Stefan Reppen: A First Glance at Lie Theory

Abstract: This text was written as a second year project and has in essence two purposes. Firstly, we aim to solve some elementary problems in Lie theory. In particular, we will (i) show that the special unitary group SU(2) is isomorphic as a Lie group to the unit sphere \(S^3\), (ii) show that the quotient manifold SO(n)/SO(n-1) is diffeomorphic to the unit sphere \(S^{(n-1 )}\), (iii) find the Lie algebra of SL_{n}(R), SU(n) and SO(n), (iv) show that the Lie algebra of SO(3) is isomorphic to the Lie algebra of SU(2), while SO(3) is not isomorphic to SU(2). Secondly, in showing these four problems, we wish to present a first glance at Lie theory.