Representations of quivers and mixed graphs

Abstract

This is a survey article for "Handbook of Linear Algebra", 2nd ed., Chapman & Hall/CRC, 2014. An informal introduction to representations of quivers and finite dimensional algebras from a linear algebraist's point of view is given. The notion of quiver representations is extended to representations of mixed graphs, which permits one to study systems of linear mappings and bilinear or sesquilinear forms. The problem of classifying such systems is reduced to the problem of classifying systems of linear mappings.

References in corpus (8)

• Canonical matrices for linear matrix problems
• Canonical forms for complex matrix congruence and *congruence
• Classification problems for system of forms and linear mappings
• Complexity of matrix problems
• Canonical matrices of bilinear and sesquilinear forms
• Canonical matrices of isometric operators on indefinite inner product spaces
• Pairs of mutually annihilating operators
• Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric, skew-symmetric, or Hermitian forms

Cited by in corpus (8)

• Wildness for tensors
• Congruence of matrix spaces, matrix tuples, and multilinear maps
• Classification of linear operators satisfying $(Au,v)=(u,A^{r}v)$ or $(Au,A^{r}v)=(u,v)$ on a vector space with indefinite scalar product
• Isometric and selfadjoint operators on a vector space with nondegenerate diagonalizable form
• Regularizing decompositions for matrix pencils and a topological classification of pairs of linear mappings
• Classification of linear mappings between indefinite inner product spaces
• Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence
• Lipschitz property for systems of linear mappings and bilinear forms