Structure of Classical Groups 69 2.1 Semisimple Elements 69 2.1.1 Toral Groups 70 2.1.2 Maximal Torus in a Classical Group 72 2.1.3 Exercises 76 2.2 Unipotent Elements 77 2.2.1 Low-Rank Examples 77 2.2.2 Unipotent Generation of Classical Groups 2.2.3 Connected Groups 81 2.2.4 Exercises 2.3 Regular Representations of SL(2, C) 2.3.1 Irreducible Representations of .5((2, C) 2.3.2 Irreducible Regular Representations of SL(2, C) 2.3.3 Complete Reducibility of SL(2, (C) 2.3.4 Exercises 2.4 The Adjoint Representation 2.4.1 Roots with Respect to a Maximal Torus 2.4.2 Commutation Relations of Root Spaces 95 2.4.3 Structure of Classical Root Systems 99 2.4.4 Irreducibility of the Adjoint Representation 2.4.5 Exercises 2.5 Semisimple Lie Algebras 2.5.1 Solvable Lie Algebras 2.5.2 Root Space Decomposition 2.5.3 Geometry of Root Systems 2.5.4 Conjugacy of Cartan Subalgebras 2.5.5 Exercises x Contents

No abstract

No abstract

No abstract

No abstract

No abstract

The structure of the commutant of Laplace operators in the enveloping and "Poisson algebra" of certain generalized "αx + b" groups leads (in this article) to a determination of classical and quantum mechanical first integrals to generalized periodic and non-periodic Toda lattices. Certain new Hamiltonian systems of Toda lattice type are also shown to fit in this framework. Finite dimensional Lax forms for the (periodic) Toda lattices are given generalizing results of Flaschke.

No abstract