Nathan Jacobson’s Lie Algebras (originally published in 1962) is considered the definitive graduate-level treatment of the subject, providing a comprehensive and self-contained exposition of structure and representation theory. Google Books Core Content & Chapter Breakdown
Background and Definition
- Pitfall: Confusing the Jacobson Lie algebra with the "Jacobson radical" of a Lie algebra. These are unrelated.
- Pitfall: Thinking the construction works for all Jordan algebras in characteristic 2. It does not; you need $\textchar \neq 2,3$ for the full power.
- Pitfall: Downloading low-resolution or incomplete PDFs. Many scans of Jacobson's 1960s papers are missing pages. Verify you have the full paper (use MathSciNet or Zentralblatt to check pagination).
The Jordan algebra approach to quantum mechanics (Jordan–von Neumann–Wigner) uses the TKJ construction to link observables (Jordan algebra) to symmetries (Lie algebra). Physicists studying supersymmetry and M-theory have rediscovered these constructions in the context of U-duality groups ($E_7(7)$ etc.).
( W(m) ) is ( \mathbbZ^m )-graded by the multidegree: [ \deg(x^(\alpha) \partial_i) = (\alpha_1, \dots, \alpha_i-1, \alpha_i - 1, \alpha_i+1, \dots, \alpha_m) ] with the convention that ( x^(-1) = 0 ).
Given a unital Jordan algebra $J$ (over a field of characteristic not 2), one can construct a 3-graded Lie algebra $L(J)$.
The core idea is this: