There will be two other parts coming soon (I hope). But I just found by chance an old manuscript on the Cornell server which has incidentally the same structure as my insights will have. It's a pdf of 172 pages and for free: https://pi.math.cornell.edu/~hatcher/Other/Samelson-LieAlg.pdfWrichik Basu said:
Yes, that's why I am reading it. I am not much find of rigorous maths as of now.fresh_42 said:
A Lie algebra is a mathematical structure that studies the algebraic properties of vector fields, which represent infinitesimal transformations on a given space. It provides a way to understand the behavior of continuous symmetries in a mathematical setting.
Lie algebras have a wide range of applications in various fields, including physics, engineering, and mathematics. They are used to study the symmetries of physical systems, to classify and solve differential equations, and to understand the geometric properties of manifolds.
Lie algebras and Lie groups are closely related mathematical structures. Lie algebras are the infinitesimal version of Lie groups, which are continuous groups of transformations. Essentially, a Lie group is a smooth manifold with a group structure, while a Lie algebra is the tangent space at the identity element of the Lie group.
Some of the basic properties of Lie algebras include closure under the Lie bracket operation, which measures the non-commutativity of the algebra, and the Jacobi identity, which ensures the associativity of the Lie bracket. Lie algebras are also equipped with a bilinear form called the Killing form, which measures the non-degeneracy of the algebra.
There are many resources available for learning about Lie algebras, including textbooks, online courses, and research articles. It is recommended to have a strong background in linear algebra and abstract algebra before delving into the study of Lie algebras. Additionally, hands-on practice with solving problems and working with examples can greatly aid in understanding this complex mathematical structure.
