U q ( gl ( m | 1 )) and canonical bases Sean Clark Northeastern - PowerPoint PPT Presentation

U q ( gl ( m | 1 )) and canonical bases Sean Clark Northeastern University / Max Planck Institute for Mathematics Algebraic Groups, Quantum Groups and Geometry Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 1 / 12

Q UANTUM ENVELOPING gl ( m | 1 ) Canonical bases Q UANTUM ALGEBRAS AND CANONICAL BASES U q ( g ) : algebra over Q ( q ) coming from root data of simple Lie algebra g . Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 2 / 12

Q UANTUM ENVELOPING gl ( m | 1 ) Canonical bases 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 simple modules Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 2 / 12

Q UANTUM ENVELOPING gl ( m | 1 ) Canonical bases Q UANTUM SUPERALGEBRAS Question: what if g is a Lie superalgebra? Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 3 / 12

Q UANTUM ENVELOPING gl ( m | 1 ) Canonical bases Q UANTUM SUPERALGEBRAS Question: what if g is a Lie superalgebra? Few examples of CBs known: ◮ 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] Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 3 / 12

Q UANTUM ENVELOPING gl ( m | 1 ) Canonical bases Q UANTUM SUPERALGEBRAS Question: what if g is a Lie superalgebra? Few examples of CBs known: ◮ 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: ◮ Different simple roots= Different U − (in general) Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 3 / 12

Q UANTUM ENVELOPING gl ( m | 1 ) Canonical bases Q UANTUM SUPERALGEBRAS Question: what if g is a Lie superalgebra? Few examples of CBs known: ◮ 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: ◮ Different simple roots= Different U − (in general) Workaround: in standard case, PBW basis → B = (CB1), (CB2) Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 3 / 12

Q UANTUM ENVELOPING gl ( m | 1 ) Canonical bases Q UANTUM SUPERALGEBRAS Question: what if g is a Lie superalgebra? Few examples of CBs known: ◮ 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: ◮ Different simple roots= Different U − (in general) Workaround: in standard case, PBW basis → B = (CB1), (CB2) ◮ if m > 1 and n > 1, this basis is not canonical! Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 3 / 12

Q UANTUM ENVELOPING gl ( m | 1 ) Canonical bases Q UANTUM SUPERALGEBRAS Question: what if g is a Lie superalgebra? Few examples of CBs known: ◮ 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: ◮ Different simple roots= Different U − (in general) Workaround: in standard case, PBW basis → B = (CB1), (CB2) ◮ if m > 1 and n > 1, this basis is not canonical! Workaround: when m = 1 or n = 1: canonical signed basis. Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 3 / 12

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 } Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 4 / 12

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; Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 4 / 12

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) Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 4 / 12

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! Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 4 / 12

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 May 25, 2016 4 / 12

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 Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 5 / 12

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 Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 5 / 12

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 May 25, 2016 5 / 12

Q UANTUM ENVELOPING gl ( m | 1 ) Quantum gl ( m | 1 ) Q UANTUM SUPERALGEBRAS Theorem (C) U − (Π std ) has a canonical basis B ; that is, a basis satisfying (CB1)-(CB3), and (CB4) for most (but not all ∗ ) finite-dimensional simple modules. Main strokes of proof: Sean Clark U q ( gl ( m | 1 )) and canonical bases May 25, 2016 6 / 12