Lie Group

A Lie group is a smooth manifold G together with a smooth map from G x G --> G that makes G into a group and such that the inverse map g -->  g^-1 is a smooth map of G to itself. 

A Lie group is a differentiable manifold G which is also a group and such that the group product G x G --> G and the inverse map g --> g^-1 are differentiable. 

Lie Algebra

A finite-dimensional real or complex Lie algebra is a finite-dimensional real or complex vector space g, together with a map [.,.] from g x g into g, with the following properties:

1. [.,.] is bilinear.

2. [X,Y] = - [Y,X] for all X,Y ∈ g (skew symmetry).

3. [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0 (Jacobi identity)

g can be any vector space and that the "bracket" operation [.,.] can be any bilinear, skew symmetric map that satisfies the Jacobi identity. Because g does not necessarily have a product operation defined on it. So [X,Y] is not necessary equal to XY-YX. For any Lie algebra, the Jacobi identity means that the bracket operation behaves as if it were XY-YX, even if it is not actually defined this way. We can think of the bracket operation as making g into an algebra in the general sense. This algebra is not associative. The Jacobi identity is to be thought of as a substitute for associativity.

If g and h are Lie algebras, then a linear map Φ : g --> h is called a Lie algebra homomorphism if ϕ([X,Y]) = [ϕ(X),ϕ(Y)] for all X, Y ∈ g . In addition, if ϕ is one-to-one and onto, then ϕ is called a Lie algebra isomorphism. A Lie algebra isomorphism of a Lie algebra with itself is called a Lie algebra automorphism.

Let g be a Lie algebra. For X ∈ g , define a linear map ad_X : g --> g by ad_X(Y) =[X,Y].  Thus, the map (X --> ad_X), "ad" can be viewed as a linear map from g into gl(g), where gl(g) denotes the space of linear operators from g to g.

ad_X(Y)² = [X,[X,Y]]. and ad_[X,Y] (Z) = [[X,Y],Z]. By Jacobi identity and skew symmetry, [[X,Y],Z] = [X,[Y,Z]] + [Y,[Z,X]] = [X,[Y,Z]] - [Y,[X,Z]] =  [ad_X ad_Y - ad_Y ad_X] (Z) =  [ad_X , ad_Y] (Z). Therefore, we have ad_[X,Y] =  [ad_X , ad_Y]. This means ad : g --> gl(g) is a Lie algebra homomorphism. 

Lie Algebra in Tangent Space

If G is a Lie group and g an element of G, define a map,
L_g : G --> G by L_g(h) = gh.
This is the "left multiplication by g" map, which is smooth since the product map of G x G to itself is assumed smooth. Then the differential (L_g)_* of L_g at a point h will be a linear map of T_h(G) to T_gh(G). A vector field X on G is called left-invariant if X satisfies (L_g)_* (X_h) = X_gh.

Let T_e(G) denote the tangent space at the identity. Then, given any  vector v  ∈ T_e(G) , there is a unique left-invariant vector field X^v with X_e^v  = v, which can be constructed by defining X_g^v  = (L_g)_* (v) .

(Note that L_g o L_h = L_gh , by chain rule, it follows that (L_gh)_*,e = (L_g)_*h, (L_h)_*,e . )

The set of all left-invariant vector fields is a real vector space whose dimension is the same as that of G, and it is isomorphic as a vector space to T_e(G) by means of evaluation at the identity. If vector fields are considered as first-order differential operators, then the commutator of two vector fields is a vector field. Therefore, the commutator of two left-invariant vector fields is, again, a left-invariant vector field. 

The Lie algebra g of a Lie group G is the tangent space at the identity with the bracket operation defined by [v, w] = [X^v, X^w ]_e. If we identify the space of left-invariant vector fields with T_e(G) by means of the map v --> X^v, then g is the space of left-invariant vector fields, which forms a Lie algebra under the commutator of vector fields.

If X^v is the left-invariant vector field, then let Φ^v_t be the associated flow. The exponential mapping is the map exp: g --> G defined by exp(v) = Φ^v₁(e). The exponential mapping is the time-one flow along a left-invariant vector field starting at the identity. This means that to compute exp(v), we first construct the left-invariant vector field X^v and we then find an integral curve Υ^t to X^v that starts at the identity. Then, exp(v) = Υ^t(1). 

Physicist's Convention
Physicists are accustomed to considering the map X --> e^iX instead of X --> e^X . Thus, a physicist would think of the Lie algebra of G as the set of all matrices X such that e^itX ∈ G for all real numbers t. In this case, physics literature does not always distinguish clearly between a matrix Lie group and its Lie algebra and the Lie algebra is frequently referred to as the space of "infinitesimal group elements."

Let g be a finite-dimensional real or complex Lie algebra, and let X₁, ..., X_n be a basis for g (as a vector space). Then for each i and j, [X_i, X_j] can be written uniquely in the form [X_i, X_j] = ∑ c_ijk X_k, for k = 1.. n. The constant c_ijk are called the structure constants of g, it determine the bracket operation on g. 

In the physics literature, the structure constants are defined as [X_i, X_j] = √-1 ∑ c_ijk X_k. The structure constants satisfy both the conditions of skew symmetry and Jacobi identity.