Representations of Typical Lie Groups and Typical Lie Algebras

Abstract

This chapter deals with representations of special Lie groups and special Lie algebras. Since a representation of a Lie algebra can be classified with the highest weight, those of a Lie group can be easily treated through those of the corresponding Lie algebra. Also, Lie algebras provide several concepts important for quantum theory. Hence, this chapter is organized so that it constructs a representation of a Lie group via that of the corresponding Lie algebra. This chapter starts with representations of Lie algebras \(\mathop {{\mathfrak {su}}}\nolimits (2)\) and \(\mathop {{\mathfrak {su}}}\nolimits (1,1)\). Since the Lie algebra \(\mathop {{\mathfrak {su}}}\nolimits (2)\) is compact and the Lie algebra \(\mathop {{\mathfrak {su}}}\nolimits (1,1)\) is not compact, they require different treatment caused in this difference. As they have unexpected common features, we handle both in a unified way. Then, we proceed to representation of the Lie algebra \(\mathop {{\mathfrak {su}}}\nolimits (r)\) by using Young diagrams. Especially, the representation of the Lie algebra \(\mathop {{\mathfrak {su}}}\nolimits (r)\) on the tensor product space is closely related to that of the permutation group on the same tensor product space. The relation is called Schur duality. We also consider what a finite subgroup of a Lie group can replace the Lie group when its representation is given. Such a problem is called design, and is discussed in this chapter.