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 $V$. Such representations are also called linear representations in the space $V$, and $V$ is called the representation space (or space of the representation). Frequently, representation theory means the theory of linear representations. If $V$ 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 $V$ over a field $k$ are in a natural one-to-one correspondence with the representations of the corresponding semi-group (group) algebra over $k$ in $V$. The representations of a Lie algebra $L$ over $k$ correspond bijectively to the linear representations of its universal enveloping algebra.

Specifying a linear representation $\phi$ of an associative algebra $A$ in a space $V$ is equivalent to specifying on $V$ an $A$-module structure; then $V$ is called the module of the representation $\phi$. When considering representations of a group $G$ or Lie algebra $L$, one also speaks of $G$-modules or $L$-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 $V$ of a representation $\phi$ is a subspace $W\subset V$ that is invariant with respect to $\phi$; the representation induced in $W$ is called a subrepresentation and the representation induced in the quotient module $V/W$ is called a quotient representation of $\phi$. 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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