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LIE GROUPS AND LIE ALGEBRAS – A FIRST VIEW ANKE D. POHL Abstract. Talk in the IRTG Student Seminar 04.04.06. This is a draft version and thought for internal use only. This exposition is based on the book “Lie groups” by J.J. Duistermaat and J.A.C. Kolk. 1. Motivation and definitions The origin of Lie theory is Lie’s idea to develop a “Galois theory” for differential equations and Klein’s “Erlangen Program” of understanding symmetry groups as the fundamental object to describe a geometry. For example, symmetric spaces can be completely described by Lie groups. Definition 1.1. A Lie group is a group G that at the same time is a finitedimensional (real) smooth manifold, such that the two group operations of G µ : G × G → G, (x, y) 7→ xy −1 ι : G → G, x 7→ x (multiplication), (inversion), are smooth mappings. Definition 1.2. A real Lie algebra is a vector space g over R together with a bilinear mapping g × g → g, (X, Y ) 7→ [X, Y ], called the Lie bracket of g, satisfying [X, Y ] = −[Y, X] [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 (antisymmetry) (Jacobi identity) for all X, Y, Z ∈ g. The group structure of a Lie group G can be completely recovered from the algebraic information contained in the Lie bracket of its Lie algebra g, at least locally near the identity element 1 ∈ G. This is very surprising since the Lie bracket of g is defined in terms of second-order derivatives at 1. This fact motivates the study of Lie groups by studying their Lie algebras as a problem in (multi-) linear algebra because in principle all local information is known. 1 2 A. POHL In view of symmetric spaces, the correspondence of the local group structure of G at 1 and the algebraic structure of g gives to the opportunity to describe geometric objects or properties purely algebraically with the help of g, e.g. geodesics have a very simple characterization in terms of certain elements of g. 2. From a Lie group to its Lie algebra Let G be a Lie group. The identity element of G will be called 1 or 1G . We denote the tangent space of the Lie group G at the identity element 1 by g := T1 G = {D : C ∞ (1) → R | D derivation}. ∼ T1 G × T1 G = g × g we obtain With the isomorphism T1,1 (G × G) = T1,1 µ : g × g → g, (X, Y ) 7→ X + Y. Further we find T1 ι : g → g, X 7→ −X. These two mappings give an additive group structure of g and (g, +) becomes a commutative Lie group. More precisely, using the scalar multiplication of derivations of C ∞ (1) turns g into a vector space of the same dimension as G. For each x ∈ G we define Ad x : G → G, y 7→ xyx−1 , the conjugation by x in G. Because (Ad x)(1) = 1, the tangent mapping of Ad x at 1 is a linear mapping Ad x := T1 (Ad x) : g → g, called the adjoint mapping of x or the infinitesimal conjugation by x in g. The mapping Ad : G → GL(g), x 7→ Ad x is a homomorphism of groups. This mapping is called the adjoint representation of G in g. Due to our definition of Lie groups this mapping is smooth. If we identify as usual the tangent space at 1 of GL(g) with End(g), then we obtain the linear mapping ad := T1 Ad : g → End(g). For each X, Y ∈ g we put [X, Y ] := (ad X)(Y ) ∈ g. Since ad is a linear mapping from g into End(g), the mapping (X, Y ) 7→ [X, Y ] is bilinear. With [·, ·] as product, g becomes an algebra over R. LIE GROUPS AND LIE ALGEBRAS 3 Definition 2.1. The tangent space g = T1 G, endowed with the mapping g × g → g, (X, Y ) 7→ [X, Y ] = (ad X)(Y ) as the product structure, is called the Lie algebra of the Lie group G. The product [X, Y ] is called the Lie bracket of X, Y ∈ g. After the following example we will see that the Lie algebra of a Lie group is in fact a Lie algebra in the sense of Definition 1.2. Example 2.2. Let V be a finite-dimensional real vector space. The general linear group G = GL(V ) is a Lie group (mainly due to Cramer’s rule) and we get g = T1 GL(V ) = End(V ). The adjoint mapping of x ∈ GL(V ) is Ad x : Y 7→ x ◦ Y ◦ x−1 and as the Lie bracket we get [X, Y ] = (ad X)(Y ) = X ◦ Y − Y ◦ X, which is the commutator in End(V ). Theorem 2.3. Let G be a Lie group and g its Lie algebra. Then for all X, Y, Z ∈ g we have [X, Y ] = −[Y, X] and [[X, Y ], Z] = [X, [Y, Z]] − [Y, [X, Z]]. The proof extensively use the following proposition which is of its own interest since it tells that a smooth homomorphism of Lie groups comes down to a smooth homomorphism of their Lie algebras. Proposition 2.4. Let G and G0 be Lie groups with Lie algebras g and g0 , respectively, and suppose that Ψ : G → G0 is a group homomorphism that is differentiable at 1 ∈ G. Let ψ := T1 Ψ. Then (a) ψ ◦ Ad x = Ad Ψ(x) ◦ ψ for each x ∈ G. (b) ψ : g → g0 is a homomorphism of Lie algebras, i.e. it is a linear mapping g → g0 and in addition ψ[X, Y ]g = [ψX, ψY ]g0 for all X, Y ∈ g. Now we see that the Lie algebra of a Lie group is really a Lie algebra. Clearly, historically the notion of “Lie algebra” derives from the properties of the Lie algebra of a Lie group. 4 A. POHL 3. From the Lie algebra of a Lie group to a Lie subgroup For each x ∈ G define the left multiplication L(x) : G → G by L(x) : y 7→ xy. A vectorfield v on G is called left invariant, if vL(x)y = Ty L(x)vy for all y ∈ G. A left invariant vector field is uniquely determined by its value X := v1 ∈ g at y = 1: (1) vx = T1 L(x)X. Conversely, since L(xy) = L(x) ◦ L(y) each tangent vector X ∈ g defines a left invariant vector field on G by (1): We have vy := T1 L(y)X and vL(x)y = T1 L(xy)X = T1 (L(x) ◦ L(y))X = Ty L(x) ◦ T1 L(y)X = Ty L(x)vy . This left invariant vector field we denote by X L . Theorem 3.1. For every X ∈ g there is a unique homomorphism h = hX : (R, +) → (G, ·) that is differentiable at t = 0 and satisfies dh dt (0) = X. L It is equal to the solution curve of X starting at the identity element of G. The flow of X L is global. Definition 3.2. For each X ∈ g we define exp(X) := Φ1 , where Φt is the flow of the vector field X L . The mapping exp : g → G, X 7→ exp X is called the exponential mapping from g into G. Proposition 3.3. There is an open neighborhood U of 0 in g and V of 1 in G such that the exponential mapping is a diffeomorphism from U onto V . Theorem 3.4. There is an open neighborhood W of (0, 0) in g × g and an smooth mapping µ : W → g which can be completely described by the algebraic structure of g such that exp(X) exp(Y ) = exp µ(X, Y ) for all (X, Y ) ∈ W . The mapping µ is given by the Baker-Campbell-Hausdorff formula (also known as Dynkin’s formula or the Taylor expansion of the product at the LIE GROUPS AND LIE ALGEBRAS origin in logarithmic coordinates): ∞ X (−1)k µ(X, Y ) = Y + X + k+1 k=1 X li ,mi ≥0, 5 1 l1 + · · · + lk + 1 li +mi >0 (ad X)l1 (ad Y )m1 (ad X)lk (ad Y )mk (X). ◦ ◦ ··· ◦ ◦ l1 ! m1 ! lk ! mk ! To get an better idea of this formula we take a look at the expansion up to order 4: 1 1 1 µ(X, Y ) = X + Y + [X, Y ] + [X, [X, Y ]] + [Y, [Y, X]] 2 12 12 1 5 + [Y, [X, [Y, X]]] + O(|(X, Y )| ) as (X, Y ) → (0, 0). 24