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Quantum Supergroups and Canonical Bases

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Quantum Supergroups and Canonical Bases Quantum supergroups and canonical bases Sean Clark University of Virginia Dissertation Defense April 4, 2014 Let g be a semisimple Lie algebra (e.g. sl(n); so(2n + 1)). Π = fαi : i 2 Ig the simple roots. ±1 Uq(g) is the Q(q) algebra with generators Ei, Fi, Ki for i 2 I, Various relations; for example, hi −1 hhi,αji I Ki ≈ q , e.g. KiEjKi = q Ej 2 2 I quantum Serre, e.g. Fi Fj − [2]FiFjFi + FjFi = 0 (here [2] = q + q−1 is a quantum integer) Some important features are: −1 −1 I an involution q = q , Ki = Ki , Ei = Ei, Fi = Fi; −1 I a bar invariant integral Z[q; q ]-form of Uq(g). WHAT IS A QUANTUM GROUP? A quantum group is a deformed universal enveloping algebra. ±1 Uq(g) is the Q(q) algebra with generators Ei, Fi, Ki for i 2 I, Various relations; for example, hi −1 hhi,αji I Ki ≈ q , e.g. KiEjKi = q Ej 2 2 I quantum Serre, e.g. Fi Fj − [2]FiFjFi + FjFi = 0 (here [2] = q + q−1 is a quantum integer) Some important features are: −1 −1 I an involution q = q , Ki = Ki , Ei = Ei, Fi = Fi; −1 I a bar invariant integral Z[q; q ]-form of Uq(g). 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 = fαi : i 2 Ig the simple roots. Some important features are: −1 −1 I an involution q = q , Ki = Ki , Ei = Ei, Fi = Fi; −1 I a bar invariant integral Z[q; q ]-form of Uq(g). 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 = fαi : i 2 Ig the simple roots. ±1 Uq(g) is the Q(q) algebra with generators Ei, Fi, Ki for i 2 I, Various relations; for example, hi −1 hhi,αji I Ki ≈ q , e.g. KiEjKi = q Ej 2 2 I quantum Serre, e.g. Fi Fj − [2]FiFjFi + FjFi = 0 (here [2] = q + q−1 is a quantum integer) 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 = fαi : i 2 Ig the simple roots. ±1 Uq(g) is the Q(q) algebra with generators Ei, Fi, Ki for i 2 I, Various relations; for example, hi −1 hhi,αji I Ki ≈ q , e.g. KiEjKi = q Ej 2 2 I quantum Serre, e.g. Fi Fj − [2]FiFjFi + FjFi = 0 (here [2] = q + q−1 is a quantum integer) Some important features are: −1 −1 I an involution q = q , Ki = Ki , Ei = Ei, Fi = Fi; −1 I a bar invariant integral Z[q; q ]-form of Uq(g). − [Lusztig, Kashiwara]: Uq(n ) has a canonical basis, which I is bar-invariant, I descends to a basis for each h. wt. integrable module, −1 I has structure constants in N[q; q ] (symmetric type). Relation to categorification: − I Uq(n ) categorified by quiver Hecke algebras [Khovanov-Lauda, Rouquier] I canonical basis $ indecomp. projectives (symmetric type) [Varagnolo-Vasserot]. CANONICAL BASIS AND CATEGORIFICATION − Uq(n ), the subalgebra generated by Fi. Relation to categorification: − I Uq(n ) categorified by quiver Hecke algebras [Khovanov-Lauda, Rouquier] I canonical basis $ indecomp. projectives (symmetric type) [Varagnolo-Vasserot]. CANONICAL BASIS AND CATEGORIFICATION − Uq(n ), the subalgebra generated by Fi. − [Lusztig, Kashiwara]: Uq(n ) has a canonical basis, which I is bar-invariant, I descends to a basis for each h. wt. integrable module, −1 I has structure constants in N[q; q ] (symmetric type). CANONICAL BASIS AND CATEGORIFICATION − Uq(n ), the subalgebra generated by Fi. − [Lusztig, Kashiwara]: Uq(n ) has a canonical basis, which I is bar-invariant, I descends to a basis for each h. wt. integrable module, −1 I has structure constants in N[q; q ] (symmetric type). Relation to categorification: − I Uq(n ) categorified by quiver Hecke algebras [Khovanov-Lauda, Rouquier] I canonical basis $ indecomp. projectives (symmetric type) [Varagnolo-Vasserot]. Example: osp(1j2) is the set of 3 × 3 matrices of the form 0 0 0 0 1 0 0 a b 1 A = @ 0 c d A + @ −b 0 0 A 0 e −c a 0 0 | {z } | {z } A0 A1 with the super bracket; i.e. the usual bracket, except [A1; B1] = A1B1+B1A1. ∼ (Note: The subalgebra of the A0 is = to sl(2).) LIE SUPERALGEBRAS g: a Lie superalgebra (everything is Z=2Z-graded). e.g. gl(mjn), osp(mj2n) LIE SUPERALGEBRAS g: a Lie superalgebra (everything is Z=2Z-graded). e.g. gl(mjn), osp(mj2n) Example: osp(1j2) is the set of 3 × 3 matrices of the form 0 0 0 0 1 0 0 a b 1 A = @ 0 c d A + @ −b 0 0 A 0 e −c a 0 0 | {z } | {z } A0 A1 with the super bracket; i.e. the usual bracket, except [A1; B1] = A1B1+B1A1. ∼ (Note: The subalgebra of the A0 is = to sl(2).) Some potential obstructions are: I Existence of isotropic simple roots: (αi; αi) = 0 I No integral form, bar involution (e.g. quantum osp(1j2)) I Lack of positivity due to super signs Experts did not expect canonical bases to exist! − Uq(n ): algebra generated by Fi satisfying super Serre relations. Is there a canonical basis a` la Lusztig, Kashiwara? OUR QUESTION Quantized Lie superalgebras have been well studied (Benkart, Jeong, Kang, Kashiwara, Kwon, Melville, Yamane, ...) Experts did not expect canonical bases to exist! Some potential obstructions are: I Existence of isotropic simple roots: (αi; αi) = 0 I No integral form, bar involution (e.g. quantum osp(1j2)) I Lack of positivity due to super signs OUR QUESTION Quantized Lie superalgebras have been well studied (Benkart, Jeong, Kang, Kashiwara, Kwon, Melville, Yamane, ...) − Uq(n ): algebra generated by Fi satisfying super Serre relations. Is there a canonical basis a` la Lusztig, Kashiwara? OUR QUESTION Quantized Lie superalgebras have been well studied (Benkart, Jeong, Kang, Kashiwara, Kwon, Melville, Yamane, ...) − Uq(n ): algebra generated by Fi satisfying super Serre relations. Is there a canonical basis a` la Lusztig, Kashiwara? Some potential obstructions are: I Existence of isotropic simple roots: (αi; αi) = 0 I No integral form, bar involution (e.g. quantum osp(1j2)) I Lack of positivity due to super signs Experts did not expect canonical bases to exist! I [KKT11]: introduce quiver Hecke superalgebras (QHSA) (Generalizes a construction of Wang (’06)) I [KKO12]: QHSA’s categorify quantum groups (Generalizes a rank 1 construction of [EKL11]) I [HW12]: QHSA’s categorify quantum supergroups (assuming no isotropic roots) INFLUENCE OF CATEGORIFICATION I [KL,R] (’08): quiver Hecke categorify quantum groups I [KKO12]: QHSA’s categorify quantum groups (Generalizes a rank 1 construction of [EKL11]) I [HW12]: QHSA’s categorify quantum supergroups (assuming no isotropic roots) INFLUENCE OF CATEGORIFICATION I [KL,R] (’08): quiver Hecke categorify quantum groups I [KKT11]: introduce quiver Hecke superalgebras (QHSA) (Generalizes a construction of Wang (’06)) I [HW12]: QHSA’s categorify quantum supergroups (assuming no isotropic roots) INFLUENCE OF CATEGORIFICATION I [KL,R] (’08): quiver Hecke categorify quantum groups I [KKT11]: introduce quiver Hecke superalgebras (QHSA) (Generalizes a construction of Wang (’06)) I [KKO12]: QHSA’s categorify quantum groups (Generalizes a rank 1 construction of [EKL11]) INFLUENCE OF CATEGORIFICATION I [KL,R] (’08): quiver Hecke categorify quantum groups I [KKT11]: introduce quiver Hecke superalgebras (QHSA) (Generalizes a construction of Wang (’06)) I [KKO12]: QHSA’s categorify quantum groups (Generalizes a rank 1 construction of [EKL11]) I [HW12]: QHSA’s categorify quantum supergroups (assuming no isotropic roots) I π = 1 non-super case. I π = −1 super case. − There is a bar involution on Q(q)[π] given by q 7! πq 1. (πq)n − q−n [n] = ; e.g. [2] = πq + q−1: πq − q−1 Note πq + q−1 has positive coefficients. (vs. −q + q−1) 2 −1 (2) (Important for categorification: e.g. Fi = (πq + q )Fi .) INSIGHT FROM [HW] Key Insight [HW]: use a parameter π2 = 1 for super signs e.g. a super commutator AB + BA becomes AB − πBA − There is a bar involution on Q(q)[π] given by q 7! πq 1. (πq)n − q−n [n] = ; e.g. [2] = πq + q−1: πq − q−1 Note πq + q−1 has positive coefficients. (vs. −q + q−1) 2 −1 (2) (Important for categorification: e.g. Fi = (πq + q )Fi .) INSIGHT FROM [HW] Key Insight [HW]: use a parameter π2 = 1 for super signs e.g. a super commutator AB + BA becomes AB − πBA I π = 1 non-super case. I π = −1 super case. (πq)n − q−n [n] = ; e.g. [2] = πq + q−1: πq − q−1 Note πq + q−1 has positive coefficients. (vs. −q + q−1) 2 −1 (2) (Important for categorification: e.g. Fi = (πq + q )Fi .) INSIGHT FROM [HW] Key Insight [HW]: use a parameter π2 = 1 for super signs e.g. a super commutator AB + BA becomes AB − πBA I π = 1 non-super case. I π = −1 super case. − There is a bar involution on Q(q)[π] given by q 7! πq 1. INSIGHT FROM [HW] Key Insight [HW]: use a parameter π2 = 1 for super signs e.g. a super commutator AB + BA becomes AB − πBA I π = 1 non-super case. I π = −1 super case. − There is a bar involution on Q(q)[π] given by q 7! πq 1.
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