Classification of complex semisimple Lie algebras by root systems - - PowerPoint PPT Presentation

Classification of complex semisimple Lie algebras by root systems

Ian Xiao Supervised by: Dr. Jeroen Schillewaert

Department of Mathematics, University of Auckland

February 27, 2019

Lie algebras

Vector space g with a bilinear map [, ] :g×g→g such that

◮ [x, x] = 0 ◮ [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0

Lie algebras

Vector space g with a bilinear map [, ] :g×g→g such that

◮ [x, x] = 0 ◮ [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0

Examples

◮ R3 with cross product. ◮ The space of n × n matrices over any field k with Lie

bracket [X, Y] = XY − YX

◮ Any associative algebra A, with Lie bracket given by

[x, y] = xy − yx

Lie algebras

More definitions

◮ Lie subalgebras: subspaces [h, h] ⊆ h ◮ Ideals: subalgebras [I, g] ⊆ I ◮ Quotient Lie algebras: g/I ◮ Lie homomorphisms: [φ(x), φ(y)] = φ([x, y]) ◮ Extensions and semidirect products ◮ Derivations:

Der(g) = {D ∈ End(g) | D(ab) = D(a)b + aD(b)}

Lie algebras

More definitions

◮ Lie subalgebras: subspaces [h, h] ⊆ h ◮ Ideals: subalgebras [I, g] ⊆ I ◮ Quotient Lie algebras: g/I ◮ Lie homomorphisms: [φ(x), φ(y)] = φ([x, y]) ◮ Extensions and semidirect products ◮ Derivations:

Der(g) = {D ∈ End(g) | D(ab) = D(a)b + aD(b)}

Basic results

◮ Der(g) is a Lie algebra ◮ Isomorphism theorems ◮ Correspondence theorem

Lie algebra representations and modules

Homomorphism ρ : g → glV, where glV = End(V) with [x, y] = xy − yx.

Lie algebra representations and modules

Homomorphism ρ : g → glV, where glV = End(V) with [x, y] = xy − yx.

Example

Adjoint representation ad : g → ad(g) given by x → (y → [x, y]). Note that ad(g) ⊆ glg.

Lie algebra representations and modules

Homomorphism ρ : g → glV, where glV = End(V) with [x, y] = xy − yx.

Example

Adjoint representation ad : g → ad(g) given by x → (y → [x, y]). Note that ad(g) ⊆ glg. A g-module is a vector space V together with a linear operator · : g × V → V which preserves the Lie bracket for glg

Lie algebra representations and modules

Homomorphism ρ : g → glV, where glV = End(V) with [x, y] = xy − yx.

Example

Adjoint representation ad : g → ad(g) given by x → (y → [x, y]). Note that ad(g) ⊆ glg. A g-module is a vector space V together with a linear operator · : g × V → V which preserves the Lie bracket for glg

Theorem

There is a one-to-one correspondence between Lie representations and Lie modules.

Lie algebra representation

More definitions

Submodules, irreducible modules, indecomposable modules, quotient modules. Let V and W be g-modules. A g-module homomorphism from V to W is a linear map φ : V → W such that φ(x · v) = x · φ(v) for all x ∈ g and v ∈ V.

Lie algebra representation

More definitions

Submodules, irreducible modules, indecomposable modules, quotient modules. Let V and W be g-modules. A g-module homomorphism from V to W is a linear map φ : V → W such that φ(x · v) = x · φ(v) for all x ∈ g and v ∈ V.

Theorem

Schur’s Lemma. Let g be a Lie algebra over C, and let V be a finite dimensional irreducible g-module. Then φ : V → V is a g-module homomorphism if and only if φ ∈ span{IV}.

Nilpotent Lie algebras

Lower central series: g = g0 ⊃ g1 ⊃ g2 ⊃ ... with g1 := [g, g] and gi+1 := [g, gi] for each i. A Lie algebra is nilpotent if its lower central series terminates.

Nilpotent Lie algebras

Lower central series: g = g0 ⊃ g1 ⊃ g2 ⊃ ... with g1 := [g, g] and gi+1 := [g, gi] for each i. A Lie algebra is nilpotent if its lower central series terminates.

Example

The space of strictly upper triangular matrices.

Nilpotent Lie algebras

Lower central series: g = g0 ⊃ g1 ⊃ g2 ⊃ ... with g1 := [g, g] and gi+1 := [g, gi] for each i. A Lie algebra is nilpotent if its lower central series terminates.

Example

The space of strictly upper triangular matrices.

Ad-nilpotency

A linear map α ∈ End(V) is nilpotent if αn = 0 for some n ∈ N. A Lie algebra g is called ad-nilpotent if ad(x) is nilpotent for each x ∈ g.

Nilpotent Lie algebras

Lower central series: g = g0 ⊃ g1 ⊃ g2 ⊃ ... with g1 := [g, g] and gi+1 := [g, gi] for each i. A Lie algebra is nilpotent if its lower central series terminates.

Example

The space of strictly upper triangular matrices.

Ad-nilpotency

A linear map α ∈ End(V) is nilpotent if αn = 0 for some n ∈ N. A Lie algebra g is called ad-nilpotent if ad(x) is nilpotent for each x ∈ g.

Engel’s Theorem

A Lie algebra is nilpotent if and only if it is ad-nilpotent.

Solvable Lie algebras

A Lie algebra g is solvable if it has a terminating derived series g = g(0) ⊃ g(1) ⊃ ... ⊃ g(r) = 0, where g(i+1) = [g(i), g(i)].

Solvable Lie algebras

A Lie algebra g is solvable if it has a terminating derived series g = g(0) ⊃ g(1) ⊃ ... ⊃ g(r) = 0, where g(i+1) = [g(i), g(i)].

Example

The space of upper triangular matrices.

Solvable Lie algebras

A Lie algebra g is solvable if it has a terminating derived series g = g(0) ⊃ g(1) ⊃ ... ⊃ g(r) = 0, where g(i+1) = [g(i), g(i)].

Example

The space of upper triangular matrices.

Lie’s Theorem

Let g ⊂ glV be a solvable Lie subalgebra where V is n-dimensional over C. There exists a basis {v1, ..., vn} of V such that every x ∈ g is represented by an upper triangular matrix.

Solvable Lie algebras

A Lie algebra g is solvable if it has a terminating derived series g = g(0) ⊃ g(1) ⊃ ... ⊃ g(r) = 0, where g(i+1) = [g(i), g(i)].

Example

The space of upper triangular matrices.

Lie’s Theorem

Let g ⊂ glV be a solvable Lie subalgebra where V is n-dimensional over C. There exists a basis {v1, ..., vn} of V such that every x ∈ g is represented by an upper triangular matrix.

Corollary

If g is a finite dimensonal solvable Lie algebra over C, then all irreducible g-modules are one-dimensional.

Solvable Lie algebras

Corollary

Let ρ : g → glV be a Lie representation where V is a n-dimensional vector space over C, and g is solvable. Then there exists a basis of V such that each ρ(x) ∈ ρ(g) is represented by an upper triangular matrix.

Solvable Lie algebras

Corollary

Let ρ : g → glV be a Lie representation where V is a n-dimensional vector space over C, and g is solvable. Then there exists a basis of V such that each ρ(x) ∈ ρ(g) is represented by an upper triangular matrix.

Proposition

Let g be a finite dimensional Lie algebra over C, then g is solvable if and only if g(1) is nilpotent.

Solvable Lie algebras

Corollary

Let ρ : g → glV be a Lie representation where V is a n-dimensional vector space over C, and g is solvable. Then there exists a basis of V such that each ρ(x) ∈ ρ(g) is represented by an upper triangular matrix.

Proposition

Let g be a finite dimensional Lie algebra over C, then g is solvable if and only if g(1) is nilpotent.

Cartan’s criterion for solvability

Let g be a finite dimensional Lie algebra over C. Then g is solvable if and only if tr(ad(x) ◦ ad(y)) = 0 for all x ∈ g and y ∈ g(1).

Killing form and semisimple Lie algebras

The killing form of a Lie algebra g over k is the symmetric bilin- ear map κ : g × g :→ k given by κ(x, y) = tr(ad(x) ◦ ad(y)). This allows us to restate Cartan’s criterion for solvability:

Killing form and semisimple Lie algebras

The killing form of a Lie algebra g over k is the symmetric bilin- ear map κ : g × g :→ k given by κ(x, y) = tr(ad(x) ◦ ad(y)). This allows us to restate Cartan’s criterion for solvability:

Cartan’s criterion for solvability

Let g be a finite dimensional Lie algebra over C. Then g is solvable if and only if κ(x, y) = 0 for all x ∈ g and y ∈ g(1).

Killing form and semisimple Lie algebras

The killing form of a Lie algebra g over k is the symmetric bilin- ear map κ : g × g :→ k given by κ(x, y) = tr(ad(x) ◦ ad(y)). This allows us to restate Cartan’s criterion for solvability:

Cartan’s criterion for solvability

Let g be a finite dimensional Lie algebra over C. Then g is solvable if and only if κ(x, y) = 0 for all x ∈ g and y ∈ g(1).

Example

The special linear Lie algebra of order n is given by sln = {x ∈ glV | tr(x) = 0}. E.g. sl2(C), with basis {e, f, h} e = 1

• , f =

1

• , and h =

1 −1

• .

Killing form and semisimple Lie algebras

The killing form of a Lie algebra g over k is the symmetric bilin- ear map κ : g × g :→ k given by κ(x, y) = tr(ad(x) ◦ ad(y)). This allows us to restate Cartan’s criterion for solvability:

Cartan’s criterion for solvability

Let g be a finite dimensional Lie algebra over C. Then g is solvable if and only if κ(x, y) = 0 for all x ∈ g and y ∈ g(1).

Example

The special linear Lie algebra of order n is given by sln = {x ∈ glV | tr(x) = 0}. E.g. sl2(C), with basis {e, f, h} e = 1

• , f =

1

• , and h =

1 −1

• .

By an easy computation, we find [e, f] = h, [f, h] = 2f, [h, e] = 2e. Hence [ad(e)] =   −2 1  , [ad(f)] =   2 −1  ,

Killing form and semisimple Lie algebras

Example continued

[ad(h)] =   2 −2  , [κ] =   4 4 8  .

Killing form and semisimple Lie algebras

Example continued

[ad(h)] =   2 −2  , [κ] =   4 4 8  . A nonzero Lie algebra g is semisimple if its largest solvable ideal is zero, or equivalently, every commutative ideal is zero.

Killing form and semisimple Lie algebras

Example continued

[ad(h)] =   2 −2  , [κ] =   4 4 8  . A nonzero Lie algebra g is semisimple if its largest solvable ideal is zero, or equivalently, every commutative ideal is zero.

Example

For any Lie algebra g, g/r(g) is semisimple where r(g) is the (unique) largest solvable ideal of g, known as the radical of g.

Killing form and semisimple Lie algebras

Example continued

[ad(h)] =   2 −2  , [κ] =   4 4 8  . A nonzero Lie algebra g is semisimple if its largest solvable ideal is zero, or equivalently, every commutative ideal is zero.

Example

For any Lie algebra g, g/r(g) is semisimple where r(g) is the (unique) largest solvable ideal of g, known as the radical of g.

Cartan’s criterion for semisimplicity

A n-dimensional Lie algebra g over C is semisimple if and only if κ is non-degenerate.

Complex semisimple Lie algebras

Theorem: Semisimple Lie algebra decomposition

Let g be a finite dimensonal Lie algebra over C. Then g is semisimple if and only if there exist simple ideals I1, ..., Ir such that g = I1 ... Ir.

Complex semisimple Lie algebras

Theorem: Semisimple Lie algebra decomposition

Let g be a finite dimensonal Lie algebra over C. Then g is semisimple if and only if there exist simple ideals I1, ..., Ir such that g = I1 ... Ir.

Corollary

Every semisimple Lie algebra over C with dimension ≤ 5 is

• simple. In particular, sl2(C) is simple.

Proof: Observe that every simple Lie algebra over C must have a dimension ≥ 3: 1-dimensional Lie algebras are abelian, and 2-dimensional Lie algebras have proper 1-dimensional ideals.

Complex semisimple Lie algebras

Theorem: Semisimple Lie algebra decomposition

Let g be a finite dimensonal Lie algebra over C. Then g is semisimple if and only if there exist simple ideals I1, ..., Ir such that g = I1 ... Ir.

Corollary

Every semisimple Lie algebra over C with dimension ≤ 5 is

• simple. In particular, sl2(C) is simple.

Proof: Observe that every simple Lie algebra over C must have a dimension ≥ 3: 1-dimensional Lie algebras are abelian, and 2-dimensional Lie algebras have proper 1-dimensional ideals.

• Theorem

DerC(g) = ad(g) if g is a complex finite dimensional semisimple Lie algebra.

Representation of sl2(C)

Let C[X, Y] be the vector space of two-variable polynomials

• ver C. For each d ∈ N, let Vd be the subspace consisting of

all homogenous polynomials of degree d, then B = {X d, X d−1Y, ..., Y d} = {v1, ..., vd+1} is a basis of Vd. It can be shown that Vd becomes a sl2(C)-module via a repre- sentation φ : sl2(C) → glVd, and every sl2(C)-module V of dimension d + 1 is isomorphic to Vd.

Representation of sl2(C)

Let C[X, Y] be the vector space of two-variable polynomials

• ver C. For each d ∈ N, let Vd be the subspace consisting of

all homogenous polynomials of degree d, then B = {X d, X d−1Y, ..., Y d} = {v1, ..., vd+1} is a basis of Vd. It can be shown that Vd becomes a sl2(C)-module via a repre- sentation φ : sl2(C) → glVd, and every sl2(C)-module V of dimension d + 1 is isomorphic to Vd.

Theorem

Let {e, f, h} be the basis of sl2(C) and let φ : sl2(C) → glVd be the linear map given by φ(e) = X δ

δY , φ(f) = Y δ δX ,

φ(h) = X δ

δX − Y δ δY . Then φ becomes a Lie representation.

Representation of sl2(C)

Let C[X, Y] be the vector space of two-variable polynomials

• ver C. For each d ∈ N, let Vd be the subspace consisting of

all homogenous polynomials of degree d, then B = {X d, X d−1Y, ..., Y d} = {v1, ..., vd+1} is a basis of Vd. It can be shown that Vd becomes a sl2(C)-module via a repre- sentation φ : sl2(C) → glVd, and every sl2(C)-module V of dimension d + 1 is isomorphic to Vd.

Theorem

Let {e, f, h} be the basis of sl2(C) and let φ : sl2(C) → glVd be the linear map given by φ(e) = X δ

δY , φ(f) = Y δ δX ,

φ(h) = X δ

δX − Y δ δY . Then φ becomes a Lie representation.

Theorem

For every d ∈ N, the sl2(C)-module Vd is irreducible.

Representation of sl2(C)

Theorem

Every d + 1-dimensional irreducible sl2(C)-module V is isomorphic to Vd.

Representation of sl2(C)

Theorem

Every d + 1-dimensional irreducible sl2(C)-module V is isomorphic to Vd.

Theorem

If ρ : sl2(C) → glV is a finite dimensional representation and w ∈ V is an eigenvector of ρ(h) such that e · w = 0, then h · w = dw for some d ∈ N, and the submodule w is isomorphic to Vd.

Representation of sl2(C)

Theorem

Every d + 1-dimensional irreducible sl2(C)-module V is isomorphic to Vd.

Theorem

If ρ : sl2(C) → glV is a finite dimensional representation and w ∈ V is an eigenvector of ρ(h) such that e · w = 0, then h · w = dw for some d ∈ N, and the submodule w is isomorphic to Vd.

Weyl’s Theorem

Every finite dimensional module of a semisimple Lie algebra g

• ver C is completely reducible.

Roots of complex semisimple Lie algebras

Let g be a complex finite dimensional semisimple Lie algebra. A Cartan subalgebra h of g is a maximal abelian subalgebra containing only semisimple elements, i.e. ad(x) is subspace- invariant for all x ∈ h.

Roots of complex semisimple Lie algebras

Let g be a complex finite dimensional semisimple Lie algebra. A Cartan subalgebra h of g is a maximal abelian subalgebra containing only semisimple elements, i.e. ad(x) is subspace- invariant for all x ∈ h.

Weight spaces and roots

◮ Given a Cartan subalgebra h and α ∈ h∗, the weight space

• f α is given by gα = {x ∈ g | [h, x] = α(h)x for all h ∈ h}.

◮ A root of g (with respect to a chosen CSA h) is α ∈ h∗ such

that gα = {0}. Denote Φ the entire set of roots of g with respect to h. Observe that |Φ| ≤ ∞ since g is finite dimensional.

Roots of complex semisimple Lie algebras

Let g be a complex finite dimensional semisimple Lie algebra. A Cartan subalgebra h of g is a maximal abelian subalgebra containing only semisimple elements, i.e. ad(x) is subspace- invariant for all x ∈ h.

Weight spaces and roots

◮ Given a Cartan subalgebra h and α ∈ h∗, the weight space

• f α is given by gα = {x ∈ g | [h, x] = α(h)x for all h ∈ h}.

◮ A root of g (with respect to a chosen CSA h) is α ∈ h∗ such

that gα = {0}. Denote Φ the entire set of roots of g with respect to h. Observe that |Φ| ≤ ∞ since g is finite dimensional.

Root space decomposition

Every complex finite dimensional semisimple Lie algebra can be expressed as g = h ⊕

α∈Φ

gα

Roots of complex semisimple Lie algebras

Lemma

(i) [gα, gβ] ⊆ gα+β. (ii) If α + β = 0, then κ(gα, gβ) = {0}. (iii) κ↾g0 is non-degenerate.

Roots of complex semisimple Lie algebras

Lemma

(i) [gα, gβ] ⊆ gα+β. (ii) If α + β = 0, then κ(gα, gβ) = {0}. (iii) κ↾g0 is non-degenerate.

Proposition

Let g = g0 ⊕

α∈Φ

gα be a root space decomposition. For each α ∈ Φ we obtain a subalgebra sl(α) := span({eα, fα, hα}) such that eα ∈ gα, fα ∈ g−α, hα ∈ h, α(hα) = 2, and sl(α) ∼ = sl2(C). In particular, if α ∈ Φ, then −α ∈ Φ.

Roots of complex semisimple Lie algebras

Lemma

(i) [gα, gβ] ⊆ gα+β. (ii) If α + β = 0, then κ(gα, gβ) = {0}. (iii) κ↾g0 is non-degenerate.

Proposition

Let g = g0 ⊕

α∈Φ

gα be a root space decomposition. For each α ∈ Φ we obtain a subalgebra sl(α) := span({eα, fα, hα}) such that eα ∈ gα, fα ∈ g−α, hα ∈ h, α(hα) = 2, and sl(α) ∼ = sl2(C). In particular, if α ∈ Φ, then −α ∈ Φ.

Proposition

For each α ∈ Φ, there exists a unique tα ∈ h such that α(h) = κ(tα, h) for all h ∈ h.

Roots of complex semisimple Lie algebras

Lemma

(i) [gα, gβ] ⊆ gα+β. (ii) If α + β = 0, then κ(gα, gβ) = {0}. (iii) κ↾g0 is non-degenerate.

Proposition

Let g = g0 ⊕

α∈Φ

gα be a root space decomposition. For each α ∈ Φ we obtain a subalgebra sl(α) := span({eα, fα, hα}) such that eα ∈ gα, fα ∈ g−α, hα ∈ h, α(hα) = 2, and sl(α) ∼ = sl2(C). In particular, if α ∈ Φ, then −α ∈ Φ.

Proposition

For each α ∈ Φ, there exists a unique tα ∈ h such that α(h) = κ(tα, h) for all h ∈ h.

Proposition

If α ∈ Φ, then dim(gα) = 1 and span(α) ∩ Φ = {−1, 1}.

Roots of complex semisimple Lie algebras

Proposition

If α, β ∈ Φ and β = ±α, then β(hα) ∈ Z and β − β(hα)α ∈ Φ.

Roots of complex semisimple Lie algebras

Proposition

If α, β ∈ Φ and β = ±α, then β(hα) ∈ Z and β − β(hα)α ∈ Φ.

Proposition

h∗ = span(Φ)

Roots of complex semisimple Lie algebras

Proposition

If α, β ∈ Φ and β = ±α, then β(hα) ∈ Z and β − β(hα)α ∈ Φ.

Proposition

h∗ = span(Φ)

Lemma

For each α, β ∈ Φ, (α, β) ∈ Q.

Roots of complex semisimple Lie algebras

Proposition

If α, β ∈ Φ and β = ±α, then β(hα) ∈ Z and β − β(hα)α ∈ Φ.

Proposition

h∗ = span(Φ)

Lemma

For each α, β ∈ Φ, (α, β) ∈ Q.

Lemma

Let {α1, ..., αl} be a basis for h∗ consisting of elements in Φ. Every root β is a linear combination of αi’s with rational coefficients.

Roots of complex semisimple Lie algebras

Proposition

If α, β ∈ Φ and β = ±α, then β(hα) ∈ Z and β − β(hα)α ∈ Φ.

Proposition

h∗ = span(Φ)

Lemma

For each α, β ∈ Φ, (α, β) ∈ Q.

Lemma

Let {α1, ..., αl} be a basis for h∗ consisting of elements in Φ. Every root β is a linear combination of αi’s with rational coefficients.

Corollary

Let E = {

l

• i=1

ciαi | c1, ..., cl ∈ R}, then Φ ⊂ E and (E, (, )) is a real inner product space.

Root systems

Let (E, (, )) be a real inner product space and let v ∈ E \ {0}. The reflection in the hyperplane normal to v is the linear map sv : E → E given by sv(v) = −v and sv(u) = u for all u ∈ E such that (v, u) = 0. By an easy calculation, sv(x) = x − 2(x,v)

(v,v) v

for all x ∈ E. It also follows that (sv(x), sv(y)) = (x, y) for all x, y ∈ E. For convenience sake we write x, y := 2(x,y)

(y,y) . Note

that , is bilinear but not symmetric.

Root systems

Let (E, (, )) be a real inner product space and let v ∈ E \ {0}. The reflection in the hyperplane normal to v is the linear map sv : E → E given by sv(v) = −v and sv(u) = u for all u ∈ E such that (v, u) = 0. By an easy calculation, sv(x) = x − 2(x,v)

(v,v) v

for all x ∈ E. It also follows that (sv(x), sv(y)) = (x, y) for all x, y ∈ E. For convenience sake we write x, y := 2(x,y)

(y,y) . Note

that , is bilinear but not symmetric.

Definition

A root system of E is a finite subset R ⊂ E such that

• 1. 0 /

∈ R and span(R) = E.

• 2. α ∈ R if and only if −α ∈ R, and the only scalar multiples
• f α in R are ±1.
• 3. if α ∈ R, then sα(β) ∈ R for all β ∈ R.
• 4. if α, β ∈ R, then α, β ∈ R.

Root systems

Example

Roots of a complex finite dimensional semisimple Lie algebra.

Root systems

Example

Roots of a complex finite dimensional semisimple Lie algebra. Angle between nonzero vectors u, v ∈ E: cos(θ) =

(u,v) u·v.

Root systems

Example

Roots of a complex finite dimensional semisimple Lie algebra. Angle between nonzero vectors u, v ∈ E: cos(θ) =

(u,v) u·v.

Lemma

Let R be a root system for E, and let α, β ∈ R be linearly independent, then α, ββ, α ∈ {0, 1, 2, 3}. Proof: Clearly α, ββ, α ∈ Z. One easily checks that 0 ≤ α, ββ, α = 4cos2(θ) ≤ 4; if α, ββ, α = 4, then cos(θ) = 1 and θ ∈ πZ, which impLies β = ±α.

Classification of root systems by Dynkin diagrams

Let α, β ∈ R be linearly independent. Without loss of generality, (β, β) ≥ (α, α)). Then |β, α| ≥ |α, β|.

Classification of root systems by Dynkin diagrams

Let α, β ∈ R be linearly independent. Without loss of generality, (β, β) ≥ (α, α)). Then |β, α| ≥ |α, β|. By the above lemma and the 4th axiom, we obtain the following table: α, ββ, α α, β β, α θ

(β,β) (α,α)

π/2 indeterminable 1 1 1 π/3 1 1 −1 −1 2π/3 1 2 1 2 π/4 2 2 −1 −2 3π/4 2 3 1 3 π/6 3 3 −1 −3 5π/6 3

Classification of root systems by Dynkin diagrams

Let α, β ∈ R be linearly independent. Without loss of generality, (β, β) ≥ (α, α)). Then |β, α| ≥ |α, β|. By the above lemma and the 4th axiom, we obtain the following table: α, ββ, α α, β β, α θ

(β,β) (α,α)

π/2 indeterminable 1 1 1 π/3 1 1 −1 −1 2π/3 1 2 1 2 π/4 2 2 −1 −2 3π/4 2 3 1 3 π/6 3 3 −1 −3 5π/6 3 We compute the last row as an example:

(α,β) (α,α) = β, α = −3 = 3(−1) = 3α, β = 3(α,β) (β,β) impLies (β,β) (α,α) = 3, and θ = cos−1( (α,β)

√

(α,α)(β,β)) = cos−1( − 3

2 (α,α)

√

(α,α)3(α,α)) =

cos−1( −

√ 3 2 ) = 5π/6.

Root system example: A2

For example, sl2(C) = span{e} span{f} span{h}.

Root system example: A2

For example, sl2(C) = span{e} span{f} span{h}.

Weyl Group

W(R) = sR

Root system example: A2

For example, sl2(C) = span{e} span{f} span{h}.

Weyl Group

W(R) = sR

Proposition

All Weyl groups are finite.

Classification of root systems by Dynkin diagrams

◮ Irreducible root system: cannot be expressed as a disjoint

union of nonempty sets R1 and R2 such that (R1, R2) = {0}.

Classification of root systems by Dynkin diagrams

◮ Irreducible root system: cannot be expressed as a disjoint

union of nonempty sets R1 and R2 such that (R1, R2) = {0}.

◮ For a reducible root system R = R1 ∪ R2, each Ri is a root

system for the subspace span(Ri).

Classification of root systems by Dynkin diagrams

◮ Irreducible root system: cannot be expressed as a disjoint

union of nonempty sets R1 and R2 such that (R1, R2) = {0}.

◮ For a reducible root system R = R1 ∪ R2, each Ri is a root

system for the subspace span(Ri).

Lemma

Every root system R for E can be expressed as a disjoint union

• f irreducible root systems Ri’s, and E = E1

... En where Ei := span(Ri).

Classification of root systems by Dynkin diagrams

◮ Irreducible root system: cannot be expressed as a disjoint

union of nonempty sets R1 and R2 such that (R1, R2) = {0}.

◮ For a reducible root system R = R1 ∪ R2, each Ri is a root

system for the subspace span(Ri).

Lemma

Every root system R for E can be expressed as a disjoint union

• f irreducible root systems Ri’s, and E = E1

... En where Ei := span(Ri). A base for R: B ⊂ R, a basis for E, every root in R is an integer linear combination of vectors in B with the same sign.

Classification of root systems by Dynkin diagrams

◮ Irreducible root system: cannot be expressed as a disjoint

union of nonempty sets R1 and R2 such that (R1, R2) = {0}.

◮ For a reducible root system R = R1 ∪ R2, each Ri is a root

system for the subspace span(Ri).

Lemma

Every root system R for E can be expressed as a disjoint union

• f irreducible root systems Ri’s, and E = E1

... En where Ei := span(Ri). A base for R: B ⊂ R, a basis for E, every root in R is an integer linear combination of vectors in B with the same sign.

Proposition

Every root system has a base.

Classification of root systems by Dynkin diagrams

Elements of a chosen base B for a root system are called sim- ple roots with respect to B.

Classification of root systems by Dynkin diagrams

Elements of a chosen base B for a root system are called sim- ple roots with respect to B.

Proposition

The angle between any pair of simple roots is obtuse.

Classification of root systems by Dynkin diagrams

Elements of a chosen base B for a root system are called sim- ple roots with respect to B.

Proposition

The angle between any pair of simple roots is obtuse.

Proposition

Let θ be the angle between α, β ∈ R. If θ > 2π/3, then (α, α) = (β, β).

Classification of root systems by Dynkin diagrams

Elements of a chosen base B for a root system are called sim- ple roots with respect to B.

Proposition

The angle between any pair of simple roots is obtuse.

Proposition

Let θ be the angle between α, β ∈ R. If θ > 2π/3, then (α, α) = (β, β).

Dynkin diagrams

Let R be a root system for E and let B be a base for R. We construct a Dynkin diagram as follows: draw a vertex for each α ∈ B and draw α, ββ, α number of edges between vertices denoting α and β. Put an > on edges between α and β corresponding to the inequality of their lengths.

Classification of root systems by Dynkin diagrams

◮ Root space isomorphism: angle-preserving vector space

isomorphism φ : E1 → E2, i.e. φ(R1) = R2, and α, β = φ(α), φ(β).

Classification of root systems by Dynkin diagrams

◮ Root space isomorphism: angle-preserving vector space

isomorphism φ : E1 → E2, i.e. φ(R1) = R2, and α, β = φ(α), φ(β).

◮ One-to-one correspondence between Dynkin diagrams

and isomorphism classes of root systems.

Classification of root systems by Dynkin diagrams

◮ Root space isomorphism: angle-preserving vector space

isomorphism φ : E1 → E2, i.e. φ(R1) = R2, and α, β = φ(α), φ(β).

◮ One-to-one correspondence between Dynkin diagrams

and isomorphism classes of root systems.

Cartan matrix

Let B = {α1, ..., αn} be a base for R. A Cartan matrix C for R is a n × n matrix with Cij = αi, αj. Observe that each diagonal entry of C is 2, and Cij ∈ {0, −1, −2, −3} for i = j.

Classification of root systems by Dynkin diagrams

◮ Root space isomorphism: angle-preserving vector space

isomorphism φ : E1 → E2, i.e. φ(R1) = R2, and α, β = φ(α), φ(β).

◮ One-to-one correspondence between Dynkin diagrams

and isomorphism classes of root systems.

Cartan matrix

Let B = {α1, ..., αn} be a base for R. A Cartan matrix C for R is a n × n matrix with Cij = αi, αj. Observe that each diagonal entry of C is 2, and Cij ∈ {0, −1, −2, −3} for i = j.

Proposition

A root system is irreducible if and only if its Dynkin diagram is connected.

Classification of Dynkin diagrams

Theorem

Let R a root system and let Γ be a Dynkin diagram of R, then Γ must fall into one of the following famiLies:

Classification of Dynkin diagrams

Theorem

Let R a root system and let Γ be a Dynkin diagram of R, then Γ must fall into one of the following famiLies: Proof: Graph theory and combinatorics.

Classification of complex semisimple Lie algebras by root systems

Conjugation Theorem

Let g be a complex semisimple Lie algebra. Then Aut(g) acts transitively on the set of Cartan subalgebras of g.

Classification of complex semisimple Lie algebras by root systems

Conjugation Theorem

Let g be a complex semisimple Lie algebra. Then Aut(g) acts transitively on the set of Cartan subalgebras of g.

Theorem

All root systems of a complex semisimple Lie algebra are isomorphic.

Classification of complex semisimple Lie algebras by root systems

Conjugation Theorem

Let g be a complex semisimple Lie algebra. Then Aut(g) acts transitively on the set of Cartan subalgebras of g.

Theorem

All root systems of a complex semisimple Lie algebra are isomorphic.

Corollary

Root systems of isomorphic complex semisimple Lie algebras are isomorphic.

Classification of complex semisimple Lie algebras by root systems

Conjugation Theorem

Let g be a complex semisimple Lie algebra. Then Aut(g) acts transitively on the set of Cartan subalgebras of g.

Theorem

All root systems of a complex semisimple Lie algebra are isomorphic.

Corollary

Root systems of isomorphic complex semisimple Lie algebras are isomorphic.

Serre’s Construction

For each isomorphism class of root systems there exists a unique construction of a complex semisimple Lie algebra.

Conclusion and future directions

There is a one-to-one correspondence between Dynkin diagrams and complex semisimple Lie algebras up to isomor- phism.

Conclusion and future directions

There is a one-to-one correspondence between Dynkin diagrams and complex semisimple Lie algebras up to isomor- phism.

Future directions

◮ Classification of real semisimple Lie algebras by Satake

diagrams via realification and complexification.

◮ Freudenthal Magic Square construction of semisimple Lie

algebras from real (and complex) composition algebras. L = (Der(A) Der(J3(B))) (A0 ⊗ J3(B)0)

A \ B R C H O R A1 A2 C3 F4 C A2 A2 × A2 A5 E6 H C3 A5 D6 E7 O F4 E6 E7 E8

References

• 1. K. Conrad. The minimal polynomial and some applications
• 2. K. Erdmann. and M. J. Wildon. Introduction of Lie

algebras.

• 3. A. Henderson. Representations of Lie algebras - an

introduction through gln

• 4. J. S. Milne. Lie algebras, algebraic groups, and Lie groups.
• 5. D. A. Vogan. Generalized eigenspaces.