Lie Algebras

Lie algebras Shlomo Sternberg April 23, 2004 2 Contents 1 The 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Campbell Baker Hausdorff Formula The problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The geometric version of the CBH formula. . . . . . . . . . . . . The Maurer-Cartan equations. . . . . . . . . . . . . . . . . . . . Proof of CBH from Maurer-Cartan. . . . . . . . . . . . . . . . . . The differential of the exponential and its inverse. . . . . . . . . The averaging method. . . . . . . . . . . . . . . . . . . . . . . . . The Euler MacLaurin Formula. . . . . . . . . . . . . . . . . . . . The universal enveloping algebra. . . . . . . . . . . . . . . . . . . 1.8.1 Tensor product of vector spaces. . . . . . . . . . . . . . . 1.8.2 The tensor product of two algebras. . . . . . . . . . . . . 1.8.3 The tensor algebra of a vector space. . . . . . . . . . . . . 1.8.4 Construction of the universal enveloping algebra. . . . . . 1.8.5 Extension of a Lie algebra homomorphism to its universal enveloping algebra. . . . . . . . . . . . . . . . . . . . . . . 1.8.6 Universal enveloping algebra of a direct sum. . . . . . . . 1.8.7 Bialgebra structure. . . . . . . . . . . . . . . . . . . . . . 1.9 The Poincar´e-Birkhoff-Witt Theorem. . . . . . . . . . . . . . . . 1.10 Primitives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Free Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.1 Magmas and free magmas on a set . . . . . . . . . . . . . 1.11.2 The Free Lie Algebra LX . . . . . . . . . . . . . . . . . . . 1.11.3 The free associative algebra Ass(X). . . . . . . . . . . . . 1.12 Algebraic proof of CBH and explicit formulas. . . . . . . . . . . . 1.12.1 Abstract version of CBH and its algebraic proof. . . . . . 1.12.2 Explicit formula for CBH. . . . . . . . . . . . . . . . . . . 2 sl(2) and its Representations. 2.1 Low dimensional L