Global Weyl modules and maximal parabolics of twisted affine Lie - - PowerPoint PPT Presentation
Motivation Background Realization of Maximal Parabolic Global Weyl Module A Global Weyl modules and maximal parabolics of twisted affine Lie algebras Matthew Lee Department of Mathematics University of California, Riverside Interactions
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
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 Aλ
For a simple Lie algebra g ⊃ h ∆ = {αi : i ∈ I} Φ+ = {
i∈I aiαi : ai ≥ 0 ∀i}
b = h ⊕
gα = h ⊕ n+ P+ dominant integral weights
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
g or g p parabolic s.s. Verma Module Category Op Parabolic Verma Module affine Global Weyl module Bimodule ???
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
g or g p parabolic 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 Aλ
g or g p parabolic 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 Aλ
Proposition (Chari–Pressley, 2001) Let V be an integrable U(g[t, t−1] ⊕ Cd)-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]), Aλ)-bimodule
Aλ = U(h[t, t−1])/AnnU(h[t,t−1])wλ
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
Proposition (Chari–Pressley, 2001) Let V be an integrable U(g[t, t−1] ⊕ Cd)-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]), Aλ)-bimodule
Aλ = U(h[t, t−1])/AnnU(h[t,t−1])wλ
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
Proposition (Chari–Pressley, 2001) Let V be an integrable U(g[t, t−1] ⊕ Cd)-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]), Aλ)-bimodule
Aλ = U(h[t, t−1])/AnnU(h[t,t−1])wλ
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
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 Aλ
g or g p parabolic s.s. Verma Module Parabolic Verma Module affine Global Weyl module (untwisted and twisted) (Untwisted) Global Weyl Module ???
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
g or g p parabolic s.s. Verma Module Parabolic Verma Module affine Global Weyl module (untwisted and twisted) (Untwisted) Global Weyl Module (Twisted)???
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
g =
k−1
gs, g0 is simple, and each gs, 1 ≤ s ≤ k − 1, is an irreducible g0-module k g g0 gk 2 A2n Bn Vg0(2θs
0)
2 A2n−1, n ≥ 2 Cn Vg0(θs
0)
2 Dn+1, n ≥ 3 Bn Vg0(θs
0)
2 E6 F4 Vg0(θs
0)
3 D4 G2 Vg0(θs
0)
σ : C[t, t−1] → C[t, t−1] by σ(f(t)) = f(ξ−1t), ξ a k-th root
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
g =
k−1
gs, g0 is simple, and each gs, 1 ≤ s ≤ k − 1, is an irreducible g0-module k g g0 gk 2 A2n Bn Vg0(2θs
0)
2 A2n−1, n ≥ 2 Cn Vg0(θs
0)
2 Dn+1, n ≥ 3 Bn Vg0(θs
0)
2 E6 F4 Vg0(θs
0)
3 D4 G2 Vg0(θs
0)
σ : C[t, t−1] → C[t, t−1] by σ(f(t)) = f(ξ−1t), ξ a k-th root
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
g =
k−1
gs, g0 is simple, and each gs, 1 ≤ s ≤ k − 1, is an irreducible g0-module k g g0 gk 2 A2n Bn Vg0(2θs
0)
2 A2n−1, n ≥ 2 Cn Vg0(θs
0)
2 Dn+1, n ≥ 3 Bn Vg0(θs
0)
2 E6 F4 Vg0(θs
0)
3 D4 G2 Vg0(θs
0)
σ : C[t, t−1] → C[t, t−1] by σ(f(t)) = f(ξ−1t), ξ a k-th root
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
The Maximal parabolic, pj =< x±
i
⊗ 1, x±
0 ⊗ t±1, x+ j
⊗ 1 >⊂ (g[t, t−1])σ Proposition (L.) For a simply laced Lie algebra g and some 0 < j ≤ n pj ≃ (g[t]σ)τ for some automorphism τ. Since the fixed points of gστ form a semisimple Lie algebra, we can define I0, ∆0, and P+
0 .
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
The Maximal parabolic, pj =< x±
i
⊗ 1, x±
0 ⊗ t±1, x+ j
⊗ 1 >⊂ (g[t, t−1])σ Proposition (L.) For a simply laced Lie algebra g and some 0 < j ≤ n pj ≃ (g[t]σ)τ for some automorphism τ. Since the fixed points of gστ form a semisimple Lie algebra, we can define I0, ∆0, and P+
0 .
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
The Maximal parabolic, pj =< x±
i
⊗ 1, x±
0 ⊗ t±1, x+ j
⊗ 1 >⊂ (g[t, t−1])σ Proposition (L.) For a simply laced Lie algebra g and some 0 < j ≤ n pj ≃ (g[t]σ)τ for some automorphism τ. Since the fixed points of gστ form a semisimple Lie algebra, we can define I0, ∆0, and P+
0 .
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
For λ ∈ P+
0 , W(λ) is generated by wλ with relations:
h.wλ = λ(h)wλ n+[t]στ.wλ = 0, (x−
i
⊗ 1)λ(hi)+1.wλ = 0. For λ ∈ P+
0 , W(λ) is a (U(g[t]στ), Aλ)-bimodule.
Aλ = U(h[t]στ)/AnnU(h[t]στ )wλ
To obtain a better description of W(λ) we need to describe Aλ
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
For λ ∈ P+
0 , W(λ) is generated by wλ with relations:
h.wλ = λ(h)wλ n+[t]στ.wλ = 0, (x−
i
⊗ 1)λ(hi)+1.wλ = 0. For λ ∈ P+
0 , W(λ) is a (U(g[t]στ), Aλ)-bimodule.
Aλ = U(h[t]στ)/AnnU(h[t]στ )wλ
To obtain a better description of W(λ) we need to describe Aλ
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
For λ ∈ P+
0 , W(λ) is generated by wλ with relations:
h.wλ = λ(h)wλ n+[t]στ.wλ = 0, (x−
i
⊗ 1)λ(hi)+1.wλ = 0. For λ ∈ P+
0 , W(λ) is a (U(g[t]στ), Aλ)-bimodule.
Aλ = U(h[t]στ)/AnnU(h[t]στ )wλ
To obtain a better description of W(λ) we need to describe Aλ
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
For λ ∈ P+
0 , W(λ) is generated by wλ with relations:
h.wλ = λ(h)wλ n+[t]στ.wλ = 0, (x−
i
⊗ 1)λ(hi)+1.wλ = 0. For λ ∈ P+
0 , W(λ) is a (U(g[t]στ), Aλ)-bimodule.
Aλ = U(h[t]στ)/AnnU(h[t]στ )wλ
To obtain a better description of W(λ) we need to describe Aλ
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
Theorem (L.) Aλ/ Jac(Aλ) ≃ C[Pi,ri : ri ≤ min{λ(hi), λ(h0)}]/ < P1,r1 · · · Pn,rn :
ai(α0)ri ≥ λ(h0) + 1 > If ai(0) ≤ 1 ∀i ∈ I0 then Jac(Aλ) = 0.
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
Theorem (L.) Aλ/ Jac(Aλ) ≃ C[Pi,ri : ri ≤ min{λ(hi), λ(h0)}]/ < P1,r1 · · · Pn,rn :
ai(α0)ri ≥ λ(h0) + 1 > If ai(0) ≤ 1 ∀i ∈ I0 then Jac(Aλ) = 0.
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
W(λ) is irreducible iff {i ∈ I0 : λ(hi) > 0} ∪ {i ∈ I0 : ai(α0) = ai(θkaj(α0) − 1)} The following are equivalent:
1
W(λ) is finite-dimensional
2
Aλ is finite-dimensional
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
W(λ) is irreducible iff {i ∈ I0 : λ(hi) > 0} ∪ {i ∈ I0 : ai(α0) = ai(θkaj(α0) − 1)} The following are equivalent:
1
W(λ) is finite-dimensional
2
Aλ is finite-dimensional
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
W(λ) is irreducible iff {i ∈ I0 : λ(hi) > 0} ∪ {i ∈ I0 : ai(α0) = ai(θkaj(α0) − 1)} The following are equivalent:
1
W(λ) is finite-dimensional
2
Aλ is finite-dimensional
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
W(λ) is irreducible iff {i ∈ I0 : λ(hi) > 0} ∪ {i ∈ I0 : ai(α0) = ai(θkaj(α0) − 1)} The following are equivalent:
1
W(λ) is finite-dimensional
2
Aλ is finite-dimensional
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
Repeat for Yangians
Matthew Lee Global Weyl modules
Motivation Background Realization of Maximal Parabolic Global Weyl Module Aλ
Matthew Lee Global Weyl modules
References For Further Reading
J.E. Humpheys Introduction to Lie Algebras and Representation Theory. Springer, 1968.
Weyl Modules for classical and quantum affine algebras Representation Theory 5 (2001), 191-223 (electronic)
. Senesi Global Weyl modules for the twisted loop algebra.
Matthew Lee Global Weyl modules
References For Further Reading
. Senesi Weyl modules for the twisted loop algebras
G Fourier and D. Kus Demazure modules and Weyl modules: The twisted current case.
Matthew Lee Global Weyl modules