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This book starts with the elementary theory of Lie groups of matrices and arrives at the definition, elementary properties, and first applications of cohomological induction, which is a recently discovered algebraic construction of group representations. Along the way it develops the computational techniques that are so important in handling Lie groups. The book is based on a one-semester course given at the State University of New York, Stony Brook in fall, 1986 to an audience having little or no background in Lie groups but interested in seeing connections among algebra, geometry, and Lie theory. These notes develop what is needed beyond a first graduate course in algebra in order to appreciate cohomological induction and to see its first consequences. Along the way one is able to study homological algebra with a significant application in mind; consequently one sees just what results in that subject are fundamental and what results are minor.

Lie groups. --- Algebra. --- Analytic group. --- Approximate identity. --- Associative algebra. --- Associativity formulas. --- Borel subalgebra. --- Borel-Weil Theorem. --- Boundary operator. --- Cartan subgroup. --- Chain map. --- Change of rings. --- Closed linear group. --- Cohomological induction. --- Complexification. --- Convolution. --- Derivation. --- Derived functor. --- Diffeomorphism. --- Divisible group. --- Dual vector space. --- Enough injectives. --- Equivalence. --- Euler-Poincare principle. --- Exponential. --- Forgetful functor. --- Free resolution. --- Functor. --- Good category. --- Haar measure. --- Highest weight. --- Homogeneous polynomial. --- Homogeneous tensor. --- I functor. --- Induction, cohomological. --- Inner derivation. --- Isotypic subspace. --- K-isotypic subspace. --- Left exact functor. --- Lie algebra. --- Lie bracket. --- Linear extension. --- Long exact sequence. --- Mackey isomorphism. --- Matrix coefficient. --- Multiplicity. --- Normalized Haar measure. --- Positive root. --- Quotient representation. --- Reductive group. --- Regular representation. --- Right exact functor. --- Schur orthogonality. --- Semidirect product. --- Spectral sequence.