Baker-Campbell-Hausdorff Formula
Baker-Campbell-Hausdorff Formula
Suppose
Specially, in the Lie algebra case, we have
Corollary
Suppose
, then
Proof This is a consequence
Lie’s Theorems
Ado's Theorem
Every finite‑dimensional Lie algebra admits a faithful finite‑dimensional representation, so it embeds as a Lie subalgebra of matrices
for some finite‑dimensional vector space .
Lie's First Theorem
If two Lie groups are isomorphic, then so their Lie algebras.
Lie's Second Theorem
If
, then and are locally isomorphic.
Lie's Third Theorem
there exists a unique simply connected Lie group with . If is simply connected then For every finite dimensional Lie algebra
The Adjoint Maps
Adjoint Map of a Lie Group
The adjoint map of a Lie group
is the map defined by where is the conjugation by . It turns out to be a group representation called the adjoint representation of .
Proposition
For matrix Lie groups, the adjoint map reduces to
for and .
Proposition
Suppose
is a Lie group homomorphism, then the following diagram commutes:
Adjoint Map of a Lie Algebra
For a Lie algebra
and an element , the adjoint map of is the linear transformation defined by:
The Adjoint Representation
The adjoint map
is a Lie algebra homomorphism. So the map given by is a representation of . It is called the adjoint representation of .
Proof The Jacobi identity
Relationship Between Adjoint Maps
for the functor .