Lectures on Selected Topics in Mathematical Physics: Introduction to Lie theory with applications by William A Schwalm

Author:William A Schwalm
Language: eng
Format: mobi
ISBN: 9781681744490
Publisher: IOP Publishing
Published: 2017-04-12T00:00:00+00:00

subject to at .

From the identification

it follows at once that the group is abelian, since

Then clearly

and the parametrization is additive, . Then notice that

and so in direct analogy with the case of the finite groups, one can see that the one-parameter Lie groups are in a sense cyclic groups generated by the element eU. Accordingly, the operator U is the infinitesimal generator of the group. It is rather like a logarithm of a group generator. Multi-parameter groups have sets of infinitesimal generators, one for each parameter. We consider only groups with finitely many parameters which hence are finitely generated. The generators need not commute as operators, and so as seen for finite groups the Lie groups need not be abelian.

It was mentioned previously that one-parameter groups can be isomorphic to the group of addition of real numbers or the group of addition modulo of angles of rotation in the plane. This is an interesting difference between the one-parameter groups and the finite cyclic groups. Each derives from a single generating element. However, for the finite case, there must be some power p of a generator g such that , the identity. This argument is based on finiteness of and it does not work when the group is infinite. Starting with some gα there may or may not be a power p ≠ 0 such that gives the identity. Of course, in any case the identity in the set is supplied by including or p = 0. This is an important distinction. When there is an element such that for some finite p it implies a sort of periodicity—periodic boundary conditions if you will—and as in the case of the Fourier series expansion it implies the possibility of expanding in a set of orthogonal functions. The distinction is in whether the range of the parameter α can be made finite or not, or more accurately, whether or not it corresponds to a closed and bounded set on the group manifold. This will come up again later.

Exercise. Find the Lie generator corresponding to the special conformal transformation

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.