1 - BASIC CONCEPTS

Summary

By a skew linear group of degree n, a positive integer, we mean a subgroup of the general linear group GL(n,D) for some division ring D. Certain aspects of the theories of skew linear and of linear groups are very similar and in this chapter we concentrate on some of these. Other aspects are very different indeed, or at least require very different proofs.

Throughout this chapter the symbols n and D have the above designation. In the first three sections we investigate how much of the linear theory of irreducibility, absolute irreducibility, and unipotence can be extended to cover skew linear groups. Intentionally these sections are in the main more elementary than the rest of the book and we hope the reader will find them comparatively easy reading. They are also fundamental for much that follows. In the fourth and final section of Chapter 1 we construct, for later use, a wide range of examples of groups with faithful skew linear representations. This section may be omitted from a first reading.

IRREDUCIBILITY

Let G be a subgroup of GL(n,D) and set V = Dⁿ, the space of row n-vectors over D. Then V is a D-G bimodule in the obvious way. We say that G is an irreducible (resp. reducible, completely reducible) subgroup of GL(n,D) whenever V is irreducible (resp. reducible, completely reducible) as D-G bimodule.