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198 pages, 900kb, year 2004

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An excellent, intermediate to advanced textbook on Lie algebras for theoretical and mathematical physicists. This recent book covers the most important aspects of Lie algebras for quantum physicists and 'elementary particle physics', such as: sl(2) and its Representations. p. 35 2.1 Low dimensional Lie algebras. . . . . . 35 2.2 sl(2) and its irreducible representations. . 36 2.3 The Casimir element. . . . 39 2.4 sl(2) is simple. . . . . 40 2.5 Complete reducibility. . . . . 41 2.6 The Weyl group. . . 42 The universal enveloping algebra. . . . . . . . . . . . . . . . . . . 19 8 Serre’s theorem. 137 8.1 The Serre relations. . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.2 The first five relations. . . . . . . . . . . . . . . . . . . . . . . . . 138 8.3 Proof of Serre's theorem. . . . . . 142 8.4 The existence of the exceptional root systems. . . 144 9 Clifford algebras and spin representations. 147 9.1 Definition and basic properties . . . . . . 147 9.1.1 Definition. . . . . . . . . . . 147 9.1.2 Gradation. . . . . . . . . . . . . 148 9.1.3 A C(p) module. . . . . . . 148 9.1.4 Chevalley's linear identification of C(p) p. 148 9.1.5 The canonical antiautomorphism. . . . . . . . 149 9.1.6 Commutator by an element of p. . . . . . . 150 9.1.7 Commutator by an element of 2p. . . . . . . . 151 9.2 Orthogonal action of a Lie algebra. . . . . . 153 9.2.1 Expressions in terms of dual bases. . . . . . . . . . . 153 9.2.2 The adjoint action of a reductive Lie algebra. . . . . . . . 153 9.3 The spin representations. . . . . .. 154 9.3.1 The even dimensional case. . . . . . 155 9.3.2 The odd dimensional case. . . . . .158 9.3.3 Spin and V p.159 10 The Kostant Dirac operator 163 10.1 Antisymmetric trilinear forms. 164 10.3 Orthogonal extension of a Lie algebra. . . . 165 10.4 The value of [v2 + l(Casr)]0. . . . . 167 10.5 Kostant's Dirac Operator. . . . . 169 10.6 Eigenvalues of the Dirac operator. . . . 172 10.7 The geometric index theorem. . . . . . 178 10.7.1 The index of equivariant Fredholm maps. . 178 10.7.2 Induced representations and Bott's theorem. . . . . . . . 179 10.7.3 Landweber's index theorem. . . . . . 1.8.1 Tensor product of vector spaces. . . . 20 1.8.2 The tensor product of two algebras. . . . . . . . . . . . . 21 1.8.3 The tensor algebra of a vector space. . . . . . . . . . . . . 21 1.8.4 Construction of the universal enveloping algebra. . 22 1.8.5 Extension of a Lie algebra homomorphism to its universal enveloping algebra. . . . . 22 1.8.6 Universal enveloping algebra of a direct sum. 22 1.8.7 Bialgebra structure. . . . .. . 23 1.9 The Poincar\'e-Birkhoff-Witt Theorem. . . . . . . 24 1.10 Primitives. . .. 28 1.11 Free Lie algebras . . Online free 900kb download at: http://www.math.harvard.edu/~shlomo/docs/lie_algebras.pdf

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http://www.math.harvard.edu/~shlomo/docs/lie_algebras.pdf Open access

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