## Algebra IX: Finite Groups of Lie Type. Finite-Dimensional Division AlgebrasThe 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. |

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### Contents

I | 3 |

III | 6 |

IV | 7 |

V | 12 |

VI | 12 |

VII | 15 |

VIII | 18 |

X | 20 |

LXXXVIII | 144 |

XC | 145 |

XCII | 146 |

XCIII | 147 |

XCIV | 149 |

XCV | 151 |

XCVII | 153 |

XCVIII | 154 |

XI | 21 |

XII | 23 |

XIII | 25 |

XIV | 27 |

XV | 28 |

XVI | 29 |

XVIII | 30 |

XIX | 32 |

XX | 34 |

XXI | 36 |

XXII | 38 |

XXIII | 39 |

XXIV | 42 |

XXV | 44 |

XXVII | 46 |

XXVIII | 47 |

XXIX | 48 |

XXXI | 50 |

XXXII | 51 |

XXXIII | 53 |

XXXIV | 58 |

XXXVI | 61 |

XXXVIII | 63 |

XXXIX | 66 |

XL | 71 |

XLI | 74 |

XLII | 78 |

XLIII | 81 |

XLIV | 84 |

XLV | 86 |

XLVI | 87 |

XLVII | 91 |

XLVIII | 95 |

XLIX | 98 |

LI | 99 |

LII | 101 |

LIII | 103 |

LIV | 109 |

123 | |

LVI | 125 |

LVII | 126 |

LIX | 127 |

LX | 128 |

LXI | 131 |

LXII | 132 |

LXIV | 133 |

LXVII | 134 |

LXX | 135 |

LXXIII | 136 |

LXXV | 137 |

LXXVI | 138 |

LXXVIII | 139 |

LXXX | 140 |

LXXXV | 142 |

LXXXVII | 143 |

XCIX | 155 |

C | 156 |

CII | 157 |

CIII | 159 |

CV | 161 |

CVI | 162 |

CVII | 163 |

CIX | 164 |

CX | 165 |

CXI | 166 |

CXII | 167 |

CXIII | 168 |

CXIV | 169 |

CXV | 170 |

CXVII | 171 |

CXIX | 172 |

CXXI | 173 |

CXXIII | 174 |

CXXIV | 175 |

CXXV | 176 |

CXXVIII | 179 |

CXXIX | 180 |

CXXX | 182 |

CXXXI | 183 |

CXXXII | 184 |

CXXXIII | 185 |

CXXXIV | 186 |

CXXXVI | 187 |

CXXXVIII | 188 |

CXL | 189 |

CXLIII | 190 |

CXLVI | 191 |

CXLIX | 193 |

CLI | 195 |

CLII | 196 |

CLIV | 197 |

CLVI | 200 |

CLVII | 205 |

CLIX | 206 |

CLXI | 207 |

CLXII | 208 |

CLXIII | 209 |

CLXVI | 210 |

CLXVII | 211 |

CLXVIII | 212 |

CLXX | 213 |

CLXXI | 214 |

CLXXII | 215 |

CLXXIII | 216 |

CLXXIV | 218 |

CLXXV | 219 |

CLXXVI | 221 |

CLXXVII | 222 |

### Other editions - View all

Algebra IX: Finite Groups of Lie Type Finite-Dimensional Division Algebras A.I. Kostrikin,I.R. Shafarevich No preview available - 2014 |

Algebra IX: Finite Groups of Lie Type Finite-Dimensional Division Algebras A.I. Kostrikin,I.R. Shafarevich No preview available - 2010 |

### Common terms and phrases

action acts algebraic groups assume automorphism bijective called cells central simple algebra centre characteristic characters of GF classes of G closed complete components condition conjugacy classes connected consider construction containing corresponding Coxeter cuspidal cyclic cyclic extension defined Definition denoted describe determined division algebra elements equivalence example exists extension F-stable fact factor finite groups follows functions Galois geometric give given group G henselian homomorphism important induced involution irreducible characters isomorphic K-algebra linear Lusztig Math maximal Moreover multiplication natural normal obtained pairs polynomial possible prime problem Proposition ramified reduced relation remark representation ring root semisimple shown shows skew field space splitting field structure subfield subgroup subset Suppose Theorem theory torus trivial unipotent characters unipotent classes unique unramified valuation variety Weyl group