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qiuhd
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I know we need metric, curvature, what is Lie algebras used for? What is the physics meaning?
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Thanks, most differential geometry and Riemann geometry textbooks contain Lie group material, I thought they have something to do with GR.bapowell said:Lie groups are heavily used in particle physics, not so much in GR as far as I know. The generators of each Lie group form an algebra, called the Lie algebra, so the Lie algebra comes along for the ride. As continuous groups, Lie groups are the symmetry groups of the standard model of particle physics. The main Lie groups of interest are SO(N), U(N), SU(N) in the standard model, but more exotic groups crop up in string theory.
OB1 said:In general, whenever you see commutation relations in physics, you can be sure there is a Lie algebra behind it. I once heard a mathematical physicist say that all of physics is just a series of Lie algebras (and superalgebras) and finding a good TOE is just about figuring out how they are all connected. I'm sure that was an oversimplification, but it gives you an idea of the importance of Lie algebras in physics.
Lie algebras provide a powerful mathematical framework for studying and understanding the behavior of continuous symmetries in physical and mathematical systems. They are also used in the development of quantum mechanics and string theory.
Lie algebras are closely related to group theory, as they are the infinitesimal version of Lie groups. Lie groups are continuous groups that preserve the structure of a smooth manifold, and Lie algebras describe the tangent space of a Lie group at the identity element.
Yes, Lie algebras have many practical applications in fields such as engineering, computer science, and economics. For example, they are used in control theory to study the behavior of dynamical systems, and in computer graphics to model the motion of objects.
Lie algebras have a wide range of applications in physics, such as in the study of symmetries in quantum mechanics and the development of gauge theories. They also play a crucial role in the study of Lie groups, which are used to describe the fundamental forces in nature.
Yes, studying Lie algebras can have practical benefits, such as helping us understand the underlying structure and symmetries of physical systems, which can lead to advancements in technology and engineering. Additionally, studying Lie algebras can also improve problem-solving skills and logical thinking abilities.