Classification of complex semisimple
Lie algebras using Dynkin diagrams
August Geelmuyden
University of Oslo
Part I
Root systems
I.
Symmetry of a vector
Definition 1. Let α 6= 0 be a vector in a vector space V. A Symmetry of the vector α is a
linear transformation sα ∈ Aut(V) satisfying
(S1) sα (α) = −α.
(S2) sα fixes a hyperplane H.
Proposition 1. There exist at most one sα .
Proof. Let α ∈ V be a non-zero vector and let s and s 0 be two symmetries for α. Since α extends
the basis for H to a basis for V every vector x ∈ V can be written in the form x = β + Cα where
β ∈ H and C is a constant from the underlying field. Then s(x) = s(β) + Cs(α) = β − Cα since s is
a symmetry for the vector α. Since s 0 is also a symmetry for α, we have s 0 (x) = β − Cα and thus
s 0 (x) = s(x). Since x was arbitrarily chosen, the proof is complete.
Proposition 2. Every sα is of the form sα (x) = x − hα∗ , xiα.
Proof. Since there exist at most one sα , it suffices to show that sα (x) = x − hα∗ , xiα is a symmetry
for α. First note that the map x 7→ hα∗ , xiα is a linear map by definition. Since hα∗ , βi = 0 for all
β ∈ H it follows that sα fixes H. Finally, since hα∗ , αi = 2, we have sα (α) = −α.
II.
Root systems
Definition 2. A set R ⊂ V is a root system if
(R1) R does not contain 0, is finite and spans V.
(R2) For all α ∈ R there is a symmetry sα with sα (β) ∈ R for all β ∈ R.
(R3) For all α, β ∈ R there is some n ∈ Z so that sα (β) − β = nα.
One one-dimensional, four two-dimensional
Definition 3. A root system R is said to be irreducible if for all α ∈ R the only distinct vector
being proportional to α is −α.
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III.
Weyl group
Proposition 3. If R is a root system, then the set W of all symmetries sα of vectors from R form
a subgroup of Aut(V) called the Weyl group.
Proposition 4. There exist a symmetric, positive definite bilinear form (., .) that is invariant under
the Weyl group.
p
Definition 4. The length of an element α ∈ R is defined as |α| = (α, α). The angle θ between
two elements α and β of a root system R is given by the relation (α, β) = |α||β| cos θ.
hα∗ , xi = 2
(α, β)
∈Z
(α, α)
Definition 5.
n(α, β) = 2
(α, β)
.
(α, α)
(1)
Proposition 5. Any non-empty subplane of a root system R is itself a root system.
Proposition 6. There is exactly seven distinct planar root systems.
Proposition 7. If α and β are elements in a root system R with (α, β) > 0, then α − β ∈ R.
Proposition 8. If R is a root system and α, β ∈ R spans an angle φ 6 π then there exist a pair of
elements γ, ρ ∈ R spanning an angle nφ for n ∈ Z.
IV.
Bases
Definition 6. A set B ⊂ R is called a basis for the root system R if it spans V and for all
P
α ∈ R there is a set of integers {nb } with the same sign satisfying α = b∈B nb b.
Choose t ∈ V ∗ so that ht, αi =
6 0∀α ∈ R and define
R+
t = α ∈ R ht, αi > 0 .
+
+
It is clear that this induces a separation R = (R+
t ) ∪ (−Rt ) and that Rt spans the same space as R.
Definition 7. A root α ∈ R+
t is said to be decomposable if there exists β, γ ∈ R so that α = β + γ.
A root that is not decomposable is said to be indecomposable.
Proposition 9. The set St of all indecomposable roots is a base for R.
Isomorphic systems and well-defined-ness
Let S and S 0 be bases for the two root systems R and R 0 respectively, being subsets of the two vector
spaces V and V 0 . Let Φ : S → S 0 be a bijection satisfying n(Φ(α), Φ(β)) = n(α, β) for all α, β ∈ S.
Then a unique isomorphism f : V → V 0 exists which is an extension of Φ to R. First note that if
α, β ∈ S then
sΦ(α) ◦ f (β) = sΦ(α) ◦ Φ(β) = Φ(β) − n (Φ(α), Φ(β)) Φ(α) = Φ(β) − n (α, β) Φ(α)
and
(f ◦ sα ) (β) = f (β − n(α, β)α) = f(β) − n(α, β)f(α) = Φ(β) − n(α, β)Φ(α).
That is, f ◦ sα = sΦ(α) ◦ f. If W and W 0 are the weyl groups of R and R 0 respectively, then we may
note that R = WS and R 0 = W 0 S 0 . Thus R 0 = W 0 S 0 = W 0 f(S) = f(WS) = f(R). Hence R ' R 0 .
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V.
Cartan matrix
Let S = {αi }i∈I be a base for a root system R. Then the matrix n(αi , αj ) determines R.
VI.
Coxeter graph
Let nij be the cartan matrix and define Nij = nij nji . Let Nij denote the number of vertices
connecting the nodes i and j.
VII.
Dynkin diagram
Part II
Semisimple Lie algebras
Lie algebra
Definition 8 (Lie algebra). A lie algebra g is a vector space over a field F equipped with a bilinear
binary operation [., .] : g × g → g, called the lie bracket, satisfying
i) (Alternativity)
[x, x] = 0 for all x ∈ g.
ii) (Jacobi identity)
[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z ∈ g.
Bilinearity may be used together with alternativity to show that the lie bracket satisfies anticommutativity
[x, y] = −[y, x] for all x, y ∈ g.
Definition 9 (Adjoint representation). If g is a lie algebra over a field F then the representation
ad : g → End(g)
given by x 7→ adx is called the adjoint representation of the algebra g.
On End(g), which is the set of all homomorphisms from g to g, the lie bracket is defined to be
[adx , ady ] = adx ◦ ady − ady ◦ adx .
For lie algebras g of finite dimension, the fact that End(g) is isomorphic to the general linear group
gl(g) is often used.
Definition 10 (Adjoint action). If x ∈ g is an element in a lie algebra over a field F then the adjoint
representation of x is the adjoint action adx : g → g where adx (y) = [x, y].
It may well be noted that the adjoint action is, by the jacobi identity, a derivation
adx [y, z] = [x, [y, z]] = [y, [x, z]] + [[x, y], z] = [adx (y), z] + [y, adx (z)]
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Definition 11 (subalgebra and ideal). A vector subspace s of a Lie algebra g is a subalgebra if
[x, y] ∈ s for all x, y ∈ s. If a Lie subalgebra s satisfies [x, y] ∈ s for all x ∈ g and s, then s is said to
be an ideal of g.
Semisimplicity
Definition 12 (Central series). For a lie algebra g the central series is a descending series of
ideals {Cn g}n>1 so that
C1 g = g
Cn g = g, Cn−1 g
It follows that C2 g = [g, g] and [Cn g, Cm g] ⊂ Cn+m g.
Definition 13 (Nilpotent). If there exist a positive integer n so that Cn g = 0 then g is said to be
nilpotent. More precisely g is said to be nilpotent of class 6 n.
Definition 14 (Derived series). For a lie algebra g the derived series is a descending series of
ideals {Dn g}n>1 so that
D1 g = g
Dn g = Dn−1 g, Dn−1 g
Definition 15 (Solvable). If there exist a positive integer n so that Dn g = 0 then g is said to be
solvable. More precisely g is said to be solvable of length 6 n.
Notice that if g is solvable of order 2, then g is abelian.
Theorem 1. Every quotient, subalgebra and extension of a solvable lie algebra is solvable.
Theorem 2. There exists a maximal solvable ideal.
Definition 16 (Radical). The maximal solvable ideal of a lie algebra g is called the radical r of g.
Definition 17 (Semisimplicity). A lie algebra g is said to be semisimple if its radial r is zero.
Definition 18 (Invariance). A symmetric bilinear form B : g × g → F is said to be invariant if
B ([x, y] , z) + B (y, [x, z]) = 0 ∀x, y, z ∈ g.
The killing form B(x, y) = Tr(adx ◦ ady ) is both invariant and symmetric.
Definition 19 (Orthogonal space). The orthogonal space a⊥ of a subset a ⊆ g relative to a bilinear
form B : g × g → F is given as the set
a⊥ = x ∈ gB(x, y) = 0 for all y ∈ a .
Definition 20 (Degenerate). A bilinear form i said to be degenerate on a Lie algebra g if there is a
nonzero element x in g⊥ . If g⊥ = 0, then the bilinear form is said to be nondegenerate.
Theorem 3 (Cartan-Killing Criterion). A Lie algebra is semisimple if, and only if, its Killing form
is nondegenerate.
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Cartan subalgebras
Definition 21 (Cartan subalgebra). A subalgebra h of g is called a Cartan subalgebra if it is
nilpotent and its own normalizer.
Definition 22 (nilspace). Let g be a complex Lie algebra and fix an element x in g. If λ ∈ C the
nilspace of adx − λI is the space
gλx = y ∈ g∃n ∈ N : (adx − λI)n y = 0 .
The space g0x is referred to as the generalized nullspace of x.
This is a decomposition of g into invariant subspaces of adx . That is, for any x ∈ g we may write
M
g=
gλx .
λ∈C
Definition 23 (regular element). An element x of a complex Lie algebra g is said to be regular if
its generalized nullspace has minimal dimension.
It should now be noted that the generalized nullspace g0x of x is a subalgebra of g. If x is regular,
then g0x is a Cartan subalgebra og g.
Factorization of complex semisimple Lie algebras
For a semisimple, complex Lie algebra any Cartan subalgebra is abelian and its own centralizer. It
so turns out that any Cartan subalgebra is a maximal abelian subalgebra.
Let x be an element of a Cartan subalgebra h of a complex semisimple Lie algebra g. Then the
endomorphism adx is diagonalizable.
Definition 24 (eigen-subspaces). Let g be a complex, semisimple Lie algebra and h a Cartan
subalgebra of g. For α ∈ h∗ let
gα = x ∈ g[y, x] = α(y)x for all y ∈ h
An element α ∈ h∗ is said to be a root of g if α 6= 0 and gα 6= 0.
Theorem 4. If R is the set of all roots of a complex, semisimple Lie algebra g, then
M
g'h⊕
gα .
α∈R
Proof. The endomorphisms ady of g for y ∈ h are diagonalizable.
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