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I am studying QFT and got stuck on one question which may be simple. Why is structure constants (in Lie algebra) real and independent on representations? Can you give me a detailed proof?
Thanks!
Thanks!
Structure constants in Lie algebra are numerical values used to define the commutation relations between basis elements of a Lie algebra. They represent the coefficients in the linear combination of basis elements that make up the Lie bracket operation.
Structure constants are typically calculated using the structure equations, which are a set of equations that describe the commutation relations between basis elements. These equations can be solved for the structure constants using various methods, such as the Cartan-Killing form or the Chevalley-Serre relations.
The structure constants of a Lie algebra provide important information about its structure and properties. They can be used to determine the dimension and type of the Lie algebra, as well as its symmetry and other structural features. They also play a crucial role in the classification and study of Lie algebras.
Yes, structure constants are an essential tool in the classification of Lie algebras. Different Lie algebras can have the same set of basis elements, but their structure constants will be different. By comparing the structure constants, we can determine if two Lie algebras are isomorphic or not.
Yes, structure constants have applications in various fields such as physics, computer science, and differential geometry. In physics, they are used to describe the symmetries of physical systems. In computer science, they are used in the development of algorithms for solving problems related to Lie algebras. In differential geometry, they are used to study the geometry of Lie groups and their actions on manifolds.