The Lie algebra of a Lie group

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 12 August 2013

Lie algebras and the exponential map

A Lie group is a group that is also a manifold, i.e. a topological group that is locally isomorphic to n.

1 U G φ:UV n 0 V Uis an openneighbourhood of1inG Vis an openneighbourhood of0inn If G is connected then G is generated by the elements of U.

The exponential map is a smooth homomorphism 𝔤 G 01 which is a homeomorphism on a neighborhood of 0.

The Lie algebra 𝔤 contains the structure of G in a neighbourhood of the identity.

A one parameter subgroup of G is a smooth group homomorphism γ:G .

0 t 1 γ(t) γ

Examples of one-parameter subgroups:

  1. Define γ: GLn() t 1+tEij =xij (t), for ij. Note that xij(t) xij(s) =xij(s+t) since ( 1t 01 ) ( 1s 01 ) = ( 1t+s 01 ) .
  2. Define γ: GLn() t ( 1                  1          et          1                  1 ) = hi(et) . Note that hi(et) hi(es ) = hi( et+s).

Let G be a Lie group. The ring of functions on G is C(G) = { f:G | f is smooth atg for allgG } where f is smooth atg if dkf dxk x=g exists for all k>0 .

Let gG.

If γ is a one-parameter subgroup of G define df(γ(t)) dt = limh0 f(γ(t+h)) - f(γ(t)) h . The following proposition says that we can identify three vector spaces

  1. {left invariant vector fields on G},
  2. {one parameter subgroups of G},
  3. {tangent vectors at 1G}.

The maps {left invariant vector fields} {tangent vectors at 1} ξ ξ1 and {one parameter subgroups} {tangent vectors at 1} γ γ1 where ξ1f =(ξf)(1) ,and γ1 =( ddtf (γ(t)) ) |t=0 , are vector space isomorphisms.

The exponential map is 𝔤G tX etX where etX = γ(t), where γ is the one-parameter subgroup corresponding to the tangent vector X.

Examples of exponential maps:

  1. The Lie algebra 𝔤𝔩n is 𝔤𝔩n = {x Mn()} with bracket [ x1,x2] =x1x2 -x2x1 . The vector space 𝔤𝔩n has basis {Eij | 1i,jn }, where Eij is the matrix with 1 in the (i,j)-entry and 0 elsewhere. The exponential map is 𝔤𝔩n GLn tX etX, where eA= 1+A+ A22! + A3 3! + for a matrix A. Then etEij = 1+tEij = ( 1     t         1 ) =xij(t), for ij, where t is in the (i,j) matrix entry, and etEii = ( 1                  1          et        1                   1 ) = hi (et) .
  2. If n=1 the exponential map × tx etx is a homeomorphism from a neighbourhood of 0 to a neighbourhood of 1. In fact, if e(t)= a0+a1t +a2t2 + and e(s+t) =e(s)e(t) , then e(s+t) = a0 +a1(s+t) +a2(s+t)2 + a3(s+t)3 + = a0+ a1s+a1t+ a2s2 +2a2st +a2t2 a3s3 +3a3s2t +3a3st2 +a3t3 + and e(s)e(t) = (a0 +a1s +a2s2 +a3s3 + ) ( a0 +a1t +a2t2 +a3t3 + ) = a02+ a0a1s +a0a1t + a0a2s2 +a12st +a0a2t2 a0a3 s3 +a2a1s2t +a1a2st2 +a0a3t3 + Hence e(s+t) =e(s)e(t) only if a0a1 =a1, 2a2=a12 , 3a3=a1 a2, 4a3 =a1a3, so that a0=1, a2 = a12 2 , a3 =a13 3! , a4 =a14 4!, and e(t)= 1+a1t + a12 2 t2 + a13 3! t3 + =ea1t . So × z ez is the "unique" smooth homomorphism × .

The functor from Lie groups to Lie algebras

Let φ:GH be a Lie group homomorphism and let 𝔤=Lie(G) and 𝔥=Lie(H). The homomorphism φ corresponds to a morphisms between the rings of functions on G and H, C(H) φ* C(G) f fφ and the differential of φ is the Lie group homomorphism 𝔤dφ 𝔥 given by dφ(ξ1) =ξ1φ* if ξ1 is a tangent vector at the identity, dφ(ξ) =ξφ* if ξ is a left invariant vector field, dφ(γ) =φγ if γ is a one parameter subgroup.

(Note: It should be checked that

  1. the map dφ is well defined,
  2. the three definitions of dφ are the same,
  3. and that dφ is a Lie algebra homomorphism.
These checks are not immediate, but are quite straightforward checks from the defiinitions.) The map the category of Lie groups the category of Lie algebras G Lie(G) φ dφ is a functor. This functor is not one to one; for example, the Lie groups On() and SOn () have the same Lie algebra. On the other hand, the Lie algebra does contain the complete structure of the Lie group in a neighbourhood of the identity. The exponential map is 𝔤 G tX etX, where etX =γ(t) is the one parameter subgroup corresponding to X𝔤. This map is a homeomorphism from a neighbourhood of 0 in 𝔤 to a neighbourhood of 1 in G.

(Lie's theorem) The functor Lie: {connected simply connected Lie groups} {Lie algebras} G 𝔤=Lie(G) =T1(G) is an equivalence of categories.

If 𝔤 is a Lie subalgebra of 𝔤𝔩n then the matrices { etX | t, X𝔤𝔩n } ,where etX =k0 tkXk k! , form a group with Lie algebra 𝔤.

The following formulas relating expansions in the Lie group to expansions in the Lie algebra can be worked out by directly expanding and multipliying the power series for the exponentials or by using techniques such as those found in [BouLie, Ch. ????, Sect. 6]. etX etY = e t(X+Y) +( t2/2) [X,Y]+ , etX etY e-tX = etY +t2 [X,Y]+ , etX etY e-tX e-tY = e t2 [X,Y]+ ,


Relating representations of Lie groups and Lie algebras

If V:G GL(V) is a homomorphism of Lie groups then the differential of V is a Lie algebra homomorphism dV:𝔤 End(V) which satisfies dV([x,y]) = [dV(x), dV(y)] =dV(x) dV(y) -dV(y) dV(x), for x,y𝔤. A representation of a Lie algebra 𝔤, or a 𝔤-module, is an action of 𝔤 on a vector space V by linear transformations, (a linear map φ:𝔤 End(V)) such that V([x,y]) =[V(x), V(y)] =V(x) V(y) - V(y) V(x) ,for all x,y𝔤, where V(x) is the linear transformation on Vdetermined by the action of x𝔤. The trivial representation of 𝔤 is the map 1: 𝔤 x 0 If V is a 𝔤-module, the dual 𝔤-module is the 𝔤-action on V* =Hom(V,) given by (xφ)(v) =φ(-xv) , for x𝔤, φV*, vV. If V and W are 𝔤-modules then the tensor product of V and W is the 𝔤-action on VW given by x(vw) =xvw +vxw, x𝔤, vV, wW.

The definitions of the trivial, dual and tensor product 𝔤-modules are accounted for by the following formulas: DOES THE SPACING OF THESE FORMULAS NEED TWEAKING?? ( ddt 1) |t=0 = ( ddt et0 )|t=0 =0, ( ddt (etX) -1 ) |t=0 = ( ddt (e-tX -1) ) |t =0 = -X, ( ddt (etX etX ) ) |t=0 = ( ddt ((1+tX+ t2X2 2! + ) (1+tX+ t2X2 2! + ) ) |t=0 = ( ddt (11+t ( X1+1X ) + ) ) |t=0 = X1+1X .

Notes and References

These notes are adapted from various lectures of Arun Ram on Representation theory, from 2008 and from Work2004/Book2003/chap41.17.03.pdf.



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