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:
“Two are better than one; because they have a good reward for their labour. For if they fall, the one will lift up his fellow: but woe to him that is alone when he falleth; for he hath not another to help him up. Again, if two lie together, then they have heat: but how can one be warm alone? And if one prevail against him, two shall withstand him; and a threefold cord is not quickly broken.”
—Bible: Hebrew Ecclesiastes, 4:9-12.