Representation theory makes abstract algebra act on vectors
Describe group elements, algebra elements, or Lie brackets as linear maps—often as concrete matrices you can multiply by hand—and hard structural questions collapse into ordinary linear algebra. Physics uses the same move to see how symmetries constrain solutions.
Representation theory studies abstract algebraic structures by realizing their elements as linear transformations of vector spaces. Matrix addition and multiplication supply concrete stand-ins for abstract operations. Groups, associative algebras, and Lie algebras are the main customers; historically groups came first, with invertible matrices multiplying as the group law. Reducing abstract problems to well-understood linear algebra clarifies properties and eases calculation. Infinite-dimensional Hilbert-space representations even import analysis into group theory. In physics, representations track how a system's symmetry group acts on the equations' solution space.
The subject permeates mathematics, with methods drawn from algebraic geometry, the theory of modules, analytic number theory, differential geometry, operator theory, combinatorics, and topology. Number theory feels the impact through automorphic forms and the Langlands program. Category theory offers a sweeping view: algebraic objects become categories and representations become functors into vector spaces—or into other well-understood target categories. That flexibility is why the same vocabulary shows up from finite groups to infinite-dimensional analysis.
Two equivalent definitions are common. One speaks of an action generalizing how matrices hit column vectors; axioms encode identity and compatibility with the algebraic operation (with suitable adjustments for algebras lacking units or for Lie brackets that are not associative products). The other is a homomorphism into endomorphisms or into GL(V). The space's dimension is the degree of the representation. Whether you start from matrices or from linear maps, the payoff is the same: abstract symmetry becomes something you can multiply.
Source: Representation theory
