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Symmetries and Group Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
A fundamental tool in examining how different systems behave under certain transformations is found in representation theory. a representation ρ of a group on a vector space V is a homomorphism from the group to the set of linear operators on V. Note that these operators form a group and it therefore makes sense to talk about such a homomorphism, see Example 4.6. The dimension of V is also referred to as the dimension of the representation. If the vector space V is finite dimensional with N dimensions, then the representation can be written down as N × N matrices. The action of a group element a on a vector x in V is given by applying the homomorphism to the vector and obtaining a new vector () x′=ρ(a)x.
Harmonic Analysis on Groups
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
The representation theory of Lie groups has many physical applications. This theory evolved with the development of quantum mechanics, and today serves as a natural language in which to articulate the quantum theory of angular momentum. There is also a very close relationship between Lie group representations and the theory of special functions [17, 57, 81, 83]. It is this connection that allows for our concrete treatment of harmonic analysis on the rotation and motion groups in Chapters 9 and 10.
Group Representations
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Representation theory is the study of the various ways a given group can be mapped into a general linear group. This information has proven to be effective at providing insight into the structure of the given group as well as the objects on which the group acts. Most notable is the central contribution made by representation theory to the complete classification of finite simple groups [Gor94]. (See also Fact 3 of Section 68.1 and Fact 5 of Section 68.6.)
Lie group method for constructing integrating factors of first-order ordinary differential equations
Published in International Journal of Mathematical Education in Science and Technology, 2023
The Lie group and Lie algebra were produced at the end of the nineteenth century. After decades of development, they became an important branch of mathematics in the 1970s. They are closely related to geometry, algebra, and analysis. The founder of Lie group theory, Marius Sophus Lie, was born on 17 December 1842 in the village Nordfjordeid in Norway (Ibragimov, 1992). Starting from 1857, he studied in Christiania (now Oslo), first in a grammar school and then (1859–1865) in a university. Later, he had a journey to Germany and France (1869–1870) and a meeting with Klein (which grew into a close friendship and a long-term partnership), Chasles, Jordan, and Darboux. During 1872–1886, Lie worked in the university of Christiania, and from 1886 to 1898 in the university of Leipzig. Lie died on 18 February 1899 in Christiania. The life, development of ideas, and work of this great Norwegian mathematician are discussed in detail in the excellent book of E.M. Polishchuk, ‘Sophus Lie’ (Nauka, Leningrad 1983). Lie's work has not received enough attention during his lifetime. It was not until the beginning of the twentieth century that his research results were carried forward due to their good applications in physics and geometry. For example, H. Weyl (1885–1955) studied the representation theory of the Lie group and used it in quantum mechanics. É. Cartan (1869–1951) used the Lie group in geometry to establish the Riemann symmetric space theory. We should also mention the use of the Lie group method for the differential equations arising in geometry, including motions in Riemannian manifolds, symmetric spaces, and invariant differential operators associated with them (see Olver, 1986 and the references therein).