-module M = λ X M λ, where M λ = { m M : K i m = q h i,λ m, } J i m = π hi,λ m.

36 REPRESENTATIONS X: integral weights, X + : dominant integral weights. A weight module is a U-module M = λ X M λ, where M λ = { m M : K i m = q h i,λ m, } J i m = π hi,λ m. Example: U q (osp(1 2)), X = Z, X + = N and M = n Z M n. Jm = π n m, Km = q n m (m M n )

37 REPRESENTATIONS Can define highest-weight (h.w.) and integrable (int.) modules. Theorem (C-Hill-Wang) For each λ X +, there is a unique simple ( π-free ) module V(λ) of highest weight λ. Any ( π-free ) h.wt. int. M is a direct sum of these V(λ). (π-free: π acts freely)

38 REPRESENTATIONS Can define highest-weight (h.w.) and integrable (int.) modules. Theorem (C-Hill-Wang) For each λ X +, there is a unique simple ( π-free ) module V(λ) of highest weight λ. Any ( π-free ) h.wt. int. M is a direct sum of these V(λ). (π-free: π acts freely) Example: U q (osp(1 2)) has simple π-free modules V(n), which are free Q(q)[π]-modules of rank n + 1. (Like sl(2)!) V(n) = V(n) }{{ π=1 } V(n) }{{ π= 1 } dim Q(q) =n+1 dim Q(q) =n+1

39 APPROACHES TO CANONICAL BASES Two potential approaches to constructing a canonical basis: [Lusztig] using geometry [Kashiwara] algebraically using crystals ( q = 0 )

40 APPROACHES TO CANONICAL BASES Two potential approaches to constructing a canonical basis: [Lusztig] using geometry [Kashiwara] algebraically using crystals ( q = 0 ) Analogous geometry for super is unknown.

41 APPROACHES TO CANONICAL BASES Two potential approaches to constructing a canonical basis: [Lusztig] using geometry [Kashiwara] algebraically using crystals ( q = 0 ) Analogous geometry for super is unknown. There are various crystal structures in modules: osp(1 2n) [Musson-Zou] ( 98) gl(m n) [Benkart-Kang-Kashiwara] ( 00), [Kwon] ( 12) for KM superalgebra with even weights [Jeong] ( 01) No examples of canonical bases.

42 WHY BELIEVE? No examples despite extensive study, experts don t believe. Why should canonical bases exist?

43 WHY BELIEVE? No examples despite extensive study, experts don t believe. Why should canonical bases exist? Because now we have a better definition of U (all h. wt. modules /Q(q));

44 WHY BELIEVE? No examples despite extensive study, experts don t believe. Why should canonical bases exist? Because now we have a better definition of U (all h. wt. modules /Q(q)); a good bar involution; a bar-invariant integral form;

45 WHY BELIEVE? No examples despite extensive study, experts don t believe. Why should canonical bases exist? Because now we have a better definition of U (all h. wt. modules /Q(q)); a good bar involution; a bar-invariant integral form; a categorical canonical basis. This motivates us to try again generalizing Kashiwara.

46 CRYSTALS We can define Kashiwara operators ẽ i, f i. Let A Q(q)[π] be the ring of functions with no pole at q = 0. V(λ) is said to have a crystal basis (L, B) if L is a A-lattice of V(λ) closed under ẽ i, f i and B L/qL satisfies B is a π-basis of L/qL; (i.e. signed at π = 1: B = B πb) ẽ i B B {0} and f i B B {0}; For b B, if ẽ i b 0 then b = f i ẽ i b. As in the π = 1 case, the crystal lattice/basis is unique.

47 CRYSTALS We can define Kashiwara operators ẽ i, f i. Let A Q(q)[π] be the ring of functions with no pole at q = 0. V(λ) is said to have a crystal basis (L, B) if L is a A-lattice of V(λ) closed under ẽ i, f i and B L/qL satisfies B is a π-basis of L/qL; (i.e. signed at π = 1: B = B πb) ẽ i B B {0} and f i B B {0}; For b B, if ẽ i b 0 then b = f i ẽ i b. As in the π = 1 case, the crystal lattice/basis is unique.

48 CRYSTALS We can define Kashiwara operators ẽ i, f i. Let A Q(q)[π] be the ring of functions with no pole at q = 0. V(λ) is said to have a crystal basis (L, B) if L is a A-lattice of V(λ) closed under ẽ i, f i and B L/qL satisfies B is a π-basis of L/qL; (i.e. signed at π = 1: B = B πb) ẽ i B B {0} and f i B B {0}; For b B, if ẽ i b 0 then b = f i ẽ i b. As in the π = 1 case, the crystal lattice/basis is unique.

49 CANONICAL BASIS We set V(λ) L(λ) = A f i1... f in v λ, B(λ) = (λ X + { }, V( ) = U ) { } π ɛ fi1... f in v λ + ql(λ)

50 CANONICAL BASIS We set V(λ) L(λ) = A f i1... f in v λ, B(λ) = (λ X + { }, V( ) = U ) { } π ɛ fi1... f in v λ + ql(λ) Theorem (C-Hill-Wang) The pairs (L(λ), B(λ)) for λ X + { } are crystal bases. Moreover, there exist maps G : L(λ)/qL(λ) L(λ) such that G(B(λ)) is a bar-invariant π-basis of V(λ). We call G(B(λ)) the canonical basis of V(λ). (π = 1: first canonical bases for quantum supergroups!)

51 MAIN OBSTACLE IN PROOF Most of Kashiwara s arguments generalize (with extra signs).

52 MAIN OBSTACLE IN PROOF Most of Kashiwara s arguments generalize (with extra signs). Kashiwara s construction of G requires ρ(l( )) L( ) where ρ is an anti-automorphism of U. Super signs cause non-positivity problems usual proof fails.

53 MAIN OBSTACLE IN PROOF Most of Kashiwara s arguments generalize (with extra signs). Kashiwara s construction of G requires ρ(l( )) L( ) where ρ is an anti-automorphism of U. Super signs cause non-positivity problems usual proof fails. New idea: a twistor (from work with Fan, Li, Wang [CFLW]). U π=1 C = U π= 1 C which is almost an algebra isomorphism. Good enough: the ρ-invariance at π = 1 transports to π = 1.

54 WHY MUST THE BASIS BE SIGNED? Example: I = I 1 = {i, j} such that a ij = a ji = 0. F i F j = πf j F i Should F i F j or F j F i be in B( )? No preferred canonical choice.

55 WHY MUST THE BASIS BE SIGNED? Example: I = I 1 = {i, j} such that a ij = a ji = 0. F i F j = πf j F i Should F i F j or F j F i be in B( )? No preferred canonical choice. This is not a bad thing! A π-basis is an honest Q(q)-basis (for π-free modules)! Categorically: represents spin states of QHSA modules.

56 CANONICAL BASES AND THE WHOLE QUANTUM GROUP Can the canonical basis on U be extended to U? Not directly: U 0 makes such a construction difficult.

57 CANONICAL BASES AND THE WHOLE QUANTUM GROUP Can the canonical basis on U be extended to U? Not directly: U 0 makes such a construction difficult. The right construction is to explode U 0 into idempotents. (Beilinson-Lusztig-McPherson (type A), Lusztig) 1 1 λ with 1 λ 1 η = δ λ,η 1 λ, K i q hi,λ 1 λ λ X λ X

58 CANONICAL BASES AND THE WHOLE QUANTUM GROUP Can the canonical basis on U be extended to U? Not directly: U 0 makes such a construction difficult. The right construction is to explode U 0 into idempotents. (Beilinson-Lusztig-McPherson (type A), Lusztig) 1 λ X 1 λ with 1 λ 1 η = δ λ,η 1 λ, K i λ X q h i,λ 1 λ U is the algebra on symbols x1 λ = 1 λ+ x x for x U, λ X. x1 λ = projection to λ-wt. space followed by the action of x.

59 RANK 1 U q (osp(1 2)) is the algebra given by Generators: E1 n = 1 n+2 E, F1 n = 1 n 2 F, 1 n Relations: 1 n 1 m = δ nm 1 n, (E1 n 2 )(F1 n ) (F1 n+2 )(E1 n ) = [n]1 n

60 RANK 1 U q (osp(1 2)) is the algebra given by Generators: E1 n = 1 n+2 E, F1 n = 1 n 2 F, 1 n Relations: 1 n 1 m = δ nm 1 n, (E1 n 2 )(F1 n ) (F1 n+2 )(E1 n ) = [n]1 n Theorem (C-Wang) U q (osp(1 2)) admits a canonical basis Ḃ = { } E (a) 1 n F (b), π ab F (b) 1 n E (a) a + b n.

61 RANK 1 U q (osp(1 2)) is the algebra given by Generators: E1 n = 1 n+2 E, F1 n = 1 n 2 F, 1 n Relations: 1 n 1 m = δ nm 1 n, (E1 n 2 )(F1 n ) (F1 n+2 )(E1 n ) = [n]1 n Theorem (C-Wang) U q (osp(1 2)) admits a canonical basis Ḃ = { } E (a) 1 n F (b), π ab F (b) 1 n E (a) a + b n. We conjectured U q (osp(1 2)) admits a categorification, and Ellis and Lauda ( 13) recently verified our conjecture.

62 CANONICAL BASIS Theorem (C) U admits a π-signed canonical basis generalizing the basis for U. For π = 1, this specializes to Lusztig s canonical basis for U π=1.

63 CANONICAL BASIS Theorem (C) U admits a π-signed canonical basis generalizing the basis for U. For π = 1, this specializes to Lusztig s canonical basis for U π=1. Idea of proof (generalizing Lusztig): Consider modules N(λ, λ ) U1 λ λ as λ, λ.

64 CANONICAL BASIS Theorem (C) U admits a π-signed canonical basis generalizing the basis for U. For π = 1, this specializes to Lusztig s canonical basis for U π=1. Idea of proof (generalizing Lusztig): Consider modules N(λ, λ ) U1 λ λ as λ, λ. Define epimorphisms t : N(λ + λ, λ + λ ) N(λ, λ ). ({N(λ, λ )} with t forms a projective system)

65 CANONICAL BASIS Theorem (C) U admits a π-signed canonical basis generalizing the basis for U. For π = 1, this specializes to Lusztig s canonical basis for U π=1. Idea of proof (generalizing Lusztig): Consider modules N(λ, λ ) U1 λ λ as λ, λ. Define epimorphisms t : N(λ + λ, λ + λ ) N(λ, λ ). ({N(λ, λ )} with t forms a projective system) Construct suitable bar involution, canonical basis on N(λ, λ ).

66 CANONICAL BASIS Theorem (C) U admits a π-signed canonical basis generalizing the basis for U. For π = 1, this specializes to Lusztig s canonical basis for U π=1. Idea of proof (generalizing Lusztig): Consider modules N(λ, λ ) U1 λ λ as λ, λ. Define epimorphisms t : N(λ + λ, λ + λ ) N(λ, λ ). ({N(λ, λ )} with t forms a projective system) Construct suitable bar involution, canonical basis on N(λ, λ ). The canonical basis is stable under the projective limit induces a bar-invariant canonical basis on U.

67 FURTHER DIRECTIONS Construction of braid group action à la Lusztig Forthcoming work with D. Hill

68 FURTHER DIRECTIONS Construction of braid group action à la Lusztig Forthcoming work with D. Hill Canonical bases for other Lie superalgebras gl(m 1), osp(2 2n) using quantum shuffles [CHW3] Open question in general; e.g. gl(2 2).

69 FURTHER DIRECTIONS Construction of braid group action à la Lusztig Forthcoming work with D. Hill Canonical bases for other Lie superalgebras gl(m 1), osp(2 2n) using quantum shuffles [CHW3] Open question in general; e.g. gl(2 2). Categorification for covering quantum groups Connection to odd link homologies (Khovanov) Tensor modules? Higher rank?

70 SOME RELATED PAPERS Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), pp Kashiwara, On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), pp Lusztig, Canonical bases in tensor products, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), pp Ellis, Khovanov, Lauda, The odd nilhecke algebra and its diagrammatics, IMRN 2014 pp Hill and Wang, Categorication of quantum Kac-Moody superalgebras, arxiv: , to appear in Trans. AMS. Fan and Li, Two-parameter quantum algebras, canonical bases and categorifications, arxiv:

71 C and Wang, Canonical basis for quantum osp(1 2), arxiv: , Lett. Math. Phys. 103 (2013), C, Hill, and Wang, Quantum supergroups I. Foundations, arxiv: , Trans. Groups. 18 (2013), C, Hill, and Wang, Quantum supergroups II. Canonical Basis, arxiv: C, Fan, Li, and Wang, Quantum supergroups III. Twistors, arxiv: , to appear in Comm. Math. Phys. C, Quantum supergroups IV. Modified form, arxiv: C, Hill, and Wang, Quantum shuffles and quantum supergroups of basic type, arxiv: Slides available at sic5ag/ Thank you for your attention!