Lie algebra | Wikipedia audio article
Channel: wikipedia tts
Date: 06/03/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Lie_algebra


00:01:44 1 History
00:02:07 2 Definitions
00:02:16 2.1 Definition of a Lie algebra
00:07:19 2.2 First example
00:10:01 2.3 Generators and dimension
00:10:59 2.4 Subalgebras, ideals and homomorphisms
00:15:06 2.5 Direct sum and semidirect product
00:18:35 2.6 Derivations
00:20:53 2.7 Split Lie algebra
00:22:49 3 Examples
00:22:58 3.1 Vector spaces
00:23:53 3.2 Associative algebra
00:26:30 3.3 Subspaces
00:27:25 3.4 Matrix Lie groups
00:31:05 3.5 Two dimensions
00:32:32 3.6 Three dimensions
00:38:51 3.7 Infinite dimensions
00:40:43 4 Representations
00:40:53 4.1 Definitions
00:42:21 4.2 Adjoint representation
00:43:35 4.3 Goals of representation theory
00:44:59 4.4 Representation theory in physics
00:46:35 5 Structure theory and classification
00:46:57 5.1 Abelian, nilpotent, and solvable
00:51:13 5.2 Simple and semisimple
00:52:42 5.3 Cartan's criterion
00:54:21 5.4 Classification
00:55:00 6 Relation to Lie groups
00:58:47 7 Lie algebra with additional structures
00:59:38 8 Lie ring
01:03:49 8.1 Examples
01:07:52 9 See also
01:08:01 10 Remarks



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SUMMARY
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In mathematics, a Lie algebra (pronounced  "Lee") is a vector space 
  
    
      
        
          
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    {\displaystyle {\mathfrak {g}}}
   together with a non-associative, alternating bilinear map 
  
    
      
        
          
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        ↦
        [
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    {\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}};\;(x,y)\mapsto [x,y]}
  , called the Lie bracket, satisfying the Jacobi identity.
Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds, with the property that the group operations of multiplication and inversion are smooth maps. Any Lie group gives rise to a Lie algebra. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering (Lie's third theorem). This correspondence between Lie groups and Lie algebras allows one to study Lie groups in terms of Lie algebras.
Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.
Lie algebras were so termed by Hermann Weyl after Sophus Lie in the 1930s. In older texts, the name infinitesimal group is used.