## Representation Theory: A First CourseThe primary goal of these lectures is to introduce a beginner to the finite dimensional representations of Lie groups and Lie algebras. Since this goal is shared by quite a few other books, we should explain in this Preface how our approach differs, although the potential reader can probably see this better by a quick browse through the book. Representation theory is simple to define: it is the study of the ways in which a given group may act on vector spaces. It is almost certainly unique, however, among such clearly delineated subjects, in the breadth of its interest to mathematicians. This is not surprising: group actions are ubiquitous in 20th century mathematics, and where the object on which a group acts is not a vector space, we have learned to replace it by one that is {e. g. , a cohomology group, tangent space, etc. }. As a consequence, many mathematicians other than specialists in the field {or even those who think they might want to be} come in contact with the subject in various ways. It is for such people that this text is designed. To put it another way, we intend this as a book for beginners to learn from and not as a reference. This idea essentially determines the choice of material covered here. As simple as is the definition of representation theory given above, it fragments considerably when we try to get more specific. |

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### Contents

IV | 3 |

V | 5 |

VI | 8 |

VII | 12 |

VIII | 15 |

IX | 18 |

X | 21 |

XI | 26 |

LIX | 273 |

LX | 282 |

LXI | 286 |

LXII | 292 |

LXIII | 294 |

LXIV | 296 |

LXV | 299 |

LXVI | 307 |

XII | 31 |

XIII | 32 |

XIV | 36 |

XV | 39 |

XVI | 44 |

XVII | 52 |

XVIII | 54 |

XIX | 63 |

XX | 67 |

XXI | 75 |

XXII | 84 |

XXIII | 89 |

XXIV | 93 |

XXV | 95 |

XXVI | 101 |

XXVII | 104 |

XXVIII | 111 |

XXIX | 114 |

XXX | 121 |

XXXI | 125 |

XXXII | 128 |

XXXIII | 131 |

XXXIV | 133 |

XXXV | 136 |

XXXVI | 139 |

XXXVII | 141 |

XXXVIII | 146 |

XXXIX | 151 |

XL | 153 |

XLI | 161 |

XLII | 175 |

XLIII | 182 |

XLIV | 185 |

XLV | 195 |

XLVI | 197 |

XLVII | 206 |

XLVIII | 211 |

XLIX | 217 |

L | 222 |

LI | 227 |

LII | 231 |

LIII | 238 |

LIV | 244 |

LV | 253 |

LVI | 259 |

LVII | 262 |

LVIII | 267 |

LXVII | 312 |

LXVIII | 317 |

LXIX | 319 |

LXX | 325 |

LXXI | 330 |

LXXII | 339 |

LXXIII | 346 |

LXXIV | 350 |

LXXV | 359 |

LXXVI | 366 |

LXXVII | 375 |

LXXVIII | 382 |

LXXIX | 395 |

LXXX | 399 |

LXXXI | 403 |

LXXXII | 415 |

LXXXIII | 419 |

LXXXIV | 424 |

LXXXV | 430 |

LXXXVI | 440 |

LXXXVII | 444 |

LXXXVIII | 451 |

LXXXIX | 453 |

XC | 462 |

XCI | 465 |

XCII | 471 |

XCIII | 472 |

XCIV | 475 |

XCV | 478 |

XCVI | 481 |

XCVII | 482 |

XCVIII | 482 |

XCIX | 483 |

C | 487 |

CI | 489 |

CII | 491 |

CIII | 493 |

CIV | 499 |

CV | 500 |

CVI | 504 |

CVII | 511 |

CVIII | 514 |

CIX | 516 |

CX | 536 |

543 | |

547 | |

### Common terms and phrases

abelian action adjoint representation algebra g automorphisms basis bracket Cartan subalgebra coefficients commutator conjugacy classes conjugate construction corresponding decompose decomposition deduce defined denote determinant diagonal dimension direct sum Dynkin diagram eigenspace eigenvalues eigenvector element endomorphism example Exercise exterior powers fact factors follows given gives GL(K GLnC group G highest weight vector homomorphism ideal identity induced integers invariant irreducible representation isomorphism kernel Killing form Lecture Lemma Lie groups linear matrix multiplication nilpotent nonzero Note partition permutation positive roots preserving proof Proposition prove quadric quaternionic quotient real Lie representations of G restriction root spaces root system scalar semisimple Lie algebra Show sI2C sI3C simple Lie algebras simple roots SL2C SLnC so2nC solvable somC sp2nC sp4C spanned standard representation subgroup subrepresentation subspace Sym"K Sym2 Sym2K symmetric polynomials symmetric powers symplectic tensor product theorem unique vector space verify weight diagram Weyl chamber Weyl group Young diagram zero

### Popular passages

Page 528 - Let g be a Lie algebra. g is semisimple if and only if its Killing form is nondegenerate.

Page viii - Mumford, from whom we learned much of what we know about the subject, and whose ideas are very much in evidence in this book.

Page 541 - The Weyl group is generated by the reflections in the simple roots, ie^ SB = $B0.