Vrepresentations

Vrepresentations, sometimes written as V-representations, denote representations of algebraic structures on a fixed vector space V over a field k. In this setting the action is realized by linear maps on V, i.e., by homomorphisms into the endomorphism algebra End_k(V). When the structure is a group G, a V-representation is a group homomorphism ρ: G → GL_k(V). When the structure is a k-algebra A, a V-representation is a k-algebra homomorphism A → End_k(V). For a Lie algebra g, a V-representation is a Lie algebra homomorphism ρ: g → End_k(V).

Given two V-representations with the same underlying space V, a morphism is a linear endomorphism T of V that intertwines the actions: T ρ(g) = ρ'(g) T for all g. If the underlying spaces vary, the usual notion of representation equivalence is isomorphism of pairs (V, ρ).

In finite dimension, V-representations can be decomposed into direct sums of irreducible subrepresentations under suitable conditions; irreducibility means no nontrivial invariant subspace. Tools such as Schur's lemma, characters, and modular representation theory apply with appropriate hypotheses.

Examples include the standard representation of a finite group on a vector space, the adjoint representation of a Lie algebra on itself, and permutation representations on function spaces. V-representations are central in studying symmetry in mathematics and physics, and they provide a common language for modules over algebras.

While the term emphasizes fixed V, in practice one often studies representations with varying underlying spaces; the broader concept is simply representation theory, with V representing the target space of the action.