Global Weyl modules and maximal parabolics of twisted affine Lie - PowerPoint PPT Presentation

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Global Weyl modules and maximal parabolics of twisted affine Lie algebras Matthew Lee Department of Mathematics University of California, Riverside Interactions of quantm affine algebras with cluster algebras, current algebras and categorification June 5, 2018 Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ For a simple Lie algebra g ⊃ h ∆ = { α i : i ∈ I } Φ + = { � i ∈ I a i α i : a i ≥ 0 ∀ i } � g α = h ⊕ n + b = h ⊕ α ∈ Φ + P + dominant integral weights Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Universal highest modules g or � g p parabolic Category O p s.s. Verma Module Parabolic Verma Module affine Global Weyl module ??? Bimodule Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Universal highest modules g or � p parabolic g s.s. Verma Module Parabolic Verma Module affine Global Weyl module ??? Bimodule Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Universal highest modules g or � p parabolic g s.s. Verma Module Parabolic Verma Module affine Global Weyl module ??? Bimodule Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Affine Lie algebras Proposition (Chari–Pressley, 2001) Let V be an integrable U ( g [ t , t − 1 ] ⊕ C d ) -module generated by a non-zero element v ∈ V + λ . Then V is a quotient of W ( λ ) , the global Weyl module. W ( λ ) is a ( U ( g [ t , t − 1 ]) , A λ ) -bimodule A λ = U ( h [ t , t − 1 ]) / Ann U ( h [ t , t − 1 ]) w λ Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Affine Lie algebras Proposition (Chari–Pressley, 2001) Let V be an integrable U ( g [ t , t − 1 ] ⊕ C d ) -module generated by a non-zero element v ∈ V + λ . Then V is a quotient of W ( λ ) , the global Weyl module. W ( λ ) is a ( U ( g [ t , t − 1 ]) , A λ ) -bimodule A λ = U ( h [ t , t − 1 ]) / Ann U ( h [ t , t − 1 ]) w λ Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Affine Lie algebras Proposition (Chari–Pressley, 2001) Let V be an integrable U ( g [ t , t − 1 ] ⊕ C d ) -module generated by a non-zero element v ∈ V + λ . Then V is a quotient of W ( λ ) , the global Weyl module. W ( λ ) is a ( U ( g [ t , t − 1 ]) , A λ ) -bimodule A λ = U ( h [ t , t − 1 ]) / Ann U ( h [ t , t − 1 ]) w λ Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Universal highest modules g or � g p parabolic s.s. Verma Module Parabolic Verma Module affine Global Weyl module ??? (untwisted and twisted) Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Universal highest modules g or � p parabolic g s.s. Verma Module Parabolic Verma Module affine Global Weyl module (Untwisted) Global (untwisted and twisted) Weyl Module ??? Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Universal highest modules g or � p parabolic g s.s. Verma Module Parabolic Verma Module affine Global Weyl module (Untwisted) Global (untwisted and twisted) Weyl Module (Twisted)??? Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Maximal parabolic k − 1 � g = g s , g 0 is simple, and each g s , 1 ≤ s ≤ k − 1, is an s = 0 irreducible g 0 -module k g g 0 g k V g 0 ( 2 θ s 2 A 2 n B n 0 ) V g 0 ( θ s 2 A 2 n − 1 , n ≥ 2 C n 0 ) V g 0 ( θ s 2 D n + 1 , n ≥ 3 B n 0 ) V g 0 ( θ s 2 E 6 F 4 0 ) V g 0 ( θ s 3 D 4 G 2 0 ) σ : C [ t , t − 1 ] → C [ t , t − 1 ] by σ ( f ( t )) = f ( ξ − 1 t ) , ξ a k -th root of unity Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Maximal parabolic k − 1 � g = g s , g 0 is simple, and each g s , 1 ≤ s ≤ k − 1, is an s = 0 irreducible g 0 -module k g g 0 g k V g 0 ( 2 θ s 2 A 2 n B n 0 ) V g 0 ( θ s 2 A 2 n − 1 , n ≥ 2 C n 0 ) V g 0 ( θ s 2 D n + 1 , n ≥ 3 B n 0 ) V g 0 ( θ s 2 E 6 F 4 0 ) V g 0 ( θ s 3 D 4 G 2 0 ) σ : C [ t , t − 1 ] → C [ t , t − 1 ] by σ ( f ( t )) = f ( ξ − 1 t ) , ξ a k -th root of unity Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Maximal parabolic k − 1 � g = g s , g 0 is simple, and each g s , 1 ≤ s ≤ k − 1, is an s = 0 irreducible g 0 -module k g g 0 g k V g 0 ( 2 θ s 2 A 2 n B n 0 ) V g 0 ( θ s 2 A 2 n − 1 , n ≥ 2 C n 0 ) V g 0 ( θ s 2 D n + 1 , n ≥ 3 B n 0 ) V g 0 ( θ s 2 E 6 F 4 0 ) V g 0 ( θ s 3 D 4 G 2 0 ) σ : C [ t , t − 1 ] → C [ t , t − 1 ] by σ ( f ( t )) = f ( ξ − 1 t ) , ξ a k -th root of unity Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Realization The Maximal parabolic, p j = < x ± ⊗ 1 , x ± 0 ⊗ t ± 1 , x + ⊗ 1 > ⊂ ( g [ t , t − 1 ]) σ i j Proposition (L.) For a simply laced Lie algebra g and some 0 < j ≤ n p j ≃ ( g [ t ] σ ) τ for some automorphism τ . Since the fixed points of g στ form a semisimple Lie algebra, we can define I 0 , ∆ 0 , and P + 0 . Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Realization The Maximal parabolic, p j = < x ± ⊗ 1 , x ± 0 ⊗ t ± 1 , x + ⊗ 1 > ⊂ ( g [ t , t − 1 ]) σ i j Proposition (L.) For a simply laced Lie algebra g and some 0 < j ≤ n p j ≃ ( g [ t ] σ ) τ for some automorphism τ . Since the fixed points of g στ form a semisimple Lie algebra, we can define I 0 , ∆ 0 , and P + 0 . Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Realization The Maximal parabolic, p j = < x ± ⊗ 1 , x ± 0 ⊗ t ± 1 , x + ⊗ 1 > ⊂ ( g [ t , t − 1 ]) σ i j Proposition (L.) For a simply laced Lie algebra g and some 0 < j ≤ n p j ≃ ( g [ t ] σ ) τ for some automorphism τ . Since the fixed points of g στ form a semisimple Lie algebra, we can define I 0 , ∆ 0 , and P + 0 . Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Global Weyl Module For λ ∈ P + 0 , W ( λ ) is generated by w λ with relations: n + [ t ] στ . w λ = 0 , ( x − ⊗ 1 ) λ ( h i )+ 1 . w λ = 0 . h . w λ = λ ( h ) w λ i For λ ∈ P + 0 , W ( λ ) is a ( U ( g [ t ] στ ) , A λ ) -bimodule. A λ = U ( h [ t ] στ ) / Ann U ( h [ t ] στ ) w λ To obtain a better description of W ( λ ) we need to describe A λ Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Global Weyl Module For λ ∈ P + 0 , W ( λ ) is generated by w λ with relations: n + [ t ] στ . w λ = 0 , ( x − ⊗ 1 ) λ ( h i )+ 1 . w λ = 0 . h . w λ = λ ( h ) w λ i For λ ∈ P + 0 , W ( λ ) is a ( U ( g [ t ] στ ) , A λ ) -bimodule. A λ = U ( h [ t ] στ ) / Ann U ( h [ t ] στ ) w λ To obtain a better description of W ( λ ) we need to describe A λ Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Global Weyl Module For λ ∈ P + 0 , W ( λ ) is generated by w λ with relations: n + [ t ] στ . w λ = 0 , ( x − ⊗ 1 ) λ ( h i )+ 1 . w λ = 0 . h . w λ = λ ( h ) w λ i For λ ∈ P + 0 , W ( λ ) is a ( U ( g [ t ] στ ) , A λ ) -bimodule. A λ = U ( h [ t ] στ ) / Ann U ( h [ t ] στ ) w λ To obtain a better description of W ( λ ) we need to describe A λ Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ Global Weyl Module For λ ∈ P + 0 , W ( λ ) is generated by w λ with relations: n + [ t ] στ . w λ = 0 , ( x − ⊗ 1 ) λ ( h i )+ 1 . w λ = 0 . h . w λ = λ ( h ) w λ i For λ ∈ P + 0 , W ( λ ) is a ( U ( g [ t ] στ ) , A λ ) -bimodule. A λ = U ( h [ t ] στ ) / Ann U ( h [ t ] στ ) w λ To obtain a better description of W ( λ ) we need to describe A λ Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ A λ Theorem (L.) A λ / Jac ( A λ ) ≃ C [ P i , r i : r i ≤ min { λ ( h i ) , λ ( h 0 ) } ] / � < P 1 , r 1 · · · P n , r n : a i ( α 0 ) r i ≥ λ ( h 0 ) + 1 > i ∈ I 0 If a i ( 0 ) ≤ 1 ∀ i ∈ I 0 then Jac ( A λ ) = 0. Matthew Lee Global Weyl modules

Motivation Background Realization of Maximal Parabolic Global Weyl Module A λ A λ Theorem (L.) A λ / Jac ( A λ ) ≃ C [ P i , r i : r i ≤ min { λ ( h i ) , λ ( h 0 ) } ] / � < P 1 , r 1 · · · P n , r n : a i ( α 0 ) r i ≥ λ ( h 0 ) + 1 > i ∈ I 0 If a i ( 0 ) ≤ 1 ∀ i ∈ I 0 then Jac ( A λ ) = 0. Matthew Lee Global Weyl modules