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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.
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The object of this monograph is to give an exposition of the real-variable theory of Hardy spaces (HP spaces). This theory has attracted considerable attention in recent years because it led to a better understanding in Rn of such related topics as singular integrals, multiplier operators, maximal functions, and real-variable methods generally. Because of its fruitful development, a systematic exposition of some of the main parts of the theory is now desirable. In addition to this exposition, these notes contain a recasting of the theory in the more general setting where the underlying Rn is replaced by a homogeneous group.The justification for this wider scope comes from two sources: 1) the theory of semi-simple Lie groups and symmetric spaces, where such homogeneous groups arise naturally as "boundaries," and 2) certain classes of non-elliptic differential equations (in particular those connected with several complex variables), where the model cases occur on homogeneous groups. The example which has been most widely studied in recent years is that of the Heisenberg group.
Functions of real variables. --- Hardy spaces. --- Lie groups. --- "admissible. --- Campanato space. --- Campbell-Hausdorff formul. --- Chebyshev's inequality. --- Constants. --- Derivatives and multiindices. --- Dilations. --- Hardy space. --- Hardy-Littlewood maximal function. --- Heisenberg group. --- Littlewood-Paley function. --- Lusin function. --- Maximal functions. --- Norms. --- Other operations on functions". --- Poisson kernel. --- Poisson-Szegö kernel. --- area integral. --- associated (to a ball). --- atom. --- atomic Hardy space. --- atomic decomposition. --- ball. --- commutative approximate identity. --- convolution. --- dilations. --- distribution function. --- graded. --- grand maximal function. --- heat kernel. --- heat semigroup. --- isotropic degree. --- kernel of type. --- lower central series. --- nilpotent. --- nonincreasing rearrangement. --- nontangential maximal function. --- p-admissible. --- polynomial. --- polyradial. --- positive operator. --- quasinorms. --- seminorms.
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