Kac-Moody quantum superalgebras and global crystal bases Sean Clark - PowerPoint PPT Presentation

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals Kac-Moody quantum superalgebras and global crystal bases Sean Clark Joint work with D. Hill and W. Wang University of Virginia Workshop on Super Representation Theory Academia Sinica, Taiwan May 10, 2013

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals Q UANTUM GROUPS Let g be a Kac-Moody algebra with quantum group U q ( g ) U q ( n − ) ⊂ U q ( g ) is related to Hall algebras, quantized shuffles. U q ( g ) has a symmetry : q �→ q − 1 .

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals C ANONICAL BASES The algebra U q ( n − ) has an extraordinarily nice basis. It is ◮ suitably independent of choice, ◮ bar-invariant, ◮ well-behaved on the playground of integrable modules, ◮ “almost-orthogonal” (and can be characterized by this), ◮ categorifiable (cf. [Rouquier, Khovanov-Lauda]), ◮ just generally awesome. For all these reasons (and more!), it deserves the honorific The Canonical Basis

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals F INDING THE CANONICAL BASES These bases were discovered through the work of Lusztig and Kashiwara. Lusztig: perverse sheaves. Kashiwara: the crystal basis at “ q = 0”.

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals Q UANTUM GROUPS FOR SUPER Is there a super version of this picture? It isn’t clear what geometry could be used. There are various crystal structures in modules: ◮ osp ( 1 | 2 n ) [Musson-Zou] ◮ gl ( m | n ) [Benkart-Kang-Kashiwara], [Kwon] ◮ q ( n ) [Grantcharov-Jung-Kang-Kashiwara-Kim] ◮ for KM superalgebra with “even” weights [Jeong] Until recently, there was doubt of existence of canonical bases.

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals I NSPIRATION FROM CATEGORIFICATION [Wang, Ellis-Lauda-Khovanov], [Kang-Kashiwara-Tsuchioka] provide a fertile setting for categorification. [W, EKL]: spin (nil)Hecke algebras [KKT]: Hecke quiver superalgebras These categorify certain quantum half KM (super)algebras and integrable modules. [Hill-Wang, Kang-Kashiwara-Oh] This is strong evidence for a canonical basis for KM super.

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals W HICH K AC -M OODY SUPERALGEBRAS ? We consider a KM superalgebra with GCM A � I 1 (simple roots) and satisfying: indexed by I = I 0 ◮ a ij ∈ Z , a ii = 2, a ij ≤ 0 ◮ there exist positive symmetrizing coefficients d i ( d i a ij = d j a ji ) ◮ (non-isotropy) a ij ∈ 2 Z for i ∈ I 1 ◮ (bar-compatibility) d i ≡ 2 p ( i )

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals E XAMPLES • < ◦ ◦ · · · ◦ ◦ ◦ ( osp ( 1 | 2 n )) • < ◦ ◦ · · · ◦ ◦ < ◦ • < ◦ ◦ · · · ◦ ◦ > • ◦ ✈ ✈ ✈ • < ◦ ◦ · · · ◦ ◦ ◦ ✈ ❍ ❍ ❍ ❍ ◦ ◦ > • < ◦ • < ◦ • < > •

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals π - QUANTUM INTEGERS There is a bar involution on Q ( q ) given by ( π 2 = 1 ) q �→ π q − 1 ◮ π = 1 � non-super case. ◮ π = − 1 � super case. We have bar-invariant quantum integers: [ n ] = ( π q ) n − q − n � n � [ n ] ! , ∈ Z [ q , q − 1 ] . π q − q − 1 , a These allow us to define quantum divided powers: ∗ ( n ) = ∗ n [ n ] ! .

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals T HE RANK 1 CASE Let U be the Q ( q ) -algebra generated by E , F , K ± 1 such that EF − π FE = K − K − 1 KEK − 1 = q 2 E , KFK − 1 = q − 2 F , π q − q − 1 . The bar involution is given by K = K − 1 , q = π q − 1 . E = E , F = F , ◮ π = 1 � U q ( sl ( 2 )) ◮ π = − 1 � “quantum osp ( 1 | 2 ) ”

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals C OMPARING SUPER VS NON - SUPER There are many nice similarities that can be deduced without choosing π . ◮ U has a triangular decomposition U = � E � ⊗ � K ± 1 � ⊗ � F � . ◮ U is a Hopf (super)algebra. ◮ U has a quasi- R -matrix and Casimir-type element. ◮ U has semi-simple finite-dimensional modules. The commutation formulas are even almost the same: � K ; 2 i − n − m � E ( m ) F ( n ) = π mn − ( i + 1 2 ) F ( m − i ) � E ( n − i ) i i

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals C OMPARING SUPER VS NON - SUPER The largest difference is in their categories of f.d. modules. Facts : ◮ ( π = 1) there is a simple U q ( sl ( 2 )) -module of dimension n for each n ≥ 0. ◮ ( π = − 1) there is a simple “quantum osp ( 1 | 2 ) ”-module of dimension n for each odd n ≥ 0. This causes headaches for crystals.

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals A DJUSTING THE DEFINITION Fact (Zou): “quantum osp ( 1 | 2 ) ” has simple modules of all even dimensions if we extend the field to Q ( √ π, q ) . Q: How can we account for all modules over Q ( q ) ? A: Modify the definition of U to get an algebra U ′ where EF − π FE = π K − K − 1 π q − q − 1 When π = − 1, U ′ has only even-dimensional simples/ Q ( q ) !

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals G LUING Theorem (C-Wang) For π = − 1 , the algebra U = U ⊕ U ′ ◮ is a Hopf (super)algebra; ◮ has finite-dimensional simple modules of each dimension; ◮ has a semisimple finite dimensional representation theory; This has a trivial canonical basis for U − . But does the modified quantum group have a canonical basis?

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals M ODIFIED QUANTUM ALGEBRA Throw in idempotents 1 n to obtain a non-unital algebra. � ( 1 � 1 n ) n The algebra ˙ U is generated by 1 n , E 1 n , F 1 n such that 1 m 1 m = δ nm 1 n , 1 n + 2 E = E 1 n , 1 n − 2 F = F 1 n − 2 ( EF − π FE ) 1 n = [ n ] 1 n Theorem (C-Wang) The algebra ˙ U has a canonical basis � π ab F ( b ) 1 n E ( a ) : n ≥ a + b � E ( a ) 1 − n F ( b ) ,

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals M ISSING LINK U ′ U (?) U � 1 2 n U � 1 2 n + 1 ˙ ˙ ˙ U

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals E XPANDING THE C ARTAN The difference between U and U ′ is EF − π FE = π p K − K − 1 π q − q − 1 where p is the parity of the “allowed weights”. If “ K = q h ”, by analogy we define “ J = π h ”. Adding these elements, we obtain the definition in [C-Hill-Wang]:

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals D EFINITION [CHW1] Let g be a KM superalgebra, A its symmetrizable GCM. Let U q ( g ) be the Q ( q ) -algebra with generators E i , F i , K ± 1 , J i i such that 2 = 1 , J i J i K i = K i J i , J i J j = J j J i , K i K j = K j K i , J i E j J − 1 = π a ij E j , K i E j K − 1 = q a ij E j , i i J i F j J − 1 = π − a ij F j , K i F j K − 1 = q − 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 ; 1 − a ij 1 − a ij ( 1 − a ij − k ) ( 1 − a ij − k ) E j E ( k ) F j F ( k ) � ( − 1 ) k π p ( k ; i , j ) E � ( − 1 ) k π p ( k ; i , j ) F = = 0 , i i i i k = 0 k = 0 where p ( k ; i , j ) = kp ( i ) p ( j ) + 1 2 k ( k − 1 ) p ( i ) .

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals T HE BAR INVOLUTION AND COPRODUCT We extend q �→ π q − 1 to U q ( g ) by setting K i = J i K − 1 E i = E i , F i = F i , , J i = J i . i We can also define a (super) coproduct ∆ by ∆( E i ) = E i ⊗ K − d i + J d i i ⊗ E i i ∆( F i ) = F i ⊗ 1 + K d i i ⊗ F i ∆( K i ) = K i ⊗ K i ∆( J i ) = J i ⊗ J i

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals R EPRESENTATIONS Let P ( P + ) be the set of (dominant) weights of g . λ ∈ P M λ , where A weight module is a U q ( g ) -module M = � M λ = � � m ∈ M : K i m = q � h i ,λ � m , J i m = π � h i ,λ � m . We can define highest-weight and integrable modules as usual to obtain a category O int . Simple modules: V ( λ ) for all λ ∈ P +

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals P ROPERTIES Proposition [CHW1]. For π = ± 1, ◮ U q ( g ) = U + ⊗ U 0 ⊗ U − . ◮ U q ( g ) is a Hopf (super)algebra. ◮ There is a quasi-R-matrix and quantum Casimir element. ◮ Each M ∈ O int is completely reducible. The question remains: is there a canonical basis for U − ?

I NTRODUCTION B ACKGROUND The rank 1 case KM quantum supergroups Crystals K ASHIWARA OPERATORS There is a left derivation operator e ′ i and a bilinear form ( − , − ) on U − such that: ( F i x , y ) = ( x , e ′ ( 1 , 1 ) = 1 , i ( y )) . Each u ∈ U − can be written x = � F ( n ) x n such that e ′ i ( x n ) = 0. i We can define Kashiwara operators ˜ F ( n + 1 ) F ( n − 1 ) � � ˜ f i x = x n , e i x = x n . i i