Lie groups and their Lie algebras  Lec 13  Frederic Schuller
This is from a series of lectures  "Lectures on the Geometric Anatomy of Theoretical Physics" delivered by Dr.Frederic P Schuller.
In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skewsymmetric bilinear forms and Hermitian or skewHermitian sesquilinear forms defined on real, complex and quaternionic finitedimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.
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Classical groups


Lie groups in physics

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In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skewsymmetric bilinear forms and Hermitian or skewHermitian sesquilinear forms defined on real, complex and quaternionic finitedimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.
The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern physics. A few examples are the following. The rotation group SO(3) is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group O(3,1) is a symmetry group of spacetime of special relativity. The special unitary group SU(3) is the symmetry group of quantum chromodynamics and the symplectic group Sp(m) finds application in hamiltonian mechanics and quantum mechanical versions of it.
This is from a series of lectures  "Lectures on the Geometric Anatomy of Theoretical Physics" delivered by Dr.Frederic P Schuller.
The second in a series of 4 lectures on Lie groups and Lie algebras (with a particular focus on physics) given by Gang Xu, a PSI Fellow, at the 20142015 PSI.
Lecture Series on Classical Physics by Prof.V.Balakrishnan, Department of Physics, IIT Madras. For more details on NPTEL visit http://nptel.iitm.ac.in.
Proof of Lemma 7.7: the Lie algebra of a Lie subgroup Section 9: closed subgroups Section 10: the groups SU(2) and SO(3) The references (section,corallary ...
In this video, we look at the linear lie algebras.