Representation theory - Wikipedia

representation-theorygroup-theorylie-algebrasabstract-algebralinear-algebra

Abstraction: Studying abstract algebra via linear transformations of vector spaces

Key points:

Reduces problems in abstract algebra to linear algebra by mapping group/algebra elements to invertible matrices
Three main algebraic objects amenable to representation: groups, associative algebras, and Lie algebras
Irreducible representations are building blocks; Schur's lemma states any equivariant map between irreducibles is zero or an isomorphism
Complete reducibility (semisimplicity) holds for finite groups (Maschke's theorem), compact groups, and semisimple Lie algebras over characteristic-zero fields
Finite-dimensional representations of semisimple Lie algebras fully classified by Cartan via weight spaces and root systems
Connects to Fourier/harmonic analysis, the Erlangen program in geometry, and the Langlands program in number theory

Connections: Representation Theory · Group Theory · Lie Algebras · Harmonic Analysis