On Lie Algebras
Given a Lie group G and its associated Lie algebra, the exponential map is a map satisfying similar properties. In fact, since R is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie group GL(n,R) of invertible n × n matrices has as Lie algebra M(n,R), the space of all n × n matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.
The identity exp(x+y) = exp(x)exp(y) can fail for Lie algebra elements x and y that do not commute; the Baker–Campbell–Hausdorff formula supplies the necessary correction terms.
Read more about this topic: Exponential Function
Famous quotes containing the word lie:
“Love is a great thing. It is not by chance that in all times and practically among all cultured peoples love in the general sense and the love of a man for his wife are both called love. If love is often cruel or destructive, the reasons lie not in love itself, but in the inequality between people.”
—Anton Pavlovich Chekhov (18601904)