## Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics, Volume 61This is an introductory text on Lie groups and algebras and their roles in diverse areas of pure and applied mathematics and physics. The material is presented in a way that is at once intuitive, geometric, applications oriented, and, most of the time, mathematically rigorous. It is intended for students and researchers without an extensive background in physics, algebra, or geometry. In addition to an exposition of the fundamental machinery of the subject, there are many concrete examples that illustrate the role of Lie groups and algebras in various fields of mathematics and physics: elementary particle physics, Riemannian geometry, symmetries of differential equations, completely integrable systems, and bifurcation theory. |

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### Contents

Lie Groups | 3 |

Lie Algebras | 21 |

The Classical Lie Algebras | 29 |

Copyright | |

18 other sections not shown

### Common terms and phrases

adjoint representation algebra g baryon basis for g boson called Cartan subalgebra Chapter classical commutation relations compact complex components construct corresponding covering group decomposition defined denoted differential equations dimensional representation direct sum Dynkin diagram eigenvalues eigenvector elements Euclidean example exercise exterior derivative follows frame bundle given group action Hamiltonian hence highest weight homomorphism identity infinite dimensional infinitesimal inner product integral invariant measure isometry isomorphic isospin Killing form left invariant Lemma Lie algebra Lie group linear Lie Lorentz mapping matrices Maurer-Cartan equations metric tensor obtained one-forms one-parameter operators orthogonal p-form particle Proof real form real Lie algebra Riemannian connexion root vectors rotation semi-simple semi-simple algebra semi-simple Lie algebra Show solution spinor representations structural equations structure constants subgroup subspace symmetry group tangent vectors tensor product Theorem theory transformation group upper triangular vector field vector figure vector space weight vector Weyl zero