Abstract
A new approach is introduced for constructing representations of Lie groups based on canonical transformations in group phase space of the Lie group which reduce the regular representation of the Lie group to canonical form. Examples are considered of the group? (semisimple, nilpotent, solvable) having application to problems in physics, and for these groups formulas are given for the generators, finite rotation operators, characters, and Plancherel measure. The Lie group is viewed as a dynamical system described by a Hamiltonian which is a linear form in the momenta with coefficients depending on the coordinates. This permits the use of the methods of quantum mechanics, in particular integrals of motion, coherent states, and path integrals, for constructing representations of Lie groups.
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Leznov, A.N., Malkin, I.A., Man’ko, V.I. (1979). Canonical Transformations and the Theory of Representations of Lie Groups. In: Basov, N.G. (eds) Problems in the General Theory of Relativity and Theory of Group Representations. The Lebedev Physics Institute Series. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-0676-4_3
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DOI: https://doi.org/10.1007/978-1-4684-0676-4_3
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