ALGEBRA OF DERIVATION ON OCTONION AS LIE ALGEBRA TYPE G2

• Main Author: Izzanardi Wijanarko, Musa
• Format: Final Project
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/55101
• Institution: Institut Teknologi Bandung

The general idea of Lie groups and Lie algebras was introduced by Sophus Lie at 1870s by the name of infinitesimal transformation groups. Lie group is a smooth manifold that is also a group, meanwhile Lie algebra is a vector space equipped with a Lie bilinear form. By observing tangent space of a Lie group, and putting all the vector space that left invariant from its vector bundle under the same set, one can obtain a Lie algebra. A simple Lie algebra is a type of Lie algebra that has no nonzero ideal besides itself. Using root system, Killing and Cartan classified simple Lie algebras into nine types, four classical simple Lie algebras, and five exceptional ones. In other words, every simple Lie algebra is isomorphic to one of the nine simple Lie algebras. Octonion is an eight dimensional vector space and also one of the four normed division algebra. We can see the set of derivations on Octonion (denoted by Der(O)), as a Lie algebra. In this thesis, we will show that Der(O) is a simple Lie algebra. Furthermore, the complexification of Der(O) is isomorphic to exceptional simple Lie algebra g2. That is, Der(O) is a Lie algebra of type G2: