A ug 2 00 4 Notes on groups and representations Stephen
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چکیده
These informal notes are concerned with the broad themes of harmonic analysis of groups and their representations. We shall follow somewhat the view of a classical analyst, with interest in various norms in particular. At the same time we shall try to notice some algebraic aspects, which includes using fields other than the complex numbers. Let G be a group. Thus G is a set with a distinguished element e and a binary operation, the group law, such that e is both a left and right identity element, the group operation satisfies the associative law, and every element of G has an inverse. If also the group operation satisfies the commutative law, then G is said to be a commutative or abelian group. A subset H of G is called a subgroup of G if it contains the identity element, if the product of any two elements of H under the group operation is also an element of H , and if the inverse of each element of H is also an element of H . In other words, H should be a group itself using the same group operations from G. Suppose that G1, G2 are groups and φ is a mapping from G1 to G2. We say that φ is a group homomorphism if φ maps the identity element of G1 to the identity element of G2 and if φ is compatible with the group operations on G1 and G2 in the sense that φ applied to a product of elements x, y of G1 is equal to the product of φ(x), φ(y) in G2 and φ applied to the inverse of an element x of G1 is equal to the inverse of φ(x) in G2. These notes are dedicated to Bob Brooks, who told me all sorts of cool stuff while we were visiting the Centre Émile Borel at the Institut Henri Poincaré in the summer of 2002.
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تاریخ انتشار 2004