View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r0814801.png" />. Such representations are also called linear representations in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r0814802.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r0814803.png" /> is called the representation space (or space of the representation). Frequently, representation theory means the theory of linear representations. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r0814804.png" /> 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|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|Representation of an associative algebra]]). More precisely, the linear representations of semi-groups (of groups) (cf. [[Representation of a group|Representation of a group]]; [[Representation of a semi-group|Representation of a semi-group]]) in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r0814805.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r0814806.png" /> are in a natural one-to-one correspondence with the representations of the corresponding semi-group (group) algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r0814807.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r0814808.png" />. The representations of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r0814809.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148010.png" /> correspond bijectively to the linear representations of its [[Universal enveloping algebra|universal enveloping algebra]].

Specifying a linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148011.png" /> of an associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148012.png" /> in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148013.png" /> is equivalent to specifying on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148014.png" /> an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148015.png" />-module structure; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148016.png" /> is called the module of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148017.png" />. When considering representations of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148018.png" /> or Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148019.png" />, one also speaks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148020.png" />-modules or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148021.png" />-modules (cf. [[Module|Module]]). Homomorphisms of modules of representations are called intertwining operators (cf. [[Intertwining operator|Intertwining operator]]). Isomorphic modules correspond to equivalent representations. A submodule of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148022.png" /> of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148023.png" /> is a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148024.png" /> that is invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148025.png" />; the representation induced in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148026.png" /> is called a subrepresentation and the representation induced in the quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148027.png" /> is called a quotient representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081480/r08148028.png" />. 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|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|Continuous representation]]; [[Representation of a topological group|Representation of a topological group]]).



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.W. Curtis,   I. Reiner,   "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR></table>