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Tag Archives: topological groups
Polynomials and Representations XXX
Representations of GLn and Un Note: all representations of topological groups are assumed to be continuous and finite-dimensional. Here, we will look at representations of the general linear group We fix the following notations: denotes for some fixed ; is the … Continue reading
Polynomials and Representations XXIX
Characters Definition. The character of a continuous G-module V is defined as: This is a continuous map since it is an example of a matrix coefficient. Clearly for any . The following are quite easy to show: The last equality, … Continue reading
Polynomials and Representations XXVIII
Starting from this article, we will look at representations of . Now, itself is extremely complicated so we will only focus on representations of particular types. Generally, for any topological group G, we want: to be a continuous homomorphism of groups. Continuous … Continue reading
Topology: Quotients of Topological Groups
Topology for Coset Space This is really a continuation from the previous article. Let G be a topological group and H a subgroup of G. The collection of left cosets G/H is then given the quotient topology. This quotient space, however, satisfies an additional … Continue reading
Posted in Notes
Tagged advanced, group quotients, open maps, quotient topology, topological groups, topology, universal properties
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Topology: Quotient Topology and Gluing
In topology, there’s the concept of gluing points or subspaces together. For example, take the closed interval X = [0, 1] and glue the endpoints 0 and 1 together. Pictorially, we get: That looks like a circle, but to prove it’s … Continue reading
Posted in Notes
Tagged advanced, gluing, klein bottle, mobius strip, quotient topology, topological groups, topology, universal properties
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Topology: Topological Groups
This article assumes you know some basic group theory. The motivation here is to consider groups whose underlying operations are continuous with respect to its topology. Definition. A topological group G is a group with an underlying topology such that: the … Continue reading
Posted in Notes
Tagged advanced, compact sets, connected components, groups, homeomorphisms, separation axioms, topological groups, topology
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Mathematics and Such 