Sachin S. Sharma - The Catholic University of America · Sachin S. Sharma Integrable modules for...

Integrable modules for Lie tori

Sachin S. Sharma

IIT Kanpur

(Joint work with S. Eswara Rao)

June 07, 2018

Sachin S. Sharma () Integrable modules for Lie tori June 07, 2018 1 / 22

Multiloop algebras

Let g be a simple finite dimensional Lie algebra and let (·|·) be anon-degenerate symmetric bilinear form on g.

Fix a positive integer n and let σ0, σ1, · · · , σn be commuting finiteorder automorphisms of g of order m0,m1, · · · ,mn respectively.

Let m = (m1, · · · ,mn), k = (k1, · · · , kn) and l = (l1, · · · , ln) arevectors in Zn. Let Γ = m1Z⊕ · · · ⊕mnZ and Γ0 = m0Z. LetΛ = Zn/Γ and Λ0 = Z/Γ0. Let k , l denote the images in Λ. For anyintegers k0 and l0, let k0 and l0 denote images in Λ0.

Multiloop algebras

Some notations

LetA = C[t±1

0 , · · · , t±1n ],

An = C[t±11 , · · · , t±1

A(m) = C[t±m11 , · · · , t±mn

n ], and

A(m0,m) = C[t±m00 , · · · , t±mn

be Laurent polynomial algebras with respective variables.

For 0 ≤ i ≤ n, let ξi denote a mi -th primitive root of unity. Let

g(k0, k) = X ∈ g |σi X = ξkii X , 0 ≤ i ≤ n.

Then⊕

(k0,k)∈Zn+1

g(k0, k)tk00 tk is called multiloop algebra.

Some notations

LetA = C[t±1

0 , · · · , t±1n ],

An = C[t±11 , · · · , t±1

A(m) = C[t±m11 , · · · , t±mn

n ], and

A(m0,m) = C[t±m00 , · · · , t±mn

Then⊕

(k0,k)∈Zn+1

Some notations

LetA = C[t±1

0 , · · · , t±1n ],

An = C[t±11 , · · · , t±1

A(m) = C[t±m11 , · · · , t±mn

n ], and

A(m0,m) = C[t±m00 , · · · , t±mn

Then⊕

(k0,k)∈Zn+1

Lie torusLet g1 be any Lie algebra and h1 be its finite dimensionalad-diagonalizable subalgebra. We set for α ∈ h∗1,

g1,α = x ∈ g1 | [h, x ] = α(h)x , h ∈ h∗1.

Then we haveg1 =

⊕α∈h∗1

g1,α.

Let ∆(g1, h1) = α ∈ h∗1 | g1,α 6= 0 which includes 0. Let∆×(g1, h1) = ∆(g1, h1)\0.For a finite dimensional simple Lie algebra g1, ∆×1 = ∆(g1, h1)× is anirreducible reduced finite root system with atmost two root lenths. Let∆×1,sh denote the set of non-zero short roots.

∆×1,en =

∆×1 ∪ 2∆×1,sh, if ∆×1 of typeBl

∆×1 , otherwise

A finite dimensional g1 module is said to satisfy condition (M) if V isirreducible of dimension> 1 and weight of V relative to h1 is contained in∆1,en .

Definition (ABFP - 2009)

A multi-loop algebra⊕

(k0,k)∈Zn+1

g(k0, k)tk00 tk is called a Lie torus and

denoted by LT if

1 g(0, 0) is a simple Lie algebra.

2 As g(0, 0) module, each g(k0, k) = U(k0, k)⊕ V (k0, k) whereU(k0, k) is trivial module and either V (k0, k) is zero or satisfy theproperty (M).

3 |〈σ0, . . . , σn〉| =∏n

i=0 |σi |.

(k0,k)∈Zn+1

denoted by LT if

3 |〈σ0, . . . , σn〉| =∏n

i=0 |σi |.

(k0,k)∈Zn+1

denoted by LT if

3 |〈σ0, . . . , σn〉| =∏n

i=0 |σi |.

(k0,k)∈Zn+1

denoted by LT if

3 |〈σ0, . . . , σn〉| =∏n

i=0 |σi |.

(k0,k)∈Zn+1

denoted by LT if

3 |〈σ0, . . . , σn〉| =∏n

i=0 |σi |.

Universal central extension of Lie torus

Let LT =⊕

(k0,k)∈Zn+1

g(k0, k)tk00 tk be a Lie torus.

Let ΩA be the vector space spanned by symbolstk00 tkKi , 0 ≤ i ≤ n, k0 ∈ Z and k ∈ Zn. Let dA be the subspace of ΩA

spanned byn∑

ki tk00 tkKi . Let Z = ΩA/dA.

Similarly define Z (n) = ΩAn/dAn, Z (m) = ΩA(m)/dA(m) andZ (m0,m) = ΩA(m0,m)/dA(m0,m).

Let LT =⊕

(k0,k)∈Zn+1

Let ΩA be the vector space spanned by symbolstk00 tkKi , 0 ≤ i ≤ n, k0 ∈ Z and k ∈ Zn. Let dA be the subspace of ΩA

spanned byn∑

ki tk00 tkKi . Let Z = ΩA/dA.

Let LT =⊕

(k0,k)∈Zn+1

Let ΩA be the vector space spanned by symbolstk00 tk Ki , 0 ≤ i ≤ n, k0 ∈ Z and k ∈ Zn. Let dA be the subspace of ΩA

spanned byn∑

ki tk00 tk Ki . Let Z = ΩA/dA.

Let LT =⊕

(k0,k)∈Zn+1

Let ΩA be the vector space spanned by symbolstk00 tk Ki , 0 ≤ i ≤ n, k0 ∈ Z and k ∈ Zn. Let dA be the subspace of ΩA

spanned byn∑

ki tk00 tk Ki . Let Z = ΩA/dA.

Let LT = LT ⊕ Z (m0,m). Define a Lie algebra structure on LT by

(a) [X (k0, k),Y (l0, l)] = [X ,Y ](k0 + l0, k + l) + (X |Y )n∑

ki tk0+l00 tk+l Ki ;

(b) Z (m0,m) is central in LT .

To see above bracket is closed, notice that (X |Y ) 6= 0⇒ k + l ∈ Γand k0 + l0 ∈ Γ0. This follows from the standard fact that (·|·) isinvariant under σi , 0 ≤ i ≤ n.

Proposition (J.Sun - 2009)

LT is the universal central extension of LT .

Both LT and LT are naturally Zn+1 graded. To reflect this fact weadd derivations. Let D be the space spanned by d0, d1, · · · , dn.

Let∼

LT = LT ⊕ D. Extend the Lie bracket in the following way:

[di ,X (k0, k)] = ki X (k0, k);

[di , tk00 tkKj ] = ki t

k00 tkKj ;

[di , dj ] = 0.

The purpose of this work is to classify irreducible integrable weight

modules for∼

Let∼

[di ,X (k0, k)] = ki X (k0, k);

k00 tkKj ;

[di , dj ] = 0.

modules for∼

Let∼

LT = LT ⊕ D.

Extend the Lie bracket in the following way:

[di ,X (k0, k)] = ki X (k0, k);

k00 tkKj ;

[di , dj ] = 0.

modules for∼

Let∼

[di ,X (k0, k)] = ki X (k0, k);

[di , tk00 tk Kj ] = ki t

k00 tk Kj ;

[di , dj ] = 0.

modules for∼

Let∼

[di ,X (k0, k)] = ki X (k0, k);

[di , tk00 tk Kj ] = ki t

k00 tk Kj ;

[di , dj ] = 0.

modules for∼

Let h(0) be a Cartan subalgebra of g(0, 0)

Let∼h= h(0)⊕

∑0≤i≤n

CKi ⊕ D which is an abelian subalgebra of∼

δi ∈∼h∗

such that δi (h(0)) = δi (Kj ) = 0 and δi (dj ) = δij , 0 ≤ i , j ≤ n.For (k0, k) ∈ Zn+1, let δk =

∑kiδi and δ(k0, k) = k0δ0 + δk .

∼LTα+δ(k0,k)=

g(k0, k , α)⊗ tk00 tk , α 6= 0,

g(k0, k , 0)⊗ tk00 tk ⊕

∑k0,k∈Γ0⊕Γ

0≤i≤n

tk0tk Ki , α = 0 and (k0, k) 6= (0, 0),

∼h= g(0, 0, 0)⊕

∑0≤i≤n

CKi ⊕ D, (k0, k , α) = (0, 0, 0).

Then∼

LT =⊕

α∈h(0)∗

(k0,k)∈Zn+1

∼LTα+δ(k0,k) is a root space decomposition with

respect to∼h and each root space is finite dimensional.

A module V of∼

LT is called weight module if

(a) V =⊕λ∈

∼h∗

Vλ,Vλ = v ∈ V | hv = λ(h)v , h ∈∼h.

(b) dim Vλ <∞.

A root α + δ(k0, k) of∼

LT is called real root if α 6= 0 and null root ifα = 0.

A weight module V of∼

LT is called integrable if every real root vector

acts locally nilpotently on V , i.e., for X ∈∼

LTα+δ(k0,k), α 6= 0 and

v ∈ V , there exists b = b(v , α + δ(k0, k)) such that X b.v = 0.

Subquotient of∼LT

Consider a Lie algebra L = LT ⊕ CK ⊗ A(m)⊕ Cd0 where K is a symbol.Let X (k0, k) ∈ g(k0, k) and Y (l0, l) ∈ g(l0, l) and define the bracketoperations on L as follows:

(a) [X (k0, k),Y (l0, l)] = [X ,Y ](k0 + l0, k + l)+(X |Y ) δl0+k0,0 k0 K⊗ t l+k ;

(b) K ⊗ A(m) is central;

(c) [d0,X (k0, k)] = k0X (k0, k).

Let∼L= L⊕ D where D is the space spanned by derivations

d1, d2, · · · , dn. Extend the Lie bracket to∼L by defining D action on L

as before.

Then Φ :∼

LT→∼L by

ΦX (k0, k) = X (k0, k),X ∈ g(k0, k);

Φ(tk00 tk Ki ) = 0 if i 6= 0 or k0 6= 0;

Φ(tkK0) = K ⊗ tk ;Φ(di ) = di , 0 ≤ i ≤ n.

is a Lie algebra homomorphism.

Subquotient of∼LT

(c) [d0,X (k0, k)] = k0X (k0, k).

as before.

Then Φ :∼

LT→∼L by

ΦX (k0, k) = X (k0, k),X ∈ g(k0, k);

Φ(tk00 tk Ki ) = 0 if i 6= 0 or k0 6= 0;

Φ(tkK0) = K ⊗ tk ;Φ(di ) = di , 0 ≤ i ≤ n.

Subquotient of∼LT

(c) [d0,X (k0, k)] = k0X (k0, k).

as before.

Then Φ :∼

LT→∼L by

ΦX (k0, k) = X (k0, k),X ∈ g(k0, k);

Φ(tk00 tk Ki ) = 0 if i 6= 0 or k0 6= 0;

Φ(tkK0) = K ⊗ tk ;Φ(di ) = di , 0 ≤ i ≤ n.

Subquotient of∼LT

(c) [d0,X (k0, k)] = k0X (k0, k).

as before.

Then Φ :∼

LT→∼L by

ΦX (k0, k) = X (k0, k),X ∈ g(k0, k);

Φ(tk00 tk Ki ) = 0 if i 6= 0 or k0 6= 0;

Φ(tkK0) = K ⊗ tk ;Φ(di ) = di , 0 ≤ i ≤ n.

Subquotient of∼LT

(c) [d0,X (k0, k)] = k0X (k0, k).

as before.

Then Φ :∼

LT→∼L by

ΦX (k0, k) = X (k0, k),X ∈ g(k0, k);

Φ(tk00 tk Ki ) = 0 if i 6= 0 or k0 6= 0;

Φ(tkK0) = K ⊗ tk ;Φ(di ) = di , 0 ≤ i ≤ n.

Subquotient of∼LT

(c) [d0,X (k0, k)] = k0X (k0, k).

as before.

Then Φ :∼

LT→∼L by

ΦX (k0, k) = X (k0, k),X ∈ g(k0, k);

Φ(tk00 tk Ki ) = 0 if i 6= 0 or k0 6= 0;

Φ(tkK0) = K ⊗ tk ;Φ(di ) = di , 0 ≤ i ≤ n.

Subquotient of∼LT

(c) [d0,X (k0, k)] = k0X (k0, k).

as before.

Then Φ :∼

LT→∼L by

ΦX (k0, k) = X (k0, k),X ∈ g(k0, k);

Φ(tk00 tk Ki ) = 0 if i 6= 0 or k0 6= 0;

Φ(tk K0) = K ⊗ tk ;Φ(di ) = di , 0 ≤ i ≤ n.

Irreducible weight module for∼LT

Irreducible weight module for∼L and L

Classification of irreducible inte-grable weight L-weight modules

Recover the original∼LT -module

from an irreducible L-module

Highest weight irreducible modules for L

Let∼h(0) = h(0)⊕ CK ⊕ Cd0, and L =

⊕α∈h(0)∗

k0∈Z

Lα+k0δ is a root space

decomposition with respect to∼h (0), where

Lα+k0δ =

⊕k∈Zn

g(k0, k, α)tk00 tk , if α + k0δ 6= 0,

⊕k∈Zn

g(0, k, 0)tk ⊕ K ⊗ A(m)⊕ Cd0, if α + k0δ = 0.

Here δ ∈∼h(0)∗ such that δ(h(0)) = 0, δ(K ) = 0 and δ(d0) = 1.

Let∼h(0) = h(0)⊕ CK ⊕ Cd0, and L =

⊕α∈h(0)∗

k0∈Z

Lα+k0δ =

⊕k∈Zn

g(k0, k, α)tk00 tk , if α + k0δ 6= 0,

⊕k∈Zn

g(0, k, 0)tk ⊕ K ⊗ A(m)⊕ Cd0, if α + k0δ = 0.

Let∼h(0) = h(0)⊕ CK ⊕ Cd0, and L =

⊕α∈h(0)∗

k0∈Z

Lα+k0δ =

⊕k∈Zn

g(k0, k, α)tk00 tk , if α + k0δ 6= 0,

⊕k∈Zn

g(0, k, 0)tk ⊕ K ⊗ A(m)⊕ Cd0, if α + k0δ = 0.

∼∆+= α + k0δ ∈

∼∆ | k0 > 0 or k0 = 0, α > 0

∼∆−= α + k0δ ∈

∼∆ | k0 < 0 or k0 = 0, α < 0.

L+ =⊕

α+k0δ>0

Lα+k0δ,

L− =⊕

α+k0δ<0

Lα+k0δ, and

L0 = L0.

Construction of highest weight module for L

Let N be an irreducible finite dimensional module for L0. Since∼h(0) + K ⊗ A(m) is central, it is easy to see they act by scalars on N.

Let U(L) denote the universal enveloping algebra of L. Define Vermamodule for L.

M(N) = U(L)⊗

L+⊕L0

where L+ acts trivially on N.

By standard arguments, M(N) admits a unique irreducible quotientsay V (N).

M(N) and V (N) are weight module with respect to∼h(0). But they

may not have finite dimensional weight spaces.

M(N) = U(L)⊗

L+⊕L0

M(N) = U(L)⊗

L+⊕L0

M(N) = U(L)⊗

L+⊕L0

M(N) = U(L)⊗

L+⊕L0

Necessary and sufficient condition for V (N) to have finitedimensional weight spaces

Let I be an ideal in A(m) and let

L0(I ) =⊕k∈Zn

g(0, k , 0)tk I ⊕ K ⊗ I ,

L(I ) =⊕k0∈Zk∈Zn

g(k0, k)⊗ tk00 tk I ⊕ K ⊗ I .

Then L0(I ) is an ideal in L0 and L(I ) is an ideal in L.

Proposition

Suppose N is finite dimensional irreducible module for L0 such thatL0(I ).N = 0 for some ideal I of A(m). Then L(I ).V (N) = 0.

L0(I ) =⊕k∈Zn

g(0, k , 0)tk I ⊕ K ⊗ I ,

g(k0, k)⊗ tk00 tk I ⊕ K ⊗ I .

Proposition

L0(I ) =⊕k∈Zn

g(0, k , 0)tk I ⊕ K ⊗ I ,

g(k0, k)⊗ tk00 tk I ⊕ K ⊗ I .

Proposition

Theorem (1)

V (N) has finite dimensional weight spaces with respect to∼h(0) if and only

if there exists a co-finite ideal I of A(m) such that L0(I ).N = 0.

We can assume that the ideal I generated by polynomials Pi invariable tmi

i , 1 ≤ i ≤ n, and the constant term is 1.

Let write Pi (tmii ) =

qi∏j=1

(tmii − ami

ij )bij for some positive integers bij

and qi , with amiij 6= ami

ij ′ for j 6= j ′.

Theorem (1)

qi∏j=1

(tmii − ami

ij ′ for j 6= j ′.

Theorem (1)

qi∏j=1

(tmii − ami

ij ′ for j 6= j ′.

Theorem (1)

qi∏j=1

(tmii − ami

ij ′ for j 6= j ′.

Integrable modules for L

Recall that a weight module V of L is called integrable if all real rootvectors are locally nilpotent on V .

As I = 〈Pi 〉, where Pi (tmii ) =

qi∏j=1

(tmii − ami

ij )bij .

Let P ′i (tmii ) =

qi∏j=1

(tmii − ami

ij ) and I ′ = 〈P ′i 〉

Then clearly I ⊆ I ′.

Theorem (2)

Let V (N) be an integrable module for L. Then L(I ′)V (N) = 0.

qi∏j=1

(tmii − ami

ij )bij .

qi∏j=1

(tmii − ami

ij ) and I ′ = 〈P ′i 〉

Theorem (2)

qi∏j=1

(tmii − ami

ij )bij .

qi∏j=1

(tmii − ami

ij ) and I ′ = 〈P ′i 〉

Theorem (2)

qi∏j=1

(tmii − ami

ij )bij .

qi∏j=1

(tmii − ami

ij ) and I ′ = 〈P ′i 〉

Theorem (2)

qi∏j=1

(tmii − ami

ij )bij .

qi∏j=1

(tmii − ami

ij ) and I ′ = 〈P ′i 〉

Theorem (2)

By Theorem (2) we see that the irreducible integrable module V (N)is actually a module for L(A(m)/I ′).

Recall that σ0 is an automorphism of order m0 and ξ0 is m0-thprimitive root of unity. Let

gk0= x ∈ g |σ0x = ξk0

DefineL(g, σ0) =

⊕k0∈Z

gk0⊗ tk0 ⊕ CK

which is known to be an affine Lie algebra.

Proposition

L′/L′(I ′) ∼=⊕

N−copies

L(g, σ0), where L′ =⊕

(k0,k)∈Zn

g(k0, k)tk00 tk ⊕K ⊗A(m)

and L′(I ′) defined similarly as before.

gk0= x ∈ g |σ0x = ξk0

DefineL(g, σ0) =

⊕k0∈Z

gk0⊗ tk0 ⊕ CK

Proposition

L′/L′(I ′) ∼=⊕

N−copies

(k0,k)∈Zn

gk0= x ∈ g |σ0x = ξk0

DefineL(g, σ0) =

⊕k0∈Z

gk0⊗ tk0 ⊕ CK

Proposition

L′/L′(I ′) ∼=⊕

N−copies

(k0,k)∈Zn

gk0= x ∈ g |σ0x = ξk0

DefineL(g, σ0) =

⊕k0∈Z

gk0⊗ tk0 ⊕ CK

Proposition

L′/L′(I ′) ∼=⊕

N−copies

(k0,k)∈Zn

Let gaff = g(0, 0)⊗ C[tm00 , t−m0

0 ]⊕ CK ⊕ Cd0 which is a subalgebraof L.

Next result shows that an irreducible L-module with finitedimensional weight space which is an integrable gaff -module is in factan highest weight module for L.

Theorem

Suppose V is an irreducible module for L with finite dimensional weight

spaces with respect to∼h(0). Further, suppose V is integrable for gaff

where the canonical central element m0K acts as positive integer, then Vis an highest weight module for L.

Now as V is an highest weight module, let λ be the top weight andput N = Vλ. Since V is irreducible, by weight argument we see thatN is irreducible L0- module. Thus V ∼= V (N).

Let gaff = g(0, 0)⊗ C[tm00 , t−m0

Theorem

Let gaff = g(0, 0)⊗ C[tm00 , t−m0

Theorem

Let gaff = g(0, 0)⊗ C[tm00 , t−m0

Theorem

Let gaff = g(0, 0)⊗ C[tm00 , t−m0

Theorem

Theorem (S.E. Rao, -)

Let V be an irreducible integrable∼

LT -module with center actingnon-trivially. Then V is a highest weight module for finitely many copiesof affine Lie algebra upto an automorphism.

Theorem (S.E. Rao, -)

Let V be an irreducible integrable∼

LT -module with center actingnon-trivially. Then V is a highest weight module for finitely many copiesof affine Lie algebra upto an automorphism.

References

[ERSS] S. Eswara Rao, Sachin S. Sharma, Integrable modules for Lie tori,Journal of Pure and Applied Algebra, 220:1074-1095 (2016) .

[ABFP] Allison, B., S. Berman, J. Faulkner and A. Pianzola, Multilooprealization of Extended Affine Lie algebras and the Lie Tori, Trans. Amer.Math. Soc., 361(2009), 4807-4842.

[E1] Eswara Rao, S., Classification of irreducible integrable modules forToroidal Lie algebras with finite dimensional weight spaces, Journal ofAlgebra, 277(2004), 318-348.

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