Text: Only a century has elapsed since 1873, when Marius Sophus Lie began his research on what has evolved into one of the most fruitful and beautiful branches of modern mathematics - the theory of Lie Groups. These researches culiminated twenty years later with the publication of landmark treatises by S. Lie and F. Engel between 1888 and 1893, and by W. Killing from 1888 and 1890. Matrices and matrix groups had been introduced by A. Cayley, Sir W. R. Hamilton, and J. J. Sylvester (1850-1859) about twenty years before the researches of Lie and Emgel began. At that time mathematicians felt that they had finally invented something of no possible use to matural scientists. However, Lie groups have come to play an increasingly important role in modern physical theories. In fact, Lie groups enter physics primarily through their infinite- and infinite-dimensional matrix representations. Certain natural questions arise. For example, just how does it happen that Lie groups play such a fundamental role in physics? And how are they used? Lie groups found their way into physics even before the development of the quntum theory. They were useful for the description of pseudo-Riemannian (locally) homogeneous symmetric spaces, being used particular in geometric theories of gravitation. But Lie groups were virtually forced into physics by the development of the modern quantum theory in 1925-1926. In this theory, physical observables appear through their hermitian matrix representatives, whereas processes producing transfomations are described by their unitary or antiunitary matrix representations. Operators that close under commutation belong to a finite-dimensional Lie algebra; transformation processes described by a finite number of continuous parameters belong to a Lie group. The kinds of applications of Lie group theory in modern physcis fall into three distinct groups: 1) A symmetry groups (1929-1960). Symmetry implies degeneracy. The greater the symmetry, the greater the degeneracy. 2) As nonsymmetry groups (1960- ). 3) Strictly speaking the third class of applications is not yet known, although its appearance is probably around the corner. It now seems possible that Lie group theory, together with differential geometry, harmonic analysis, and some devious arguments, might be able to predict some of Nature's dimensionless numbers. In retrospect, it seems clear that the application of group theory to physical problems represents the dividing line between kinematics and dynamics.

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