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Lie Algebras [Sternberg] - mklabgr - 06-13-2026
Lie Algebras
Shlomo Sternberg
Summary
"Lie Algebras" by Shlomo Sternberg (2004) is a rigorous academic text detailing the structure, classification, and representation theory of finite-dimensional complex Lie algebras. The book establishes the foundational structure theory of nilpotent, solvable, and semisimple algebras via Engel's and Lie's theorems, alongside Cartan's criteria. It provides a complete classification of simple Lie algebras into the four classical families ($A_n, B_n, C_n, D_n$) and five exceptional types ($E_6, E_7, E_8, F_4, G_2$) utilizing root systems and Dynkin diagrams. Furthermore, the text develops representation theory using $\mathfrak{sl}(2)$ as a building block for highest weight modules, Verma modules, and the Weyl character formula, while concluding with advanced geometric applications including the Campbell-Baker-Hausdorff formula, Clifford algebras, and the Kostant-Dirac operator.
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