Quantum supergroups and canonical bases Sean Clark University of - PowerPoint PPT Presentation

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

W HAT IS A QUANTUM GROUP ? A quantum group is a deformed universal enveloping algebra.

W HAT 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 ( 2 n + 1 ) ). � Π = { α i : i ∈ I } the simple roots.

W HAT 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 ( 2 n + 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 for i ∈ I , i Various relations; for example, = q � h i ,α j � E j ◮ K i ≈ q h i , e.g. K i E j K − 1 i ◮ 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)

W HAT 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 ( 2 n + 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 for i ∈ I , i Various relations; for example, = q � h i ,α j � E j ◮ K i ≈ q h i , e.g. K i E j K − 1 i ◮ 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 , E i = E i , F i = F i ; i ◮ a bar invariant integral Z [ q , q − 1 ] -form of U q ( g ) .

C ANONICAL BASIS AND CATEGORIFICATION U q ( n − ) , the subalgebra generated by F i .

C ANONICAL 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).

C ANONICAL 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].

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

L IE SUPERALGEBRAS g : a Lie superalgebra (everything is Z / 2 Z -graded). e.g. gl ( m | n ) , osp ( m | 2 n ) Example: osp ( 1 | 2 ) is the set of 3 × 3 matrices of the form     0 0 0 0 a b A = 0 c d + − b 0 0     e − c a 0 0 0 � �� � � �� � A 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 ) .)

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

O UR 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 ` a la Lusztig, Kashiwara?

O UR 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 ` a 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!

I NFLUENCE OF CATEGORIFICATION ◮ [KL,R] (’08): quiver Hecke categorify quantum groups

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

I NFLUENCE 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])

I NFLUENCE 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 super groups (assuming no isotropic roots)

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

I NSIGHT FROM [HW] Key Insight [HW]: use a parameter π 2 = 1 for super signs e.g. a super commutator AB + BA becomes AB − π BA ◮ π = 1 � non-super case. ◮ π = − 1 � super case.

I NSIGHT FROM [HW] Key Insight [HW]: use a parameter π 2 = 1 for super signs e.g. a super commutator AB + BA becomes AB − π BA ◮ π = 1 � non-super case. ◮ π = − 1 � super case. There is a bar involution on Q ( q )[ π ] given by q �→ π q − 1 .

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

A NISOTROPIC KM � I 1 (simple roots), parity p ( i ) with i ∈ I p ( i ) . I = I 0 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 ∈ 2 Z for i ∈ I 1 ; ◮ (bar-compatibility) d i = p ( i ) mod 2, where i ∈ I p ( i )

E XAMPLES (F INITE AND A FFINE ) ( • =odd root) • < ◦ ◦ · · · ◦ ◦ ◦ ( osp ( 1 | 2 n )) • < ◦ ◦ · · · ◦ ◦ < ◦ • < ◦ ◦ · · · ◦ ◦ > • ◦ ✈ ✈ ✈ • ◦ ◦ · · · ◦ ◦ ◦ < ✈ ❍ ❍ ❍ ❍ ◦ ◦ > • < ◦ • < ◦ • < > •

F INITE TYPE The only finite type covering algebras have Dynkin diagrams • < ◦ ◦ · · · ◦ ◦ ◦

F INITE TYPE The only finite type covering algebras have Dynkin diagrams • < ◦ ◦ · · · ◦ ◦ ◦ This diagram corresponds to ◮ the Lie superalgebra osp ( 1 | 2 n )

F INITE TYPE The only finite type covering algebras have Dynkin diagrams • < ◦ ◦ · · · ◦ ◦ ◦ This diagram corresponds to ◮ the Lie superalgebra osp ( 1 | 2 n ) ◮ the Lie algebra so ( 1 + 2 n )

F INITE TYPE The only finite type covering algebras have Dynkin diagrams • < ◦ ◦ · · · ◦ ◦ ◦ This diagram corresponds to ◮ the Lie superalgebra osp ( 1 | 2 n ) ◮ the Lie algebra so ( 1 + 2 n ) These algebras have similar representation theories. ◮ osp ( 1 | 2 n ) irreps ↔ half of so ( 2 n + 1 ) irreps.

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

R ANK 1 [CW]: U q ( osp ( 1 | 2 )) / Q ( q ) can be tweaked to get all reps. EF − π FE = 1 K − K − 1 π K − K − 1 or π q − q − 1 π q − q − 1 � �� � � �� � even h.w. odd h.w.

R ANK 1 [CW]: U q ( osp ( 1 | 2 )) / Q ( q ) can be tweaked to get all reps. EF − π FE = π h K − K − 1 ( h the Cartan generator ) ( ∗ ) π q − q − 1

R ANK 1 [CW]: U q ( osp ( 1 | 2 )) / Q ( q ) can be tweaked to get all reps. EF − π FE = π h K − K − 1 ( h the Cartan generator ) ( ∗ ) π q − q − 1 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 − π F j E i = JK − K − 1 ( ∗ ′ ) π q − q − 1 ; (If h is the Cartan element, K = q h and J = π h .)

D EFINITION 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 , J i and relations i 2 = 1 , J i J i K i = K i J i , J i J j = J j J i J i E j J − 1 = π a ij E j , J i F j J − 1 = π − a ij F j . i i J d i i K d i i − K − d i E i F j − π p ( i ) p ( j ) F j E i = δ ij i ( π q ) d i − q − d i ; and others (super quantum Serre, usual K relations).