ALG `EBRES DE LIE
Contents
1. Day 1: Lie algebras 1
1.1. Definition 1
1.2. Motivation: Lie groups 2
1.3. Basic properties 4
1.4. Classical Lie algebras 5
1. Day 1: Lie algebras
• (a) Definition
• (b) Motivation: Lie groups
• (c) Basic properties
• (d) Classical Lie algebras
1.1. Definition. LetK be a field. A Lie algebra overK is a vector spaceg overK with an operation, denoted [,] (bracket), that satisfies the following three axioms.
Axiom 1 [,] :g×g→ g is bilinear over K.
Axiom 2 [X, X] = 0 for all X∈g.
Axiom 3 (Jacobi identity) For all X, Y, Z ∈g,
[X,[Y, Z]] + [Y,[Z, X]] + [Z,[X, Y]] = 0.
Note that Axioms 1 + 2, applied to 0 = [X+Y, X+Y] = [X, X]+[Y, X]+
[X, Y] + [Y, X], implies that the bracket is alternating:
[X, Y] =−[Y, X]
Moreover, the alternating property implies Axiom 2 ifK is of characteristic 6= 2. (In this course we almost always assume K of characteristic 0.)
A Lie subalgebra h ⊂ g is a K-subspace of g that is stable under the bracket.
The main example of a Lie algebra is End(V), the algebra of endomor- phisms of the K-vector spaceV, with bracket
[X, Y] =XY −Y X.
Bilinearity and the Jacobi identity are left as an exercise!
Classification of subalgebras ofEnd(V) up to appropriate equivalence re- lations is the subject of representation theory. The first part of the course
1
develops the classification of finite-dimensional Lie algebras, and this is fol- lowed by their finite-dimensional representation theory. This theory is due primarily to Elie Cartan and Hermann Weyl and was one of the main inno- vations of the first third of the 20th century. The remainder of the course treats various infinite-dimensional representation theories.
1.2. Motivation: Lie groups. So what did (Sophus) Lie do? One way to answer that question is to take a detour through the non-commutative parts of commutative algebra.
Definition 1.1. Let R be a commutative ring, S an R algebra (not nec- essarily commutative nor even associative). A derivationof S over R is an R-linear mapD:S → S such that, for all X, Y ∈S,
D(XY) =D(X)Y +XD(Y).
In other words, a derivation is a map that satisfies the Leibniz rule and commutes with multiplication by R. The set of derivations of S over R is denoted DerR(S). It is anR-module by definition.
Proposition 1.2. Let Di, i= 1,2, be two derivations of S over R. Then [D1, D2] =D1◦D2−D2◦D1 is a derivation. The bracket satisfies the Jacobi identity and makes DerR(S) a Lie algebra overR (obvious definition).
Proof. We prove the first part.
[D1, D2](XY) =D1(D2(XY))−D2(D1(XY))
=D1(D2(X)Y +XD2(Y))−D2(D1(X)Y +XD1(Y))
= (D1D2)(X)Y +D2(X)D1(Y) +D1(X)D2(Y) +X(D1D2)(Y)
−(D2D1)(X)Y −D2(Y)D1(X)−D2(X)D1(Y)−X(D2D1)(Y)
=(D1D2)(X)Y +X(D1D2)(Y)−(D2D1)(X)Y −X(D2D1)(Y)
=[D1, D2](X)Y +X[D1, D2](Y).
(1.3)
The Jacobi identity holds for any associative algebra (see below) and is left as an exercise.
Example 1.4. Let R=C, S =C∞(R), the ring of complex valued smooth functions on the line. LetD= dxd. This is the original Leibniz rule.
Example 1.5. Let R = C, S = C∞(Rn). Let Di = dxd
i. Then each Di∈DerR(S) and they generate a Lie algebra with trivial bracket, because
∂2
∂xi∂xj = ∂2
∂xj∂xi.
We can also define derivations of Lie algebras by the same property.
Lemma 1.6. Let g be a Lie algebra over K, X ∈g. Define ad(X) :g → g by
ad(X)Y = [X, Y].
Then ad(X)∈DerK(g).
Proof. We need to show that, for allY, Z ∈g,
ad(X)[Y, Z] = [ad(X)Y, Z] + [Y, ad(X)Z], in other words
[X,[Y, Z]] = [[X, Y], Z] + [Y,[X, Z]] =−[Z,[X, Y]]−[Y,[Z, X]]
which is just the Jacobi identity.
A Lie algebra with trivial bracket is calledabelian.
Example 1.7. More generally, let M be a smooth manifold, R = C, S = C∞(M). Then any smooth vector field X ∈ V ect(M) on M defines a derivation of S. The bracket operation makes V ect(M) into an infinite- dimensional Lie algebra. Moreover, V ect(M) is a module overS under left multiplication. Since this doesn’t commute with right multiplication, it’s easy to see that V ect(M) is not an abelian Lie algebra, even whenM =R.
[ d dx, f d
dx]g= d dx(f d
dx)g−fd2g dx2 = df
dx dg
dx+fd2g
dx2 −fd2g
dx2 =f0dg dx; thus [dxd, fdxd] =f0·dxd.
Now suppose M = G is a Lie group, i.e. a smooth manifold with a group structure whose operations are smooth. Inside V ect(G) is the sub- spaceLie(G) of left-invariant vector fields; i.e., smooth vector fieldsX that commute with left translation byG:
XLg(f) =Lg(X(f)),∀g∈G, f ∈C∞(G).
Restriction to the identity defines an isomorphismLie(G) −→T∼ e(G), and thus Lie(G) is a finite-dimensional subspace of V ect(G). Moreover, the defining relation shows thatLie(G) is stable under bracket. In this way we obtain a map
Lie groups ⇒Lie algebras
and by functoriality of vector fields, this is even a functor.
Here are some of the basic properties of this functor.
Theorem 1.8(Cartan-von Neumann). Every closed subgroup of a Lie group is a Lie subgroup.
Theorem 1.9. Every Lie subalgebra of Lie(G) is the Lie algebra of a Lie subgroup.
This is defined by means of the exponential map from Lie(G) to G: if X ∈ Lie(G) then t7→ exp(tX) is a 1-parameter subgroup; in other words, this is a theorem in ordinary differential equations on manifolds. But the Theorem needs to be interpreted carefully. Let K = R, G be the torus R2/Z2. Then Lie(G) = R2 is abelian. Every subspace of Lie(G) is thus a Lie subalgebra (with zero bracket). But if h is the subspace spanned by
(a, b) with ba irrational, then the exponential of his a dense subgroup ofG that is locally of dimension 1.
Nevertheless, the following theorem holds (with definition of HomLA in the next lecture).
Theorem 1.10. Let LGp be the category of Lie groups, LA the category of Lie algebras. The functorial map
HomLgp(G, G0) → HomLA(Lie(G), Lie(G0))
is injective if G is connected, and is bijective if G is simply connected.
In particular, the functor defines an equivalence between the category of simply connected Lie groups and the category of Lie algebras.
1.3. Basic properties.
Example 1.11. LetA be an associative algebra. Then the bracket[X, Y] = XY −Y X, X, Y ∈A makes A into a Lie algebra.
Proof.
X(Y Z−ZY)−(Y Z−ZY)X+Y(ZX−XZ)
−(ZX−XZ)Y +Z(XY −Y X)−(XY −Y X)Z
=XY Z−XZY −Y ZX+ZY X+Y ZX−Y XZ
−ZXY +XZY +ZXY −ZY X−XY Z+Y XZ = 0 (1.12)
Definition 1.13. Let g be a Lie algebra. A K-subspace h of g is a Lie subalgebra if it is stable under the bracket; i.e., if [X, Y]∈hfor allX, Y ∈h.
A K-subspace hofg is a Lie ideal if [X, Y]∈hfor all X∈g, Y ∈h.
The correspondence of the last section associates Lie subalgebras (resp.
Lie ideals) ofLie(G) to Lie subgroups (resp. normal subgroups) ofG.
Lemma 1.14. Letg be a Lie algebra. LetDg= [g,g]be the linear subspace generated by {[X, Y], X, Y ∈ g}. Then Dg is a Lie ideal of g, called the derived subalgebra of g.
Proof. LetX, X0, Y0∈g. We need to see that ad(X)[X0, Y0]∈Dg.
But
ad(X)[X0, Y0] = [ad(X)X0, Y0] + [X0, ad(X)Y0]
is a sum of two brackets, so this is clear.
We have the relation
D(Lie(G)) =Lie(Gder)
whereGder is the derived subgroup of G(generated by commutators).
The conjugation action ofGon itself fixes the tangent space at the identity and preserves the brackets of vector fields, as well as left-invariant vector
fields. This defines a homomorphism Ad:G→ Aut(g), which is trivial on the centerZofG. One shows thatdAd:g→ End(g) =TAut(g),eis identified withad.
Supposegis finite dimensional, and letXi, i= 1, . . . , N, be a basis. Then for all i, j we have
[Xi, Xj] =
N
X
k=1
akijXk.
Theakij ∈K are thestructure constantsand they determine gup to isomor- phism. These have to be compatible with Axioms 2 and 3, namely
akii= 0, akij =−akji,∀i, j, k and
X
k
(akijamk`+akj`amki+ak`iamkj)
for all i, j, `, m. Moreover, these conditions suffice to define a Lie algebra.
For example ifg=sl(2), we can choose as basis the three matrices e=
0 1 0 0
, f =
0 0 1 0
, h=
1 0 0 −1
. Then one computes:
[h, e] = 2e; [h, f] =−2f; [e, f] =h.
These standard relations determine sl(2) uniquely; the entire theory of semisimple Lie algebras is based on these relations.
For any vector space V a triple of endomorphisms of V {h, e, f} satis- fying these relations is called an sl(2)-triple; they generate a Lie algebra isomorphic to sl(2).
1.4. Classical Lie algebras. LetV be ann-dimensional vector space over K. The Lie algebra of GL(V) is gl(V) := End(V) with the commutator bracket. This can be seen by observing that
[exp(tX), exp(tY)] =t[X, Y] +O(t2)
and applying the definition of the Lie algebra, but we can just take this to be the definition.
Next, the subspacesl(V) ={X∈End(V) |T r(X) = 0} is clearly stable under bracket. Indeed, for any X, Y ∈ gl(V), T r([X, Y]) = T r(XY) − T r(Y X) = 0, so we can write
[gl(V),gl(V)]⊂sl(V).
The matrix algebra has some other distinguished Lie subalgebras. Let b be the space of upper triangular matrices,n⊂bthe subspace with diagonal entries = 0, and t ⊂b the subset of diagonal matrices. One checks that if
X, Y ∈b (or n, ort) then so is [X, Y]. Moreover, t is abelian,n is an ideal inb, and it follows by bilinearity (since b=t+u as vector spaces) that
D(b)⊂n.
It’s easy enough to prove using elementary matrices that D(b) =n.
Let V be a vector space over K with a non-degenerate bilinear form h,i. Let O(V) be the orthogonal group of this bilinear form, SO(V) = O(V) ∩SL(V). This is a closed subgroup of GL(V), and so the above theorems show that its Lie algebra is a subalgebra ofgl(V); but we can also compute it. The defining property ofO(V):
hg(v), g(w)i=hv, wi,∀v, w∈V can be applied to g=exp(tX)
hv+tXv, w+tXwi+O(t2) =hv, wi implies that
d
dthv+tXv, w+tXwit=0 = 0 i.e.
hXv, wi+hv, Xwi= 0,∀v, w∈V.
In other words Lie(O(V)) = Lie(SO(V)) = so(V) is given by endomor- phisms that areskew-symmetricrelative toh,i. For example, ifhv, wi=tvw is the standard dot product onKn,
O(V) ={g∈GL(n, K) |tg=g−1}.
Then
so(n) ={X ∈M(n, K) |tX=−X}.
(Take the equation texp(tX) = exp(ttX) = exp(−X), differentiate with respect tot, and sett= 0.)
The same applies if h,i is alternating. Then the symmetry group is the symplectic group Sp(V). Say dimV = 2nand the alternating form is given by
Jn=
0 In
−In 0
so that Sp(2n) ={g∈GL(2n, K)|tgJng=Jn}.Then sp(2n) ={X ∈M(2n, K) |tXJn+JnX = 0}.
In both of the last two examples, the Lie algebra is defined by explicit linear equations in the matrix algebra. One sees that if dimV = n then dimgl(V) =n2, dimsl(V) =n2−1. To compute dimsp(2n), write
X=
A B C D
withA, B, C, D all n×nblock matrices.
Over K = R there are additional families of classical groups and Lie algebras defined by symmetries of hermitian and skew-hermitian forms on
vector spaces overCor the quaternions (various kinds of unitary groups and Lie algebras).
