Matches in DBpedia 2015-10 for { <http://dbpedia.org/resource/Representation_theory> ?p ?o }
- Representation_theory abstract "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.A feature of representation theory is its pervasiveness in mathematics. There are two sides to this. 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.The second aspect is the diversity of 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.A representation should not be confused with a presentation.".
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- Representation_theory wikiPageWikiLink Abstract_algebra.
- Representation_theory wikiPageWikiLink Abstract_harmonic_analysis.
- Representation_theory wikiPageWikiLink Adelic_algebraic_group.
- Representation_theory wikiPageWikiLink Affine_representation.
- Representation_theory wikiPageWikiLink Affine_space.
- Representation_theory wikiPageWikiLink Alexander_Grothendieck.
- Representation_theory wikiPageWikiLink Algebra.
- Representation_theory wikiPageWikiLink Algebra_homomorphism.
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- Representation_theory wikiPageWikiLink Algebraic_group.
- Representation_theory wikiPageWikiLink Algebraic_operation.
- Representation_theory wikiPageWikiLink Algebraic_structure.
- Representation_theory wikiPageWikiLink Algebraic_variety.
- Representation_theory wikiPageWikiLink Algebraically_closed.
- Representation_theory wikiPageWikiLink Algebraically_closed_field.
- Representation_theory wikiPageWikiLink Analytic_function.
- Representation_theory wikiPageWikiLink Analytic_number_theory.
- Representation_theory wikiPageWikiLink Annals_of_Mathematics.
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- Representation_theory wikiPageWikiLink Banach_space.
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- Representation_theory wikiPageWikiLink Category:Representation_theory.
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- Representation_theory wikiPageWikiLink Category_of_modules.
- Representation_theory wikiPageWikiLink Category_of_sets.
- Representation_theory wikiPageWikiLink Category_of_topological_spaces.
- Representation_theory wikiPageWikiLink Category_of_vector_spaces.
- Representation_theory wikiPageWikiLink Category_theory.
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- Representation_theory wikiPageWikiLink Classification_of_finite_simple_groups.
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- Representation_theory wikiPageWikiLink Crystal_basis.
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- Representation_theory wikiPageWikiLink Dimension_(vector_space).
- Representation_theory wikiPageWikiLink Dimension_of_a_vector_space.
- Representation_theory wikiPageWikiLink Direct_sum.
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- Representation_theory wikiPageWikiLink Direct_sum_of_representations.
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- Representation_theory wikiPageWikiLink Division_algebra.
- Representation_theory wikiPageWikiLink Doublet_representation.
- Representation_theory wikiPageWikiLink Dual_space.
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- Representation_theory wikiPageWikiLink Eigenspace.
- Representation_theory wikiPageWikiLink Eigenvalues_and_eigenvectors.
- Representation_theory wikiPageWikiLink Element_(mathematics).
- Representation_theory wikiPageWikiLink Element_(set_theory).
- Representation_theory wikiPageWikiLink Endomorphism.
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- Representation_theory wikiPageWikiLink Euclidean_group.
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- Representation_theory wikiPageWikiLink Eugene_Wigner.
- Representation_theory wikiPageWikiLink Exactly_solvable_model.
- Representation_theory wikiPageWikiLink Factorization.
- Representation_theory wikiPageWikiLink Faithful_representation.
- Representation_theory wikiPageWikiLink Felix_Klein.
- Representation_theory wikiPageWikiLink Field_(mathematics).
- Representation_theory wikiPageWikiLink Finite_field.
- Representation_theory wikiPageWikiLink Finite_groups_of_Lie_type.