Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory C ANONICAL BASES AND QUANTUM SHUFFLE SUPERALGEBRAS OF BASIC TYPE Sean Clark University of Virginia (joint with David Hill and Weiqiang Wang) arXiv:1310.7523 12/20/13
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory Q UANTUM GROUPS AND CANONICAL BASES Let g = n − ⊕ h ⊕ n be a simple Lie algebra. [Lusztig, Kashiwara]: U q ( n ) admits a canonical basis B ; that is, q = q − 1 ; ◮ B is invariant under q �→ ¯ ◮ B equals (any choice of) a PBW basis mod q ; ◮ B is orthonormal (mod q ) with respect to a bilinear form. This basis holds a remarkable amount of information about ◮ geometry; ◮ representation theory; ◮ categorification.
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory L IE SUPERALGEBRAS Question : do quantized Lie superalgebras admit canonical bases? (e.g. gl ( m | n ) , osp ( m | 2 n ) , Kac-Moody superalgebras) No adequate setting for geometric construction à la Lusztig Theorem (C-Hill-Wang) The half-quantum supergroup U q ( n ) associated to a Kac-Moody superalgebra with no isotropic roots admits a canonical basis. Example: osp ( 1 | 2 n ) (finite type). Sketch: ◮ Set a parameter π 2 = 1 and a bar involution ¯ q = π q − 1 ◮ Use π to interpolate between super and non-super ◮ Use Kashiwara’s algebraic (crystal) approach to obtain CB
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory B ASIC TYPE An important class are Lie superalgebras of basic type ( gl , osp , simple Lie algebras) ◮ Isotropic simple roots ⇒ no root strings ◮ In particular, Kashiwara’s algebraic strategy fails ◮ [Benkart-Kang-Kashiwara, Kwon] Many interesting gl ( m | n ) -modules admit crystal bases. ◮ [Khovanov] gl ( 1 | 2 ) admits a categorification Conclusion: We will have to try some new methods
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory A QUANTUM SHUFFLE APPROACH Non-super U q ( n ) has been studied using quantum shuffles [Ram-Lalonde, Rosso, Green, Leclerc, . . . ] Algebra structure � combinatorics of words. ◮ Order on simple roots induces lexicographic order. ◮ Certain words (dominant Lyndon) ↔ positive roots. ◮ Get distinguished bases from word combinatorics. Fact: [Leclerc] Canonical bases can be constructed using quantum shuffles
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory S UPER CANONICAL BASES Goal: Use quantum shuffles to construct PBW/canonical bases. For this, we need: ◮ a quantum shuffle presentation of the quantum group; ◮ to study super word combinatorics; ◮ a PBW basis for any order on roots; ◮ a suitable integral form; This is a nontrivial generalization: ◮ Super shuffles lack positivity. ◮ Super word combinatorics are less well behaved. ◮ Leclerc’s construction is not self-contained (need Lusztig’s PBW).
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory L IE S UPERALGEBRAS Let g = g ¯ 0 ⊕ g ¯ 1 be a simple Lie superalgebra of basic type ( A − G ). ◮ [ x , y ] = xy − ( − 1 ) p ( x ) p ( y ) yx . Fix a triangular decomposition g = n − ⊕ h ⊕ n + : 1 = { α ∈ h ∗ | g α � = 0 } ; ◮ Root system: � Φ = � 0 ⊔ � Φ ¯ Φ ¯ ◮ � 1 = � Φ iso ⊔ � Φ ¯ Φ n − iso ; ◮ Simple roots: Π = Π ¯ 0 ⊔ Π ¯ 1 = { α i | i ∈ I } , ◮ Dynkin diagram: (Γ , I ) , ◮ I = I ¯ 0 ⊔ I ¯ 1 = I iso ⊔ I n − iso . 1 , I ¯ Unlike Lie algebras, (Γ , I ) depends on h , and � Φ may be unreduced (e.g. type BC ). Reduced root system: Φ = { α ∈ � ∈ � Φ | 1 2 α / Φ }
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory (Types A-D) gl ( m | n ) � · · · � � · · · � ⊗ ⊗ � � ⊙ ⊙ · · · ⊙ ⊙ ⊙ ⊙ · · · ⊙ ⊙ osp ( 2 m + 1 | 2 n ) ⊙ · · · ⊙ ⊙ · · · ⊙ ⊙ ⊙ ⊙ > � ⊙ ⊙ · · · ⊙ ⊙ ⊙ < � � ✈ ✈ osp ( 2 n | 2 m ) ⊙ ⊙ · · · ⊙ ⊙ ⊙ ⊙ · · · ⊙ ⊙ ✈ ❍ ❍ ❍ � ⊗ ⊗ ✈ ✈ ⊙ · · · ⊙ ⊙ · · · ⊙ ✈ ⊙ ⊙ ⊙ ⊙ ❍ ❍ ❍ ⊗ ⊗
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory T HE HALF - QUANTUM SUPERGROUP Let U q = U q ( n + ) = Q ( q ) � e i | i ∈ I � ; ◮ This is a bialgebra under the multiplication: ( u ⊗ v )( x ⊗ y ) = ( − 1 ) p ( v ) p ( x ) q − ( | v | , | x | ) ( ux ⊗ vy ) , where | v | , | x | ∈ Q + = ⊕ i ∈ I Z + α i ; ◮ [Yamane] Subject to (exotic) Serre-type relations determined by subdiagrams of Γ . Example: � ⊗ � 1 2 3 e 1 e 2 e 3 e 2 + e 3 e 2 e 1 e 2 + e 2 e 1 e 2 e 3 + e 2 e 3 e 2 e 1 = ( q + q − 1 ) e 2 e 1 e 3 e 2
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory B ILINEAR FORM F = Q ( q ) � I � , the free algebra on I, i = ( i 1 i 2 . . . i d ) = i 1 · i 2 · · · i d , | i | = α i 1 + . . . + α i d There is a canonical surjection F − → U q , i �→ e i . Facts: [Lusztig, Yamane] ◮ U q ∼ = F / Rad ( · , · ) ; ◮ U q is equipped with a nondegenerate bilinear form ( · , · ) satisfying 1. ( e i , e j ) = δ ij ; 2. ( xy , z ) = ( x ⊗ y , ∆( z )) ; ◮ the bilinear form on U q ⊗ U q can be given by ( u ⊗ v , x ⊗ y ) = ( u , x )( v , y ) .
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory Q UANTUM SHUFFLE EMBEDDING Coproduct on F induces a product on F ∗ . Dualizing F ։ U q induces an injective homomorphism → F ∗ ∼ Ψ : U q ∼ = U ∗ q ֒ = ( F , ⋄ ) where ⋄ is the quantum shuffle product : ( i · i ) ⋄ ( j · j ) = ( i ⋄ ( j · j )) · i +( − 1 ) ( p ( i )+ p ( i )) p ( j ) q − ( | i | + α i ,α j ) (( i · i ) ⋄ j ) · j . (quantum shuffles approach is dual to Lusztig’s bialgebra approach) We shall study U q through its image U = Ψ( U q ) ⊂ F .
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory E XAMPLE : gl ( 3 | 2 ) ⊗ � � � 1 2 3 4 2 − 1 0 0 − 1 − 1 2 0 0 = { 1 , 2 , 4 } , 1 = I iso = { 3 } , A = I ¯ I ¯ 0 − 1 0 1 − 2 0 0 1 Ψ( e 1 e 2 ) = 1 ⋄ 2 = ( 21 ) + q ( 12 ) Ψ( e 3 e 3 ) = 3 ⋄ 3 = ( 33 ) + ( − q 0 )( 33 ) = 0 4 ⋄ ( 13 ) = ( 134 ) + q − 1 ( 143 ) + q − 1 ( 413 ) (In fact, ( 13 ) �∈ U : Ψ( e 1 e 3 ) = ( 13 ) + ( 31 ) )
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory T OWARDS PBW BASES Theorem: [Yamane] Any basic type Lie superalgebra admits a PBW basis for a particular (standard) ordering of the roots. But we need PBW bases associated to an arbitary ordering of the simple roots. Idea: Order on simple roots induces a lexicographic order on words. We can then use the combinatorics of the word basis of F .
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory C OMBINATORICS OF W ORDS F has a word basis W = ⊔ n ≥ 0 I n ⊂ F , so we can learn things about elements of U by writing them in this basis, e.g. Ψ( e i e j ) = i ⋄ j = ( ji ) + ( − 1 ) p ( i ) p ( j ) q − ( α i ,α j ) ( ij ) . ◮ Fix the lexicographic order on W relative to some ordering ( I , ≤ ) . ◮ Let W + be the set of dominant words , i.e words of the form i = max ( u ) for some u ∈ U . e.g. if i < j , then max ( i ⋄ j ) = ( ji ) .
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory M ONOMIAL BASIS Proposition: Let i = ( i 1 , . . . , i d ) ∈ W and ε i = Ψ( e i 1 . . . e i d ) . Then { ε i = i 1 ⋄ · · · ⋄ i d | i = ( i 1 , . . . , i d ) ∈ W + } is a basis of U . We want to refine this basis. Let L denote the set of Lyndon words : i = ( i 1 , . . . , i d ) ∈ L ⇔ i < ( i k , . . . , i d ) for k > 1 . Let L + = L ∩ W + .
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory D OMINANT L YNDON WORDS AND POSITIVE ROOTS Theorem: [C-Hill-Wang] 1. The map i = ( i 1 , . . . , i d ) �→ α i 1 + · · · + α i d = | i | is a bijection L + − → Φ + ; 2. Every i ∈ W + has a canonical factorization i = i 1 · · · i n , where i 1 , . . . , i n ∈ L + , i 1 ≥ · · · ≥ i n and i k > i k + 1 if | i k | ∈ Φ + iso . Sketch: ◮ Induct on height of roots ◮ Yamane’s PBW theorem gives dimensions of root spaces ◮ If no bijection, then ii ∈ W + with | i | ∈ Φ iso ◮ This yields a contradition.
Motivation Quantum Supergroups of Basic Type Lyndon Theory PBW Bases Canonical Bases Representation Theory E XAMPLE : gl ( 2 | 1 ) ⊗ � 1 2 L + = { ( 1 ) , ( 12 ) , ( 2 ) } W + = { ( 2 ) a ( 12 ) b ( 1 ) c : c ∈ N , a , b ∈ { 0 , 1 }} Words which are dominant: Words which are not dominant: ◮ (2121) ◮ (112) ◮ (21111) ◮ (22111) ◮ (121) ◮ (1212) ◮ (1111) ◮ (1121)
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