Abstract:
Lie algebras are important linear objects, they are closely related to Lie groups via the Lie Correspondence Theorems, and play a central role in modern mathematics and physics. This thesis describes the classification and the construction (via Tits’ magic square) of complex semisimple Lie algebras. First we review the theory of Lie algebras and study the classification of complex semisimple Lie algebras by root systems; in Theorem 2.59 we present an original proof which shows that this classification is well-defined. Next we examine the ingredients of Tits’ construction, namely composition algebras, Jordan algebras, and their derivations; in Remark 3.9 we identity a typographical error in [Sch66] which has accumulated in the literature. Finally we give a complete description of Tits’ construction; denote H the split quaternion algebra over C, from Proposition 4.11 onward we independently prove that both Der(H) and Der(H3(C)) are isomorphic to the Lie algebra sl2(C) and that Der(H3(C ⊕ C)) is isomorphic to sl3(C); we give an original construction for each of Der(H3(C ⊕ C)) and Der(H3(H)) and compute their Lie brackets, leading the construction of all classical semisimple Lie algebras in Tits’ magic square (Table 4.1); in the future one can take the same approach to construct each of the five exceptional simple Lie algebras.
