UC Santa Cruz

This thesis presents a unified classification of semisimple Lie algebras and Kac-Moodyalgebras through their shared foundation in Cartan matrices and Dynkin diagrams. Motivated by the systematic classification of finite-dimensional Lie theory and inspired by Professor Chongying Dong’s insight, this work systematically explores the algebraic and geometric frameworks underpinning both classifications. For semisimple Lie algebras, we establish the classification via root systems and Dynkin diagrams, emphasizing Cartan’s criterion and the Killing form. Extending these principles, Kac-Moody algebras are constructed through generalized Cartan matrices, revealing infinite-dimensional symmetries critical to modern theoretical physics. By bridging finite and infinite dimensions, this thesis highlights applications in string theory, conformal field theory, and quantum gravity, while demonstrating how combinatorial tools like Dynkin diagrams unify disparate algebraic structures. The representation theories of both algebras are examined, culminating in the Weyl and Weyl-Kac character formulas, which underpin physical systems from atomic spectra to vertex operator algebras.