A journey through representation theory. From finite groups to quivers via algebras (Q1667458)

A journey through representation theory -- this title is quite appropriate for a 200 pages enterprise into the vast field of representation theory. The first half gives an account on classical representations of finite and compact groups, providing a piece of module theory and homological algebra as far as required for the second part on quiver representations and root systems. Both main parts are divided by a comparatively long, central chapter, written by Laurent Gruson, that opens a window for a glimpse into the world of Hopf algebras and Zelevinsky's book on representations of finite classical groups. The title also reflects a serious attempt of the authors to break with the ingrained habit of so many textbooks to present a topic in a purified, isolated, and self-sufficient manner, cutting connections to the historical roots and vital interactions with other mathematical subjects. To combine such an attempt with the promise not to go beyond linear algebra and the basics in group and ring theory inevitably leads to some compromise, unless some proofs are only sketched or omitted. Apart from L. Gruson's Chapter 6, the book offers complete proofs. In their own words, the authors could not ``resist the temptation to prove [the] Stone-von Neumann theorem'' on the irreducible representations of the continuous Heisenberg group. To profit from this exercise, the reader should take it as an invitation to learn more about the history and quantum mechanical relevance of this important result, as well as its mathematical role within operator algebras and Lie theory. The middle Chapter 6 starts with a brief introduction into representations of symmetric groups and Schur-Weyl duality, followed by an interpretation in terms of Hopf algebras, to be ready for an encounter with Zelevinsky's PSH algebras. Given that the student has got a feeling for the interplay between representations of discrete and continuous groups, this chapter provides a lot of inspiration and insight. The semisimple category of all representations with simple objects corresponding to partitions is considered, and its Grothendieck group is shown to be a PSH algebra of rank one, hence with a basic primitive element, given by the underlying complex vector space. The chapter ends with a summary of highlights from Zelevinsky's intriguing book. The second part gives an introduction to quivers and their representations, path algebras, first perceptions of representation varieties, Coxeter diagrams and root systems, reflection functors and the Weyl group. In this framework, the preprojective/preinjective partition, regular modules and tubes are explained without use of Auslander-Reiten theory. Applications deal with quivers and relations, abelian categories, and the tame-wild dichotomy. The journey ends with a more sketchy account on Harish-Chandra modules over \(\mathfrak{sl}_2(\mathbb C)\). In total, the book touches a lot of material in relation to its size. The goal to present mathematical ideas in their most elementary incarnation has been reached at several places where serious students may be inclined to delve into further study. The terminology is standard, with two exceptions. The group algebra \(k[G]\) (alias \(kG\)) over a field \(k\) is denoted by \(k(G)\), inspired perhaps by the \(C^\ast\)-algebra \(C^\ast(G)\), and representation-finite algebras are called ``finitely represented'', which is quite appropriate, but could be misread as ``finitely presented''. For eagle-eyed graduate students -- to be sure! -- this should not cause any problem.