_Representations_ describe how we take an axiomatic system and represent it in terms of structure that obeys particular rules of its own. In particular, a representation is a [map](Maps.md) of an _[[algebraic structure]]_ onto a [set](Sets.md). Encompassed in a representation may be sets of operators as well as combinations of these operators that may also form equations. Given an equation, the resulting _space of solutions_ invariant under a particular symmetry may be a way in which a representation is manifested. # Faithful representations If the representation is a one-to-one homomorphism (i.e. an [isomorphism](Isomorphisms.md)), it is considered _faithful_. Hence we say the goal of _representation theory_ is to understand the ways in which an [algebraic structure](algebraic%20structure) can act as a group of matrices or numbers _up to an isomorphism_ - i.e. the way in which we can _represent_ groups and other algebraic structures as faithful representations. # Representations of algebraic structures * [Group representation](Group%20representation.md) * [[Algebra representation]] * [Lie algebra representation](Lie%20algebra%20representation.md) %%The most general definition of a representation seems to be that of a representation of an associative algebra - See Introduction to representation theory by P. Etingof et. al.%% %%In addition there's the notion of linear representations. It appears that most representations are linear representations or permutation representations. Linear representations are also vector spaces - see Dummit and Foote's text.%% %%Where we refer to linear representations it seems to be that the notion of direct sums of representations naturally arises - see Woit's text on this and in addition see the lecture notes by E. Kowalski at ETH Zurich%% %%Other notions are representations of quivers - which seem to be distinct from representations of groups%% #MathematicalFoundations/Algebra/AbstractAlgebra/RepresentationTheory