U q ( gl ( m | 1 )) and canonical bases Sean Clark Northeastern University / Max Planck Institute for Mathematics 2 nd US-Mexico Conference on Representation theory, Categorification, and Noncommutative Algebra Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 1 / 23
Q UANTUM ENVELOPING gl ( m | 1 ) Some history Q UANTUM ALGEBRAS AND CANONICAL BASES U q ( g ) : algebra over Q ( q ) coming from root data of simple Lie algebra g . ( ∼ 1990) Lusztig and Kashiwara: miraculous bases for U − q ( g ) = U q ( n − ) (CB1) B is a Z [ q , q − 1 ] -basis of the integral form A U − q ( g ) ; (CB2) For any b ∈ B , b = b ( · is natural involution q �→ q − 1 ); (CB3) PBW → B is q Z [ q ] -unitriangular for any PBW (CB4) B induces a basis on suitable modules Connections to geometry, combinatorics, categorification, . . . Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 2 / 23
Q UANTUM ENVELOPING gl ( m | 1 ) Some history Q UANTUM SUPERALGEBRAS Question: what if g is a Lie superalgebra? E.g. gl ( m | n ) : linear maps on vector superspace C m | n . Super complications: ◮ Different simple roots= Different U − (in general) Workaround: Work with a standard choice, e.g. α 1 = ǫ 1 − ǫ 2 , α 2 = ǫ 2 − ǫ 3 , . . . . ◮ Finite-dimensional representations = not semisimple (in general) ⊕ ⊂ V ⊗ t Workaround: Work with a nice subcategory, e.g. polynomial V ( λ ) vec ◮ “Chirality” in parameter q ; e.g. U q ( gl ( m | n )) 0 ∼ = U q ( gl ( m )) ⊗ U q − 1 ( gl ( n )) Workaround: Restrict rank; e.g. gl ( m | 1 ) Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 3 / 23
Q UANTUM ENVELOPING gl ( m | 1 ) Some history S OME KNOWN RESULTS Few examples of CBs known: ◮ easy small rank examples on U − ; e.g. gl ( 1 | 1 ) , standard gl ( 2 | 1 ) , osp ( 1 | 2 ) ◮ osp ( 1 | 2 n ) and anisotropic Kac-Moody super [C-Hill-Wang, ’13] ◮ Partial results for gl ( m | n ) , osp ( 2 | 2 n ) [C-Hill-Wang, ’13; Du-Gu ’14] Why partial? Problems for gl ( m | n ) : ◮ Only considers standard simple roots case; ◮ In standard case, PBW basis → B = (CB1), (CB2) ◮ If m > 1 and n > 1, this basis is not canonical! (Depends on PBW) ◮ When m = 1 or n = 1, get pseudo-canonical basis (a signed basis) Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 4 / 23
Q UANTUM ENVELOPING gl ( m | 1 ) Quantum gl ( m | 1 ) R OOT DATA FOR gl ( m | 1 ) � m ⊕ Z ǫ m + 1 with ( ǫ i , ǫ i ) = ( − 1 ) p ( ǫ i ) , P ∨ , �· , ·� as usual. P = i = 1 Z ǫ i ���� ���� even odd � � Φ = ǫ i − ǫ j | 1 ≤ i � = j ≤ m + 1 with simple roots Π = { α i | i ∈ I } Standard choice: Π std = { ǫ 1 − ǫ 2 , . . . , ǫ m − ǫ m + 1 } Note: ◮ odd roots are isotropic; e.g. ( ǫ m − ǫ m + 1 , ǫ m − ǫ m + 1 ) = 1 + − 1 = 0; ◮ no odd simple reflection! (Still obvious S m + 1 action: Weyl groupoid) ◮ different choices of simple roots may have different GCMs! 2 − 1 0 2 − 1 0 GCMs for m = 3: − 1 2 − 1 , − 1 0 1 0 − 1 0 0 1 0 � �� � � �� � standard choice Π= { ǫ 1 − ǫ 2 ,ǫ 2 − ǫ 4 ,ǫ 4 − ǫ 3 } Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 5 / 23
Q UANTUM ENVELOPING gl ( m | 1 ) Quantum gl ( m | 1 ) U q ( gl ( m | 1 )) � E i , F i , q h | i ∈ I , h ∈ P ∨ � Fix a choice of Π . We define U = U q (Π) = Q ( q ) subject to usual relations: e.g. q h i − q − h i q h E i q − h = q � h ,α i � E i , = E i F j − ( − 1 ) p ( i ) p ( j ) F j E i = δ ij [ E i , F j ] q − q − 1 � �� � super commutator and both usual and unusual Serre relations: if p ( i ) = 1, E 2 i = F 2 i = 0 , [ E i − 1 , [ E i , [ E i + 1 , E i ] q ] q ] q = 0 � �� � super q − commutators This has standard structural features (integral form, triangular decomposition, bar-involution, . . . ) NOTE: Different Π yield different U − = Q ( q ) � F i | i ∈ I � in general! Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 6 / 23
Q UANTUM ENVELOPING gl ( m | 1 ) Quantum gl ( m | 1 ) E XAMPLES FOR m = 2 � 2 � � 0 � − 1 1 Let A = B = both GCMs for gl ( 2 | 1 ) . , , − 1 0 1 0 U − ( A ) has generators F 1 , F 2 subject to the relation F 2 1 F 2 + F 2 F 2 1 = ( q + q − 1 ) F 1 F 2 F 1 ; F 2 2 = 0 1 F y So U − ( A ) has basis of form F x 2 F a 1 (where a ∈ Z ≥ 0 and x , y ≤ 1) U − ( B ) has generators F 1 , F 2 subject to the relations F 2 1 = F 2 2 = 0 . So U − ( B ) has basis e.g. F 1 F 2 F 1 F 2 F 1 F 2 . Different GCMs typically yield non-isomorphic half-quantum groups! Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 7 / 23
Q UANTUM ENVELOPING gl ( m | 1 ) Quantum gl ( m | 1 ) T HE GOAL Goal: Construct (non-signed!) canonical basis for U − = U − q (Π std ) Strategy: 1. Construct crystal on U − using Benkart-Kang-Kashiwara; 2. Globalize using pseudo-canonical basis ⇒ get canonical B satisfying (CB1), (CB2), and (CB4) in many cases 3. Prove (CB3) using a braid group action. Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 8 / 23
Results Crystals U - MODULES AND CRYSTALS As usual, can consider finite-dimensional weight representations. λ ∈ P + : weights in P which are gl ( m ) -dominant K ( λ ) : induced module from U q ( gl ( m )) -rep V gl ( m ) ( λ ) V ( λ ) : simple quotient Crystal basis is a pair ( L , B ) where ◮ L is a lattice over A ⊂ Q ( q ) (no poles at 0) ◮ B is a basis of L / qL ↔ nodes on a colored digraph e i , ˜ ◮ E i , F i � operators ˜ f i on L which, mod q, move along the arrows Theorem (Benkart-Kang-Kashiwara, Kwon) Let λ ∈ P + . Then K ( λ ) has a crystal basis ( L K ( λ ) , B K ( λ )) . If additionally λ is a polynomial, V ( λ ) admits a crystal basis ( L ( λ ) , B ( λ )) , combinatorially realized by super semistandard Young tableaux. Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 9 / 23
Results Crystals S UPER S EMISTANDARD T ABLEAUX FOR gl ( 2 | 1 ) � � Super alphabet: S = { 1 , 2 } ∪ 3 Semistandard tableaux in S means a Young diagram colored by S such that: ◮ even (odd) letters are (strictly) increasing along rows ◮ odd (even) letters are (strictly) increasing along columns “Read” tableaux from right-to-left and top-to-bottom to get element of V ⊗ t std The ˜ f i , ˜ e i act via the “tensor product rule” Vector representation: 1 1 2 ⇒ 2 ⊗ 1 ⊗ 1 ⊗ 3 ⊗ 3 α 1 α 2 3 1 2 3 3 Simple rule for odd root: Always apply ˜ f 2 to first 2 or 3 encountered. Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 10 / 23
Results Crystals C RYSTAL FOR U − Theorem (C) U − = U − (Π std ) admits a crystal basis B which is compatible with those on modules. Moreover, these crystals “globalize” to a basis satisfying (CB1), (CB2), and (CB4) for ◮ the half-quantum enveloping algebra U − ; ◮ the Kac modules K ( λ ) for any λ ∈ P + ; ◮ the simple modules V ( λ ) for polynomial λ ∈ P + . corresponds to 1 1 1 • • 2 • • • • • • • • • • • • • • • • • • • • • • • corresponds to 2 2 3 • 3 Crystal for U − q ( gl ( 2 | 1 )) Crystal for V ( 3 ǫ 1 + ǫ 2 ) Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 11 / 23
Results Crystals I DEAS IN PROOF ◮ Simplify [BKK] results: for the m ≥ n = 1 case, ◮ no upper crystal part ( ↔ gl ( n ) part); ◮ no fake highest weights: ˜ e i x = 0 for all i implies x = v λ ◮ not signed basis (odd operators never pass odd-colored boxes). ◮ Construct crystal inductively using (truncated) “grand loop” and [BKK]. ◮ Characterize lattice and (signed) basis with bilinear form. ◮ Construct integral form of lattice in usual way. ◮ Existence of globalizations follows from pseudo-canonical basis Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 12 / 23
Results Braid Group Action O DD REFLECTIONS To relate CB and PBW, want a braid group action (following Lusztig, Saito, Tingley). Non-super: automorphism T i : U q ( g ) → U q ( g ) “lifting s i action on weights” T i U ν ⊂ U s i ( ν ) We want an analogue for super, but there is a Problem: Odd isotropic simple root α i ⇒ no odd reflection s i ∈ W ! (There is a formal odd “reflection” in a Weyl groupoid .) Consequence: No odd braid auto morphism, but what about iso morphisms? Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 13 / 23
Results Braid Group Action P ERSPECTIVE ON BRAIDS : N ON - SUPER Usual definition of braid action: automorphism T i : U q ( g ) → U q ( g ) e.g. for U q ( sl ( 3 )) T 2 ( F 1 ) = F 2 F 1 − qF 1 F 2 ∈ U − q ( sl ( 3 )) s 2 · α 1 Interpretation: T i is weight-preserving translation between choices of simples Conjugacy of Borels ↔ presentation is “unique”; e.g. F ǫ 1 − ǫ 2 , F ǫ 2 − ǫ 3 , . . . Π = { ǫ 1 − ǫ 2 , ǫ 2 − ǫ 3 } F ǫ 1 − ǫ 3 , F ǫ 3 − ǫ 2 , . . . Π = { ǫ 1 − ǫ 3 , ǫ 3 − ǫ 2 } all essentially the same F 1 , F 2 , . . . Π = { α 1 , α 2 } Braid operator is automorphism sending e.g. T 2 ( F 1 ) = F 2 F 1 − qF 1 F 2 ⇒ T 2 ( F ǫ 1 − ǫ 2 ) = F ǫ 3 − ǫ 2 F ǫ 1 − ǫ 3 − qF ǫ 1 − ǫ 3 F ǫ 3 − ǫ 2 Sean Clark U q ( gl ( m | 1 )) and canonical bases June 1, 2016 14 / 23
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