Weyl orbits of π-systems in Kac-Moody algebras

Krishanu Roy

The Institute of Mathematical Sciences, Chennai, India

(Joint with Lisa Carbone, K N Raghavan, Biswajit Ransingh and Sankaran Viswanath)

June 4, 2018

π-systems

Definition: Let ∆ be a root system. A non-empty subset Σ ⊂ ∆ is called aπ-system if α− β 6∈ ∆ for all α, β ∈ Σ.

π-systems of B2:SingletonsTwo roots making a 3π/4 angle (simple roots of ∆)Two long roots making a π/2 angle.Two roots making a π angleTwo long and one short root, making angles 3π/4, 3π/4, π/2

π-systems give rise to GCMs

Definition: An integer matrix C is a generalized Cartan matrix (GCM) if(i) cii = 2 (1 ≤ i ≤ m) (ii) cij ≤ 0 for i 6= j (iii) cij = 0 iff cji = 0 .

Lemma: Let Σ = {β1, β2, · · · , βm} be a π-system in ∆. Then the matrixB =

[2(βi|βj)

(βi|βi)

]ij

is a GCM.

Examples for ∆ = B2:

Singletons: B = [2];

Simple roots of ∆: B =

[2 −2−1 2

]=: A(∆).

Two long roots making a π/2 angle: B =

[2 00 2

].

Two roots making a π angle: B =

[2 −2−2 2

]Two long and one short root, making angles 3π/4, 3π/4, π/2:

B =

2 −1 0−2 2 −20 −1 2

π-systems in Kac-Moody algebras

Let A be a GCM, and g(A) the corresponding Kac-Moody algebra.

g(A) = h⊕⊕

α∈∆(A)

gα

∆(A) = ∆re(A) t∆im(A)

g(A)α are one-dimensional for real roots α.

Σ ⊂ ∆re(A) is a π-system if α− β 6∈ ∆(A) for all α, β ∈ Σ.

Theorem

Let A be a GCM and {βi}mi=1 ⊂ ∆re(A) be a π-system in ∆(A) with GCM

B.

Let e±βi ∈ g′(A)±βi such that [eβi , e−βi ] = β∨i .

Then:

there exists a unique Lie algebra homomorphism i : g′(B)→ g′(A) suchthat ei 7→ eβi , fi 7→ e−βi , α

∨i 7→ β∨i .

this map is injective iff the π-system is linearly independent.

π-systems in Kac-Moody algebras

Lemma

Conversely, given a Lie algebra homomorphism φ : g′(B)→ g′(A) satisfying0 6= φ(ei) ∈ g′(A)βi , 0 6= φ(fi) ∈ g′(A)−βi for some real roots βi of g′(A). Then,the set {βi} is a π-system of type B in A.

The image of g′(B) will be called a regular subalgebra of g′(A).

If A has a π-system of type B and B has a π-system of type C, then A hasa π-system of type C.

Weyl conjugacy of π-systems

Consider π-systems in ∆ with a fixed GCM, say B =

[2 00 2

].

Pair of orthogonal long roots.Any two such π-systems can be transformed one into the other by areflection associated to one of the roots in ∆.Let W(∆) be the group generated by reflections associated to the roots(the Weyl group of ∆).In this case, W(∆) acts transitively on such π-systems.

General Question (in the setting of Kac-Moody algebras)

Let A be a GCM. Consider the set of all π-systems in A of a fixed type B. Howmany W(A)-orbits does this split into ?

Theorem

Let A,B be symmetrizable GCMs and Σ a linearly independent π-system oftype B in A. If B is indecomposable, then:

1 There exists w ∈ W(A) such that wΣ ⊂ ∆re+(A) or wΣ ⊂ ∆re

−(A).

2 There exist w1,w2 ∈ W(A) such that w1Σ ⊂ ∆re+(A) and w2Σ ⊂ ∆re

−(A) ifand only if B is of finite type.

Let m(B,A) denote the number of W(A)-orbits of π-systems in A of type B(this could be infinity in general).

When A,B are of finite type, Dynkin determined all these numbersm(B,A).

Dynkin: On Semisimple subalgebras of semisimple Liealgebras (1951)

Dynkin (1951): the exceptional Lie algebras

π-systems of affine type

Theorem

Let A be a symmetrizable GCM and B be a GCM of affine type. Suppose Σ isa linearly independent π-system of type B in A. Then,

1 There exists an affine subdiagram Y of S(A) and w ∈ W(A) such thatevery element of wΣ is supported in Y.

2 Suppose (Y ′,w′) is another such pair, i.e., with Y ′ a subdiagram of affinetype, w′ ∈ W(A) such that w′Σ is supported in Y ′. Then Y = Y ′ andw′w−1 ∈ W(Y t Y⊥).

3 m(B,A) =∞.

Let A be a symmetrizable GCM such that S(A) has no subdiagrams of affinetype. Then A contains no linearly independent π-systems of affine type.

Overextended Dynkin diagrams (EXT type)

X(1)1

where X(1) is untwisted affine (i.e, one of the following) and the marked vertexis special. We denote this diagram X++.

The main theorem in general

Theorem

Let X be a simply-laced Dynkin diagram (↔ symmetric GCM) and let K be asimply-laced Dynkin diagram of EXT type. Then:

1 There exists a π-system in X of type K if and only if there exists an EXTtype subdiagram Z of X such that Z◦ has a π-system of type K◦.

2 The number of W(X) orbits of π-systems of type K in X is given by:

m (K,X) = 2∑Z⊆X

Z∈EXT

m(K◦, Z◦) (1)

where K◦, Z◦ denote their finite parts.

If K = X++, then we write X = K◦.

Corollaries

The theorem reduces the computation of the multiplicity of K in X to asum of multiplicities involving only finite type diagrams.

The latter are completely known (Dynkin).

Corollary

Let K be a simply-laced Dynkin diagram of EXT type. Then,

1 m(K,X) is finite for all simply-laced diagrams X.

2 m(K,X) = 2m(K◦,X◦) for all X ∈ HYP ∩ EXT.

The main theorem: a special case

A1++ :

GCM: B =

2 −2 0−2 2 −10 −1 2

For K = A++

1 , K◦ is of type A1. Since any Z◦ occurring on the right hand sideof (1) is simply laced, we have m(K◦, Z◦) = 1. So this reduces exactly to thefollowing theorem:

Theorem

Let X be a simply laced Dynkin diagram (↔ symmetric GCM). Then:

1 X has a π-system of type A++1 if and only if it contains a subdiagram of

EXT type.

2 The number of W(X)-orbits of π-systems of type A++1 in A is twice the

number of such subdiagrams (and is, in particular, finite).

In particular, if X is itself (simply laced) of EXT type, then X contains aπ-system of type A++

1 .

Thank You