# Representation theory

A theory studying homomorphisms of semi-groups (in particular, groups), algebras or other algebraic systems into corresponding endomorphism systems of suitable structures. Most often one considers linear representations, i.e. homomorphisms of semi-groups, groups, associative algebras, or Lie algebras into a semi-group, a group, an algebra, or a Lie algebra of linear transformations of a vector space . Such representations are also called linear representations in the space , and is called the representation space (or space of the representation). Frequently, representation theory means the theory of linear representations. If is finite-dimensional, then its dimension is called the dimension or degree of the representation, and the representation itself is called finite-dimensional. Thus, one distinguishes between finite-dimensional and infinite-dimensional representations. A representation is called faithful if it is injective (cf. Injection).

The study of linear representations of semi-groups, groups and Lie algebras also leads to the study of linear representations of associative algebras (cf. Representation of an associative algebra). More precisely, the linear representations of semi-groups (of groups) (cf. Representation of a group; Representation of a semi-group) in a space over a field are in a natural one-to-one correspondence with the representations of the corresponding semi-group (group) algebra over in . The representations of a Lie algebra over correspond bijectively to the linear representations of its universal enveloping algebra.

Specifying a linear representation of an associative algebra in a space is equivalent to specifying on an -module structure; then is called the module of the representation . When considering representations of a group or Lie algebra , one also speaks of -modules or -modules (cf. Module). Homomorphisms of modules of representations are called intertwining operators (cf. Intertwining operator). Isomorphic modules correspond to equivalent representations. A submodule of a module of a representation is a subspace that is invariant with respect to ; the representation induced in is called a subrepresentation and the representation induced in the quotient module is called a quotient representation of . Direct sums of modules correspond to direct sums of representations, indecomposable modules to indecomposable representations, simple modules to irreducible representations, and semi-simple modules to completely-reducible representations. The tensor product of linear representations, as well as the exterior and symmetric powers of a representation, also yield linear representations (cf. Tensor product of representations).

Next to abstract (or algebraic) representation theory there is a representation theory of topological objects, e.g., topological groups or Banach algebras (cf. Continuous representation; Representation of a topological group).

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#### References

[a1] | A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) |

[a2] | C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) |

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Representation theory.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Representation_theory&oldid=13728