Lie Algebras-Classical and Relativistic Mechanics-Lecture Handout, Exercises of Classical and Relativistic Mechanics

Download Lie Algebras-Classical and Relativistic Mechanics-Lecture Handout and more Exercises Classical and Relativistic Mechanics in PDF only on Docsity! k! ∈ SO(n), (t ∈ R) In fact, any Lie group G has a Lie algebra g and there is a map called exponentiation exp: g→ G such that exp((s+ t)A) = exp(sA) exp(tA). For example, if g = R (a vector spce) and G = R+ (the Lie group of positive real numbers, with multiplication as the group operation), then exp : R→ R+ is the usual exponential function and I am just saying exp(s+ t) = exp(s) exp(t). Here a slide rule (or table of logarithms) converts problems in the Lie group into problems in the Lie algebra. Definition 1 A Lie algebra is a (real) vector space g equipped with an operation [·, ·]: g× g→ g called the Lie bracket which has the following properties: 1. bilinearity 2. antisymmetry 3. the Jacobi identity We’ve seen one type of Lie algebra so far: if X is a Poisson manifold, C∞(X) is a Lie algebra with the Poisson bracket as its bracket. But in fact, every manifold gives a Lie algebra, in a different way: 1 docsity.com