I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form A canonical basis for covering quantum groups Sean Clark Joint work with D. Hill and W. Wang University of Virginia AMS Fall Western Sectional Meeting University of California, Riverside November 2, 2013
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form Q UANTUM GROUPS q : a generic parameter; g : a Kac-Moody algebra with simple roots Π = { α i : i ∈ I } . U q ( g ) is the Q ( q ) algebra with generators E i , F i , K ± 1 for i ∈ I . i U q ( n − ) , the subalgebra generated by F i . U q ( n − ) has many interesting properties, e.g. ◮ Lusztig-Kashiwara canonical basis ; ◮ categorifications of Khovanov-Lauda and Rouquier; U q ( g ) admits a categorification for its modified form [L, KL, R].
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form H ALF QUANTUM SUPERGROUPS g : an anisotropic Kac-Moody superalgebra with Z / 2 Z -graded simple roots Π = Π 0 ⊔ Π 1 = { α i : i ∈ I } U q ( n − ) : algebra generated by F i satisfying super Serre relations. Was not expected to admit a canonical basis. Super KLR= quiver Hecke superalgebras (Ellis-Khovanov-Lauda in rank 1, Kang-Kashiwara-Tsuchioka independently defined the general construction) [Hill-Wang] U q ( n − ) is categorified by QHSA’s. ⇒ It has a categorical canonical basis. Is there an intrinsic canonical basis ` a la Lusztig, Kashiwara?
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form I NSIGHT FROM [HW] Anisotropic super and non-super are formally similar Key Insight [HW]: use a parameter π 2 = 1 for super signs ◮ π = 1 � non-super case. ◮ π = − 1 � super case. There is a bar involution on Q ( q ) π = Q ( q , π ) / ( π 2 − 1 ) given by ( π 2 = 1 ) q �→ π q − 1 and quantum integers [ n ] = ( π q ) n − q − n � n � [ n ] ! , ∈ Z [ q , q − 1 ] . π q − q − 1 , a giving U q ( n − ) a suitable bar-invariant integral form.
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form A NISOTROPIC KM 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 ) ◮ (anisotropy) a ij ∈ 2 Z for i ∈ I 1 We call these “of anisotropic type”. We will also impose: ◮ (bar-compatibility) d i ≡ 2 p ( i )
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form E XAMPLES ( • =odd root) • < ◦ ◦ · · · ◦ ◦ ◦ ( osp ( 1 | 2 n )) • < ◦ ◦ · · · ◦ ◦ < ◦ • < ◦ ◦ · · · ◦ ◦ > • ◦ ✈ ✈ ✈ • ◦ ◦ · · · ◦ ◦ ◦ < ✈ ❍ ❍ ❍ ❍ ◦ ◦ > • < ◦ • < ◦ • < > •
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form K NOWN FACTS ABOUT KM S UPER Quantized Lie superalgebras have been well studied (Benkart, Jeong, Kang, Kashiwara, Kwon, Musson, Zou, ...) Some key coincidences exist for anisotropic KM: ◮ osp ( 1 | 2 n ) reps “=” half of so ( 2 n + 1 ) reps (R.B. Zhang, Lanzmann) ◮ Over C ( q ) , U q ( osp ( 1 | 2 n )) miraculously has the missing reps. (Musson-Zou) [CW]: U q ( osp ( 1 | 2 )) / Q ( q ) can be tweaked to get all reps. EF − π FE = K − K − 1 π K − K − 1 or π q − q − 1 π q − q − 1 � �� � � �� � even h.w. odd h.w.
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form D EFINITION [CHW1] Let g be a KM superalgebra of anisotropic type, A its symmetrizable GCM. Let U = 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 i − K − d i J 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 M OTIVATION C OVERING QUANTUM GROUPS Modified Form R ANK 1 For U q ( osp ( 1 | 2 ))) 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 .) We call this covering quantum osp ( 1 | 2 ) or sl ( 2 ) ( π = 1 case)
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form 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 ) (NB. There is no ”covering sl ( n ) ” in this construction)
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form S TRUCTURES IN A COVERING QUANTUM GROUP U has all the nice features you could hope for: ◮ U = U − ⊗ U 0 ⊗ U + ; ◮ U − admits a nondegenerate bilinear form; ◮ there is a Hopf superalgebra structure (super sign �→ π ); ◮ there is a bar involution ( K �→ JK − 1 ); ◮ there is a quasi- R -matrix and Casimir-type operator;
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form R EPRESENTATIONS Let P ( P + ) be the set of (dominant) weights of g . A weight module is a U q ( g ) -module M = � λ ∈ P M λ , where � � 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 semi-simple category O int . Simple modules: V ( λ ) for all λ ∈ P + (Same character as in classical case)
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form C RYSTALS To construct a CB, we use the algebraic approach with crystals. Specifically, we construct a covering analogue for e i , ˜ ◮ Kashiwara operators ˜ f i ; ◮ the crystal lattice; ◮ the action of the q -Boson algebra; ◮ the polarizations (= deformed Shapovalov forms); ◮ the tensor product rule; Kashiwara’s grand loop argument can be extended to the covering case. Moreover, this crystal basis admits globalization to a canonical basis.
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form C ANONICAL B ASIS Theorem (C-Hill-Wang) U − and the integrable modules admit compatible canonical bases. Let B be the canonical basis of U − . ◮ If v λ is the highest weight vector of V ( λ ) , bv λ = 0 or is a CB element. ◮ B | π = 1 = the Lusztig-Kashiwara CB ◮ B is typically π -signed: b ∈ B implies π b ∈ B . Example: a ij = 0, p ( i ) = p ( j ) = 1 F i F j = π F j F i (Categorically: M is not isomorphic to its parity shift Π M .)
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form M ODIFIED F ORM Basic idea: 1 � � λ ∈ P 1 λ with 1 λ 1 η = δ λ,η 1 λ For x ∈ U , let | x | be the weight. ˙ U is the algebra on symbols x 1 λ = 1 λ + | x | x for x ∈ U , λ ∈ P satisfying J µ K ν 1 λ = π � µ,λ � q � ν,λ � 1 λ ( xy ) 1 λ = x 1 λ + | y | y 1 λ , Any weight U -module M is a ˙ U module: x 1 λ acts as projection to M λ followed by the U -action of x .
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form S OME PROPERTIES ˙ U has some additional useful properties: ◮ Automorphisms of U extend to ˙ U ; v . s . = U − ⊗ U + ; ◮ ˙ U 1 λ Theorem (C.) There is a non-degenerate symmetric bilinear form on ˙ U which: ◮ extends the form on U − ; ◮ is invariant under our favorite maps; ◮ is a limit of polarizations; For π = 1, this is Lusztig’s form on ˙ U .
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form R ANK 1 ˙ U q ( osp ( 1 | 2 )) is the algebra given by Generators: E 1 n = 1 n + 2 E , F 1 n = 1 n − 2 F , 1 n Relations: 1 n 1 m = δ nm 1 n and EF 1 n − FE 1 n = [ n ] 1 n Theorem (C-Wang) ˙ U q ( osp ( 1 | 2 )) admits a canonical basis � � E ( a ) 1 n F ( b ) , π ab F ( b ) 1 n E ( a ) | a + b ≥ n ˙ B = . (In rank 1, the basis need not be π -signed) Ellis and Lauda have categorified ˙ U q ( osp ( 1 | 2 )) .
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form C ONSTRUCTING THE CB ◮ ˙ U 1 λ − λ ′ projects “nicely” onto N ( λ, λ ′ ) (highest weight ⊗ lowest weight); ◮ N ( λ, λ ′ ) has a bar involution (Lusztig quasi- R -matrix); ◮ N ( λ, λ ′ ) admits a CB (bar involution + CB on simples); ◮ The CB of N ( λ, λ ′ ) is compatible with N ( λ + λ ′′ , λ ′′ + λ ′ ) ; These facts allow us to build a basis for ˙ U .
I NTRODUCTION M OTIVATION C OVERING QUANTUM GROUPS Modified Form C ANONICAL B ASIS Theorem (C) ˙ U admits a π -signed canonical basis generalizing the basis for U − . This basis is π -almost orthonormal under the bilinear form. For π = 1 , this specializes to Lusztig’s canonical basis for ˙ U | π = 1 .
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