Studies of beam dynamics in relativistic klystron two-beam accelerators Page: 78 of 229
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X - adX is a representation of the Lie algebra g with g itself playing the role of the
vector space of transformations. This representation is called the adjoint representation.
This representation always provides a matrix representation of the Lie algebra. To see
this, consider a basis {E;} of the algebra. Then
adE (Ej) = C Ek,
(where Ct are the structure constants of the algebra), and
(Mi). =Cik
is an element of the operator matrix in the adjoint representation.
We have already discussed that a Lie algebra is the tangent space of the underlying
Lie group manifold. This is formalized by theorem (see [70] for details). Let G be
a matrix Lie group, and let g be the set of tangent vectors to all curves in G at the
identity. Then g is a Lie algebra of matrices with the commutator as the Lie product.
The converse is also true. For a linear Lie algebra g, the exponentials generate a Lie
group G. Heuristically, G (t) = exp (tg). There is a much deeper result that any Lie
algebra over the real or complex numbers generates a Lie group [70].
These same arguments can be made to apply to the symplectic mappings 0. Hence,
the set of symplectic maps defines a group, and this a Lie group with an associated Lie
algebra. For symplectic matrices, the Lie algebra is finite dimensional. For symplectic
mappings, the Lie algebra is infinite dimensional.
6.4 Lie Algebraic Tools
6.4.1 Lie Operators and Lie Transformations
A Lie operator can be constructed from any scalar function on the phase space, f(()
f(q, p), and the Poisson bracket, [-, -]. We denote this operator by : f :, and define it by
the action of the Poisson bracket,
S (Of 0 Of 0
- [f ]. (6.7)60
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Lidia, Steven M. Studies of beam dynamics in relativistic klystron two-beam accelerators, thesis or dissertation, November 1, 1999; California. (https://digital.library.unt.edu/ark:/67531/metadc724108/m1/78/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.