4 - Lie theory

Summary

In Chapter 4, we recall some important concepts from the theory of Lie algebras and Weyl groups; these will play an important role in subsequent chapters. Although it would be possible to develop some of these concepts in the heap framework, there are already good introductory texts dealing with this material. We will therefore omit most of the proofs, which are elementary (although not trivial).

In Section 4.1, we recall the definition of the derived algebra of an affine Kac–Moody algebra in terms of generators and relations. We then use the Local Structure Theorem (Theorem 2.3.15) to obtain a representation of the derived algebra of an affine Kac–Moody algebra from a full heap; this is proved in Theorem 4.1.6.

In Sections 4.2 and 4.3, we review some key definitions and results from the theories of Lie algebras and Weyl groups, respectively. These sections are not self-contained and do not contain proofs: for details, we provide references to the books of Kac [37] and Carter [11] for results on Lie algebras, and to Humphreys' book [36] for results on Weyl groups.

Finally, Section 4.4 introduces the notion of a strongly orthogonal set. We will call a set S of mutually orthogonal positive roots “strongly orthogonal” if each root s ∈ S is orthogonal to every other positive root that is orthogonal to all members of S\{s}; see Definition 4.4.1 for the full definition.