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Bowles
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What Lie groups are also Riemann manifolds?
thanks
thanks
Bowles said:What Lie groups are also Riemann manifolds?
The joy of others is my award.Bowles said:I honestly hope you get paid for your knowledge.
Yes, that is conjugate transpose. For real matrices it reduces to ordinary transpose.But why xx*=1 (conjugate), shouldn't it be the transpose?
A Lie group is a mathematical concept that combines the properties of a group (a set of elements that can be combined in a specific way) and a smooth manifold (a space that is locally Euclidean, meaning it can be approximated by flat spaces).
Lie groups can be thought of as a special type of Riemann manifold, where the group operations are smooth and the group elements can be continuously varied. This allows for the application of differential geometry techniques to study the structure of Lie groups.
Examples of Lie groups include the rotation group in three dimensions (SO(3)), the special linear group (SL(2,R)), and the general linear group (GL(n,R)). Lie groups also have applications in physics, such as the Lorentz group in special relativity and the symmetry groups in quantum mechanics.
Lie groups have many applications in mathematics, including differential geometry, representation theory, and algebraic topology. They are also used in physics and engineering, particularly in the study of symmetries and transformations.
Lie groups as Riemann manifolds have a unique structure that allows for the application of differential geometry techniques, making them useful for studying symmetry and other mathematical concepts. They also have important applications in physics, particularly in the study of symmetries and transformations in physical systems.