Representation space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-30

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Representation_theory


00:02:19 1 Definitions and concepts
00:03:47 1.1 Definition
00:07:49 1.2 Terminology
00:08:34 1.3 Equivariant maps and isomorphisms
00:10:15 1.4 Subrepresentations, quotients, and irreducible representations
00:11:54 1.5 Direct sums and indecomposable representations
00:12:49 1.6 Complete reducibility
00:13:32 1.7 Tensor products of representations
00:19:10 2 Branches and topics
00:20:34 2.1 Finite groups
00:24:58 2.2 Modular representations
00:25:49 2.3 Unitary representations
00:27:44 2.4 Harmonic analysis
00:29:25 2.5 Lie groups
00:30:41 2.6 Lie algebras
00:32:03 2.6.1 Infinite-dimensional Lie algebras
00:33:03 2.6.2 Lie superalgebras
00:33:28 2.7 Linear algebraic groups
00:34:06 2.8 Invariant theory
00:35:23 2.9 Automorphic forms and number theory
00:36:59 2.10 Associative algebras
00:37:35 2.10.1 Module theory
00:38:02 2.10.2 Hopf algebras and quantum groups
00:38:51 3 Generalizations
00:39:00 3.1 Set-theoretic representations
00:40:14 3.2 Representations in other categories
00:41:29 3.3 Representations of categories
00:42:43 4 See also



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SUMMARY
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Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies 
modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. Furthermore, the vector space on which a group (for example) is represented can be infinite-dimensional, and by allowing it to be, for instance, a Hilbert space, methods of analysis can be applied to the theory of groups. Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system.Representation theory is pervasive across fields of mathematics, for two reasons. First, the applications of representation theory are diverse: in addition to its impact on algebra, representation theory:

illuminates and generalizes Fourier analysis via harmonic analysis,
is connected to geometry via invariant theory and the Erlangen program,
has an impact in number theory via automorphic forms and the Langlands program.Secondly, there are diverse approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology.The success of representation theory has led to numerous generalizations. One of the most general is in category theory. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. This description points to two obvious generalizations: first, the algebraic objects can be replaced by more general categories; second, the target category of vector spaces can be replaced by other well-understood categories.