ABSTRACT
One of the most important aspects of a Lie algebra is its representation in terms of operators or matrices. One type of representation is called the highest weight representation. An important bilinear form defined for any Lie algebra which helps in the classification of its properties is the Cartan-Killing form. This chapter argues that the exponential map leads from the Lie algebra to the Lie group, and describes a class of finite dimensional Lie groups. An infinite dimensional Lie group is a group whose base manifold in infinite dimensional and modeled on a complete, locally convex topological vector space. The chapter considers the corresponding Virasoro group. The Virasoro group is similar to finite dimensional noncompact groups, and loop groups are similar to compact groups. The chapter focuses on the construction of coadjoint orbits. The method of orbit is the most suitable one for the straightforward derivation of the symplectic form and bi-Hamiltonian structure, which was quite difficult in earlier formalisms.
