SLIDE 1 Block Decomposition of a Class of Integrable Representations of Toroidal Lie Algebras
Tanusree Khandai
Indian Institute of Science Education and Research, Mohali
Interactions of quantum affine algebras with cluster algebras, current algebras and categorification
- Conference celebrating 60th birthday of Vyjayanthi Chari
SLIDE 2 Notations
Let g a complex finite-dimensional simple Lie algebra ; h a Cartan subalgebra of g. {αi : 1 ≤ i ≤ n} := simple roots of g, {ω1, · · · , ωn}:= fundamental weights of g, Qfin = n
i=1 Zαi the root lattice, Pfin = n i=1 Zωi, weight lattice and
P+
fin = n i=1 Z+ωi dominant integral weights of g
θ the highest root of g and θ∨ the corresponding co-root;
- gaff affine Kac-Moody algebra associated with g
haff a Cartan subalgebra of gaff (.|.) the Killing form on g; {αi : 1 ≤ 0 ≤ n} := simple roots of gaff {Λ1, · · · , Λn, Λ0} := fundamental weights of gaff Qaff = n
i=0 Zαi the root lattice Paff = n i=0 ZΛi, weight lattice and
P+
aff = n i=0 Z+Λi dominant integral weights of gaff
SLIDE 3 Notations
Let g a complex finite-dimensional simple Lie algebra ; h a Cartan subalgebra of g. {αi : 1 ≤ i ≤ n} := simple roots of g, {ω1, · · · , ωn}:= fundamental weights of g, Qfin = n
i=1 Zαi the root lattice, Pfin = n i=1 Zωi, weight lattice and
P+
fin = n i=1 Z+ωi dominant integral weights of g
θ the highest root of g and θ∨ the corresponding co-root;
- gaff affine Kac-Moody algebra associated with g
haff a Cartan subalgebra of gaff (.|.) the Killing form on g; {αi : 1 ≤ 0 ≤ n} := simple roots of gaff {Λ1, · · · , Λn, Λ0} := fundamental weights of gaff Qaff = n
i=0 Zαi the root lattice Paff = n i=0 ZΛi, weight lattice and
P+
aff = n i=0 Z+Λi dominant integral weights of gaff
SLIDE 4 Toroidal Lie Algebra
Definition A k-toroidal Lie algebra associated with g is a Lie algebra with underlying vector space Tk(g) := g ⊗ C[t±1
1 , · · · , t±k k ] ⊕ Dk ⊕ Z,
where, Dk is the space spanned by k derivations d1, · · · , dk, Z is an infinite-dimensional space spanned by Zk-graded central elements {tmci, m ∈ Zk, 1 ≤ i ≤ k}, together with the relation k
i=1 ritrci = 0; and
Lie bracket : [x ⊗ tm, y ⊗ ts] = [x, y] ⊗ tm+s +
k
mitm+sci (x|y), di(x ⊗ tm) = mix ⊗ tm, ∀ x ∈ g. Let htor := h ⊕ Dk ⊕ Cc1 ⊕ · · · ⊕ Cck where c1, · · · , ck are the zero graded central elements in Tk(g).
SLIDE 5 Toroidal Lie Algebra
Definition A k-toroidal Lie algebra associated with g is a Lie algebra with underlying vector space Tk(g) := g ⊗ C[t±1
1 , · · · , t±k k ] ⊕ Dk ⊕ Z,
where, Dk is the space spanned by k derivations d1, · · · , dk, Z is an infinite-dimensional space spanned by Zk-graded central elements {tmci, m ∈ Zk, 1 ≤ i ≤ k}, together with the relation k
i=1 ritrci = 0; and
Lie bracket : [x ⊗ tm, y ⊗ ts] = [x, y] ⊗ tm+s +
k
mitm+sci (x|y), di(x ⊗ tm) = mix ⊗ tm, ∀ x ∈ g. Let htor := h ⊕ Dk ⊕ Cc1 ⊕ · · · ⊕ Cck where c1, · · · , ck are the zero graded central elements in Tk(g).
SLIDE 6 Integrable Representation of Lie algebra
Definition
A Tk(g)-module V is said integrable if V = ⊕
µ∈h∗
tor
Vµ, where Vµ = {v ∈ V : h.v = µ(h)v, for all h ∈ h}. the root vectors corresponding to the real roots of Tk(g) act nilpotently on every non-zero vector of V.
For an integer k, let Ik
fin be the category of integral Tk(g)-modules with
finite-dimensional weight spaces.
SLIDE 7 Irreducible objects of I1
fin
For k = 1, Tk(g) = gaff (untwisted affine Kac-Moody algebra associated with g):
The simple objects of I1
fin on which the center acts trivially are of the form
V(
→
λ,
→
b , s) := Vb1(λ1) ⊗ · · · ⊗ Vbr(λr) ⊗ tsC[t±m
1
]
where,
→
λ = (λ1, · · · , λr) ∈ (P+
fin)r , →
b = (b1, · · · , br) ∈ (C×)r for r ∈ Z+, m ∈ Z+ and 0 ≤ s ≤ m − 1,
and the simple objects of I1
fin on which the center acts non-trivially are :
standard modules of the form X(Λ), with Λ ∈ P+
aff or
restricted duals of standard modules.
Here, for λ ∈ P+, b ∈ C×, Vb(λ) is the evaluation module for the loop algebra g ⊗ C[t±1
1 ], with underlying vector space V(λ), and g ⊗ C[t±1 1 ] action:
x ⊗ tr
1.v = brx.v,
∀ x ∈ g, v ∈ V(λ).
SLIDE 8 Irreducible objects of I1
fin
For k = 1, Tk(g) = gaff (untwisted affine Kac-Moody algebra associated with g):
The simple objects of I1
fin on which the center acts trivially are of the form
V(
→
λ,
→
b , s) := Vb1(λ1) ⊗ · · · ⊗ Vbr(λr) ⊗ tsC[t±m
1
]
where,
→
λ = (λ1, · · · , λr) ∈ (P+
fin)r , →
b = (b1, · · · , br) ∈ (C×)r for r ∈ Z+, m ∈ Z+ and 0 ≤ s ≤ m − 1,
and the simple objects of I1
fin on which the center acts non-trivially are :
standard modules of the form X(Λ), with Λ ∈ P+
aff or
restricted duals of standard modules.
Here, for λ ∈ P+, b ∈ C×, Vb(λ) is the evaluation module for the loop algebra g ⊗ C[t±1
1 ], with underlying vector space V(λ), and g ⊗ C[t±1 1 ] action:
x ⊗ tr
1.v = brx.v,
∀ x ∈ g, v ∈ V(λ).
SLIDE 9 Irreducible objects of I1
fin
For k = 1, Tk(g) = gaff (untwisted affine Kac-Moody algebra associated with g):
The simple objects of I1
fin on which the center acts trivially are of the form
V(
→
λ,
→
b , s) := Vb1(λ1) ⊗ · · · ⊗ Vbr(λr) ⊗ tsC[t±m
1
]
where,
→
λ = (λ1, · · · , λr) ∈ (P+
fin)r , →
b = (b1, · · · , br) ∈ (C×)r for r ∈ Z+, m ∈ Z+ and 0 ≤ s ≤ m − 1,
and the simple objects of I1
fin on which the center acts non-trivially are :
standard modules of the form X(Λ), with Λ ∈ P+
aff or
restricted duals of standard modules.
Here, for λ ∈ P+, b ∈ C×, Vb(λ) is the evaluation module for the loop algebra g ⊗ C[t±1
1 ], with underlying vector space V(λ), and g ⊗ C[t±1 1 ] action:
x ⊗ tr
1.v = brx.v,
∀ x ∈ g, v ∈ V(λ).
SLIDE 10 Block Decomposition of the category I1
fin[0]
Let I1
fin[0] be the subcategory of level zero objects of gaff .
- V. Chari, J. Greenstein, A. Moura
The objects of I1
fin[0] can be written as direct sum of indecomposables.
SLIDE 11 Block Decomposition of the category I1
fin[0]
Let I1
fin[0] be the subcategory of level zero objects of gaff .
- V. Chari, J. Greenstein, A. Moura
The objects of I1
fin[0] can be written as direct sum of indecomposables.
An indecomposable object in I1
fin[0] has finitely many simple constituents
which are non-trivial as modules over g ⊗ C[t1, t−1
1 ].
SLIDE 12 Block Decomposition of the category I1
fin[0]
Let I1
fin[0] be the subcategory of level zero objects of gaff .
- V. Chari, J. Greenstein, A. Moura
The objects of I1
fin[0] can be written as direct sum of indecomposables.
An indecomposable object in I1
fin[0] has finitely many simple constituents
which are non-trivial as modules over g ⊗ C[t1, t−1
1 ].
Given two irreducible gaff -modules, V(
→
λ,
→
b , s) and V(
→
µ,
→
a , p), with
→
λ,
→
µ ∈ (P+
fin)r and →
b ,
→
a ∈ (C×)r, there exists a sequence of indecomposable gaff -modules V1, V2, · · · , Vr in I1
fin[0] such that
Hom(Vi, Vi+1) = 0,
Hom(Vi+1, Vi) = 0, if and only if
→
b = s.
→
a for some s ∈ C∗ and
→
λ −
→
µ ∈ (Qfin)r.
SLIDE 13 Block Decomposition of the category I1
fin[0]
Let I1
fin[0] be the subcategory of level zero objects of gaff .
- V. Chari, J. Greenstein, A. Moura
The objects of I1
fin[0] can be written as direct sum of indecomposables.
An indecomposable object in I1
fin[0] has finitely many simple constituents
which are non-trivial as modules over g ⊗ C[t1, t−1
1 ].
The category I1
fin[0] decomposes into blocks which are parametrized by
- rbits for the natural action of the group C× on
Ξ := {f : C× → Pfin/Qfin : f(a) = 0 for all but finitely many a ∈ C×}. Blocks in I1
fin[0] ←
→
Equivalence classes of indecomposable objects in I1
fin[0] with respect to the equivalence relation ∼
defined as follows.
X ∼ Y if there exists a sequence of indecomposable modules X = X1, · · · , Xk = Y in I1
fin[0] such that for 1 ≤ i ≤ k − 1,
Hom(Xi, Xi+1) = 0,
SLIDE 14 Irreducible Modules in Ik
fin for k > 1
For k ≥ 2,
the simple objects in Ik
fin on which the central elements act trivially are of the form,
V(
→
λ,
→
a , s) := Va1(λ1) ⊗ · · · ⊗ Var(λr) ⊗ tsC[tG(
→
λ ,
→
a )]
where,
→
λ = (λ1, · · · , λr) ∈ (P+
fin)r , →
a = (a1, · · · , ar) ∈ ((C×)k)r for r ∈ Z+, G(
→
λ,
→
a ) is a subgroup of Zk of rank k and ts ∈ C[t±1
1
, · · · , t±1
k
]. Va(λ) is the evaluation module for g ⊗ C[t±1
1
, · · · , t±1
k
], with underlying vector space V(λ), with λ ∈ P+
fin, a ∈ (C×)k.
SLIDE 15 Irreducible Modules in Ik
fin for k > 1
For k ≥ 2,
the simple objects in Ik
fin on which the central elements act trivially are of the form,
V(
→
λ,
→
a , s) := Va1(λ1) ⊗ · · · ⊗ Var(λr) ⊗ tsC[tG(
→
λ ,
→
a )]
where,
→
λ = (λ1, · · · , λr) ∈ (P+
fin)r , →
a = (a1, · · · , ar) ∈ ((C×)k)r for r ∈ Z+, G(
→
λ,
→
a ) is a subgroup of Zk of rank k and ts ∈ C[t±1
1
, · · · , t±1
k
]. Upto an isomorphism, the irreducible objects of Ik
fin on which the center acts
non-trivially, are of the form X(
→
Λ,
→
m, r0) := Xm1(Λ1) ⊗ · · · Xmr(Λr) ⊗ tr0C[tG(
→
Λ ,
→
m)],
where Λ = (Λ1, · · · , Λr) ∈ P+
aff , →
m = (m1, · · · , mr) ∈ (max C[t±1
2
, · · · , t±1
k
])r, G(Λ,
→
m) is a subgroup of Zk−1 of rank k − 1 and tr0 ∈ C[t±1
2
, · · · , t±1
k
]. Xm(Λ) is the evaluation module for the algebra g′
aff ⊗ C[t±1 2
, · · · , t±1
k
], with underlying vector space X(Λ), with Λ ∈ P+
aff , m ∈ max C[t±1 2
, · · · , t±1
k
].
SLIDE 16
The Category J+
fin,k Let J+
fin,k be the full subcategory of positive level objects in Ik fin which have finite length.
SLIDE 17
The Category J+
fin,k Let J+
fin,k be the full subcategory of positive level objects in Ik fin which have finite length.
Irr(J+
fin,k) := { X( →
Λ,
→
m, r0) :
→
Λ ∈ (P+
aff )s, →
m ∈ (max C[t±1
2
, · · · , t±1
k
])s, r0 ∈ Zk−1}
SLIDE 18 The Category J+
fin,k Let J+
fin,k be the full subcategory of positive level objects in Ik fin which have finite length.
Irr(J+
fin,k) := { X( →
Λ,
→
m, r0) :
→
Λ ∈ (P+
aff )s, →
m ∈ (max C[t±1
2
, · · · , t±1
k
])s, r0 ∈ Zk−1} For X(
→
Λ,
→
m, r0) ∈ Irr(J+
fin,k) and m ∈ max C[t±1 2
, · · · , t±1
k
] let, supp(X(
→
Λ,
→
m, r0)) = {m1, · · · , mr} wt
X(
→
Λ ,
→
m,r0)(m) =
if Xm(Λ) is a tensor component of X(
→
Λ,
→
m, r0)
SLIDE 19 The Category J+
fin,k Let J+
fin,k be the full subcategory of positive level objects in Ik fin which have finite length.
Irr(J+
fin,k) := { X( →
Λ,
→
m, r0) :
→
Λ ∈ (P+
aff )s, →
m ∈ (max C[t±1
2
, · · · , t±1
k
])s, r0 ∈ Zk−1} For X(
→
Λ,
→
m, r0) ∈ Irr(J+
fin,k) and m ∈ max C[t±1 2
, · · · , t±1
k
] let, supp(X(
→
Λ,
→
m, r0)) = {m1, · · · , mr} wt
X(
→
Λ ,
→
m,r0)(m) =
if Xm(Λ) is a tensor component of X(
→
Λ,
→
m, r0)
Recall Λ ∈ Paff is of the form Λ = Λ(c1)Λ0 + Λ|h + (Λ|Λ0)δ1
SLIDE 20 The Category J+
fin,k
For X(
→
Λ,
→
m, r0) ∈ Irr(J+
fin,k) and m ∈ max C[t±1 2 , · · · , t±1 k ]
let, supp(X(
→
Λ,
→
m, r0)) = {m1, · · · , mr} wt
X(
→
Λ ,
→
m,r0)(m) =
if Xm(Λ) is a tensor component of X(
→
Λ,
→
m, r0)
Recall Λ ∈ Paff is of the form Λ = Λ(c1)Λ0 + Λ|h + (Λ|Λ0)δ1 Type I and Type II Irreducibles in J+
fin
X(
→
Λ,
→
m, r0) ∈ Irr(J+
fin,k) is said to be of
Type I, if wt
X(
→
Λ ,
→
m,r0)(m)(c1) < θ(θ∨) for all m ∈ supp(X( →
Λ,
→
m, r0)); Type II, otherwise.
SLIDE 21
Results
Using results by Adamovich on the irreducibility of the tensor product of a highest weight irreducible integrable gaff -module and the loop modules for gaff , we prove:
Result Let X(
→
Λ,
→
m, r0) be an irreducible Tk(g) of type I. Then (g ⊗ C[t±1
1 ]) ⊗ Xm(Λ) is an irreducible gaff -module for all
m ∈ supp(X(
→
Λ,
→
m, r0)). The local Weyl module corresponding to X(
→
Λ,
→
m, r0) is irreducible. Ext1(X(
→
Λ,
→
m, r0), X(
→
Λ′,
→
m′, s0)) = 0, if X(
→
Λ,
→
m, r0) ∼ =Tk(g) X(
→
Λ′,
→
m′, s0) Theorem 1. If V is an indecomposable Tk(g)-module in J+
fin,k, such that a type I irreducible
Tk(g)-module, X(
→
Λ,
→
m, r0), is an irreducible constituent of V, then every irreducible constituent of V is isomorphic to X(
→
Λ,
→
m, r0).
SLIDE 22
Results
Using results by Adamovich on the irreducibility of the tensor product of a highest weight irreducible integrable gaff -module and the loop modules for gaff , we prove:
Result Let X(
→
Λ,
→
m, r0) be an irreducible Tk(g) of type I. Then (g ⊗ C[t±1
1 ]) ⊗ Xm(Λ) is an irreducible gaff -module for all
m ∈ supp(X(
→
Λ,
→
m, r0)). The local Weyl module corresponding to X(
→
Λ,
→
m, r0) is irreducible. Ext1(X(
→
Λ,
→
m, r0), X(
→
Λ′,
→
m′, s0)) = 0, if X(
→
Λ,
→
m, r0) ∼ =Tk(g) X(
→
Λ′,
→
m′, s0) Theorem 1. If V is an indecomposable Tk(g)-module in J+
fin,k, such that a type I irreducible
Tk(g)-module, X(
→
Λ,
→
m, r0), is an irreducible constituent of V, then every irreducible constituent of V is isomorphic to X(
→
Λ,
→
m, r0).
SLIDE 23
Results
Using results by Adamovich on the irreducibility of the tensor product of a highest weight irreducible integrable gaff -module and the loop modules for gaff , we prove:
Result Let X(
→
Λ,
→
m, r0) be an irreducible Tk(g) of type I. Then (g ⊗ C[t±1
1 ]) ⊗ Xm(Λ) is an irreducible gaff -module for all
m ∈ supp(X(
→
Λ,
→
m, r0)). The local Weyl module corresponding to X(
→
Λ,
→
m, r0) is irreducible. Ext1(X(
→
Λ,
→
m, r0), X(
→
Λ′,
→
m′, s0)) = 0, if X(
→
Λ,
→
m, r0) ∼ =Tk(g) X(
→
Λ′,
→
m′, s0) Theorem 1. If V is an indecomposable Tk(g)-module in J+
fin,k, such that a type I irreducible
Tk(g)-module, X(
→
Λ,
→
m, r0), is an irreducible constituent of V, then every irreducible constituent of V is isomorphic to X(
→
Λ,
→
m, r0).
SLIDE 24
Results
Using results by Adamovich on the irreducibility of the tensor product of a highest weight irreducible integrable gaff -module and the loop modules for gaff , we prove:
Result Let X(
→
Λ,
→
m, r0) be an irreducible Tk(g) of type I. Then (g ⊗ C[t±1
1 ]) ⊗ Xm(Λ) is an irreducible gaff -module for all
m ∈ supp(X(
→
Λ,
→
m, r0)). The local Weyl module corresponding to X(
→
Λ,
→
m, r0) is irreducible. Ext1(X(
→
Λ,
→
m, r0), X(
→
Λ′,
→
m′, s0)) = 0, if X(
→
Λ,
→
m, r0) ∼ =Tk(g) X(
→
Λ′,
→
m′, s0) Theorem 1. If V is an indecomposable Tk(g)-module in J+
fin,k, such that a type I irreducible
Tk(g)-module, X(
→
Λ,
→
m, r0), is an irreducible constituent of V, then every irreducible constituent of V is isomorphic to X(
→
Λ,
→
m, r0).
SLIDE 25
Theorem 2. Let Tk(g) be a toroidal Lie algebra where the underlying finite-dimensional Lie algebra is of type An, Dn, E7, E8 or F4. Given two irreducible Tk(g)-modules of type II, X(
→
Λ,
→
m, r0), and X(
→
Λ′,
→
m′, s0), such that
→
m′ = c.
→
m for some c ∈ (C∗)k−1, (
→
Λ −
→
Λ′)(c1) = 0, and (
→
Λ −
→
Λ′)|h ∈ (Qfin)r, there exists a sequence X1, X2, · · · , Xr of irreducible Tk(g)-modules of type II, with Xr = X(
→
Λ′,
→
m′, s0) such that upto tensoring by one-dimensional modules X1 ∼ =Tk(g) X(
→
Λ,
→
m, r0) and for each 1 ≤ j ≤ r − 1, Exti(Xj, Xj+1) = 0 or Exti(Xj+1, Xj) = 0, for i = 0 or 1.
SLIDE 26
Theorem 2. Let Tk(g) be a toroidal Lie algebra where the underlying finite-dimensional Lie algebra is of type An, Dn, E7, E8 or F4. Given two irreducible Tk(g)-modules of type II, X(
→
Λ,
→
m, r0), and X(
→
Λ′,
→
m′, s0), such that
→
m′ = c.
→
m for some c ∈ (C∗)k−1, (
→
Λ −
→
Λ′)(c1) = 0, and (
→
Λ −
→
Λ′)|h ∈ (Qfin)r, there exists a sequence X1, X2, · · · , Xr of irreducible Tk(g)-modules of type II, with Xr = X(
→
Λ′,
→
m′, s0) such that upto tensoring by one-dimensional modules X1 ∼ =Tk(g) X(
→
Λ,
→
m, r0) and for each 1 ≤ j ≤ r − 1, Exti(Xj, Xj+1) = 0 or Exti(Xj+1, Xj) = 0, for i = 0 or 1.
SLIDE 27 References
- D. Adamovic, An application of U(g)-bimodules to representation theory of affine
Lie algebras, Algebra Representation Theory 7 (2004), no. 4, 457-469.
- V. Chari, Integrable representations of affine Lie-algebras, Invent. Math. 85
(1986), 317-335.
- V. Chari and J. Greenstein, Graded Level Zero Integrable Representations of
Affine Lie Algebras, Transactions of AMS. 360 (2008), no. 6, 2923-2940.
- V. Chari, T. Le, Representations of Double Affine Lie Algebras, In a Tribute to C.
- S. Seshadri (Chennai,2002) Trends Math., Birkhauser, Basel, 2003, 199-219.
- V. Chari, A. Moura, Spectral Characters of finite-dimensional representations of
Affine Algebras, J. Algebra 279 (2004), no.2, 820-839.
- V. Chari, A. Pressley, New unitary representations of loop algebras, Math. Ann.
275, (1986), no.1 87-104.
- V. Chari, A. Pressley, A new family of Irreducible, Integrable Modules for Affine Lie
algebras, Math. Ann. 277, (1987), 543-562.
- S. E. Rao, Classification of irreducible integrable modules for toroidal Lie algebras
with finite dimensional weight spaces, Journal of Algebra 277 (2004), 318-348.
SLIDE 28
Thank you !
