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Representation theory depends upon the type of algebraic object being represented. There are several different classes of groups, associative algebras and Lie algebras, and their representation theories all have an individual flavour. Representation theory depends upon the nature of the vector space on which the algebraic object is represented.
In representation theory, the category of representations of some algebraic structure A has the representations of A as objects and equivariant maps as morphisms between them. . One of the basic thrusts of representation theory is to understand the conditions under which this category is semisimple; i.e., whether an object decomposes into simple objects (see Maschke's theorem for the case of ...
This is a glossary of representation theory in mathematics. The term "module" is often used synonymously for a representation; for the module-theoretic terminology, see also glossary of module theory. See also Glossary of Lie groups and Lie algebras, list of representation theory topics and Category:Representation theory.
Collective representations are concepts, ideas, categories and beliefs that do not belong to isolated individuals, but are instead the product of a social collectivity. [1] Émile Durkheim (1858-1917) originated the term "collective representations" to emphasise the way that many of the categories of everyday use–space, time, class, number etc–were in fact the product of collective social ...
In representation theory, a branch of mathematics, Artin's theorem, introduced by E. Artin, states that a character on a finite group is a rational linear combination of characters induced from all cyclic subgroups of the group.
In representation theory, a subrepresentation of a representation (,) of a group G is a representation (|,) such that W is a vector subspace of V and | = |.. A nonzero finite-dimensional representation always contains a nonzero subrepresentation that is irreducible, the fact seen by induction on dimension.
Definition (4) evidently implies definition (3). To show the converse, let G be a locally compact group satisfying (3), assume by contradiction that for every K and ε there is a unitary representation that has a (K, ε)-invariant unit vector and does not have an invariant vector. Look at the direct sum of all such representation and that will ...
Rather than study the representation theory of these Lie groups, it is often preferable to study the closely related representation theory of the corresponding Lie algebras, which are usually simpler to compute. Now, representations of the Lie algebra correspond to representations of the universal cover of the original group. [6]