# Lie Algebras

Written by: Shlomo Sternberg of Harvard University

A math professor from Harvard University, Shlomo Sternberg is offering his textbook on Lie Algebras in PDF form.

Sternberg’s table of contents provides an almost glossary-like list of topics which should make it easier for students searching for help with those topics find what they are looking for. He has also written two other advanced mathematics textbooks – Semi-Riemann Geometry and General Relativity and Theory of Functions of a Real Variable.

1. The Campbell Baker Hausdorff Formula
1.1 The problem
1.2 The geometric version of the CBH formula
1.3 The Maurer-Cartan equations
1.4 Proof of CBH from Maurer-Cartan
1.5 The differential of the exponential and its inverse
1.6 The averaging method
1.7 The Euler MacLaurin Formula
1.8 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 Lie algebras
2.2 sl(2) and its irreducable representations
2.3 The Casimir element
2.4 sl(2) is simple
2.5 Complete reducibility
2.6 The Weyl group
3. The classical simple algebras
3.2 sl(n+1)
3.3 The orthogonal algebras
3.4 The symplectic algebras
3.5 The root structures
3.5.1 An=sl(n+1)
3.5.2 Cn=sp(2n),n≥2
3.5.3 Dn=o(2n),n≥3
3.5.4 Bn=o(2n+1)n≥2
class=”inner-dd”3.5.5 Diagrammatic presentation
3.6 Low dimensional coincidences
3.7 Extended diagrams
4. Engel-Lie-Cartan-Weyl
4.1 Engel’s theorem
4.2 Solvable Lie algebras
4.3 Linear algebra
4.4 Cartan’s criterion
4.6 The Killing form
4.7 Complete reducibility
5. Conjugacy of Cartan subalgebras
5.1 Derivations
5.2 Cartan subalgebras
5.3 Solvable case
5.4 Toral subalgebras and Cartan subalgebras
5.5 Roots
5.6 Bases
5.7 Weyl chambers
5.8 Length
5.9 Conjugacy of Borel subalgebras
6. The simple finite dimensional algebras
6.1 Simple Lie algebras and irreducible root systems
6.2 The maximal root and the minimal root
6.3 Graphs
6.4 Perron-Frobenius
6.5 Classification of their reducible ∆
6.6 Classification of the irreducible root systems
6.7 The classification of the possible simple Lie algebras
7. Cyclic highest weight modules
7.1 Verma modules
7.2 When is dim Irr(λ)<∞?
7.3 The value of the Casimir
7.4 The Weyl character formula
7.5 The Weyl dimension formula
7.6 The Kostant multiplicity formula
7.7 Steinberg’s formula
7.8 The Freudenthal-de Vries formula
7.9 Fundamental representations
7.10 Equal rank subgroups
8. Serre’s theorem
8.1 The Serre relations
8.2 The first five relations
8.3 Proof of Serre’s theorem
8.4 The existence of the exceptional root systems
9. Clifford algebras and spin representations
9.1 Definition and basic properties
9.1.1 Definition
9.1.3 ∧p as a C(p) module
9.1.4 Chevalley’s linear identification of C(p) with ∧p
9.1.5 The canonical antiautomorphism
9.1.6 Commutator by an element of p
9.1.7 Commutator by an element of ∧2p
9.2 Orthogonal action of a Lie algebra
9.2.1 Expression for ν in terms of dual bases
9.2.2 The adjoint action of a reductive Lie algebra
9.3 The spin representations
9.3.1 The even dimensional case
9.3.2 The odd dimensional case
10. The Kostant Dirac operator
10.1 Anti symmetric trilinear forms
10.2 Jacobi and Clifford
10.3 Orthogonal extension of a Lie algebra
10.4 The value of [v2+ν(Casr)]0
10.5 Kostant’s Dirac Operator
10.6 Eigenvalues of the Dirac operator
10.7 The geometric index theorem
10.7.1 The index of equivariant Fredholm maps
10.7.2 Induced representations and Bott’s theorem
10.7.3 Landweber’s index theorem
11. The center of U(g).
11.1 The Harish-Chandra isomorphism
11.1.1 Statement
11.1.2 Example of sl(2)
11.1.3 Using Verma modules to prove that γH : Z(g) → U(h)W
11.1.4 Outline of proof of bijectivity
11.1.5 Restriction from S(g∗)g toS(h∗)W
11.1.6 From S(g)g to S(h)W
11.1.7 Completion of the proof
11.2 Chevalley’s theorem
11.2.1 Transcendence degrees
11.2.2 Symmetric polynomials
11.2.3 Fixed fields
11.2.4 Invariants of finite groups
11.2.5 The Hilbert basis theorem
11.2.6 Proof of Chevalley’s theorem

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Lie Algebras 