Lie algebras: infinite generalizations and deformations

Abstract

There are many applications of Lie algebras to theoretical physics. This thesis is a study of some new mathematical structures which also are applicable to current physical ideas. The structures studied are Lie algebras of infinite dimension and the deformations of Lie algebras known as quantum algebras. The approach is algebraic, although physical applications are indicated. Chapter 1 The mathematics of finite and infinite dimensional Lie algebras is reviewed, together with an indication of well established uses in physics. The terms and notation used in the rest of the thesis are introduced. Chapter 2 Explicit examples of new infinite dimensional algebras of a type related to the algebras of conformal transformations on arbitrary genus Riemann surfaces are given. The relationship of these algebras to the Virasoro algebra is discussed. Chapter 3 The sine algebra is introduced and its relationship to the Moyal bracket discussed. The finite Lie algebras are given in a trigonometric basis. The many applications of the Moyal algebra are reviewed. Chapter 4 An original proof of the uniqueness of the Moyal algebra is presented. It is shown that the Moyal bracket is the most general Lie bracket of functions of two variables, and thus that the underlying associative star product is unique. It follows that all 2-index Lie algebras correspond to the Moyal algebra in some basis. Chapter 5 Quantum deformations of Lie algebras, or quantum algebras, are introduced. The many deformations of su(2) are described and the associativity conditions are discussed. Some new higher dimensional and infinite dimensional quantum algebras are given. Chapter 6 Quantum groups are discussed as groups of transformations of the quantum plane. Higher dimensional quantum groups and quantum supergroups are also described.