Algebra IX: Finite Groups of Lie Type Finite-Dimensional Division Algebras (Encyclopaedia of Mathematical Sciences (77))

The Finite Groups Of Lie Type Are Of Central Mathematical Importance And The Problem Of Understanding Their Irreducible Representations Is Of Great Interest. The Representation Theory Of These Groups Over An Algebraically Closed Field Of Characteristic Zero Was Developed By P.deligne And G.lusztig In 1976 And Subsequently In A Series Of Papers By Lusztig Culminating In His Book In 1984. The Purpose Of The First Part Of This Book Is To Give An Overview Of The Subject, Without Including Detailed Proofs. The Second Part Is A Survey Of The Structure Of Finite-dimensional Division Algebras With Many Outline Proofs, Giving The Basic Theory And Methods Of Construction And Then Goes On To A Deeper Analysis Of Division Algebras Over Valuated Fields. An Account Of The Multiplicative Structure And Reduced K-theory Presents Recent Work On The Subject, Including That Of The Authors. Thus It Forms A Convenient And Very Readable Introduction To A Field Which In The Last Two Decades Has Seen Much Progress. A.i. Kostrikin, I.r. Shafarevich (eds.). Translation Of: Algebra 9, Issued In Serial: Itogi Nauki I Tekhniki. Serii︠a︡ Sovremennye Problemy Matematiki. Fundamentalʹnye Napravlenii︠a︡. Includes Bibliographical References (p. 224-233) And Indexes. The finite groups of Lie type are of central mathematical importance and the problem of understanding their irreducible representations is of great interest. The representation theory of these groups over an algebraically closed field of characteristic zero was developed by P. Deligne and G. Lusztig in 1976 and subsequently in a series of papers by Lusztig culminating in his book in 1984. The purpose of the first part of this book is to give an overview of the subject, without including detailed proofs. The second part is a survey of the structure of finite-dimensional division algebras with many outline proofs, giving the basic theory and methods of construction and then goes on to a deeper analysis of division algebras over valuated fields. An account of the multiplicative structure and reduced K-theory presents recent work on the subject, including that of the authors. Thus it forms a convenient and very readable introduction to a field which in the last two decades has seen much progress From the fields, commutative rings and groups studied in university mathematics courses, through Lie groups and algebras to category theory, this text shows how the origins of each algebraic concept can be related to attempts to model phenomena in physics or in other branches of mathematics. The first contribution by Carter covers the theory of finite groups of Lie type, an important field of current mathematical research. In the second part, Platonov and Yanchevskii survey the structure of finite-dimensional division algebras, including an account of reduced K-theory. Front Matter....Pages i-vii On the Representation Theory of the Finite Groups of Lie Type over an Algebraically Closed Field of Characteristic 0....Pages 1-120 Finite-Dimensional Division Algebras....Pages 121-233 Back Matter....Pages 235-243 On the representation theory of the finite groups of lie type over an algebraically closed field of characteristic 0 / R.W. Carter Finite-dimensional division algebras / V.P. Platonov, V.I. Yanchevskii In this article we shall be describing the representation theory of a certain class of finite groups.