Representation theory MOC

Representation

A (linear) representation of a structure 𝐴 is a homomorphism from 𝐴 to a structure of the same kind on a set of linear maps, rep which for single-typed structures is typically a subset of End⁡𝑉 for some vector space 𝑉 called the carrier space. A representation may thus be considered a functor.

Relation to modules

If the represented structure is an K-monoid, a representation is equivalent to a Module over a unital associative algebra, and indeed in most cases one can move from representations of an algebraic structure to representations of a related associative algebra (e.g. group ring, Universal enveloping algebra) over a field without loss of information. This is because End⁡𝑉 itself is an Endomorphism ring.

This yields the three equivalent ways of viewing a representation of a gadget 𝐺:

1. A gadget homomorphism from 𝐺 to a gadget of vector spaces;
2. A functor from a gadget-as-category to Category of vector spaces;
3. A module over a K-monoid related to 𝐺.

Represented object

• Group representation
  • Representation of a finite group
  • Representation of an abelian group
  • Representation of a compact Lie group
• Algebras
  • Module over a unital associative algebra
  • ∗-representation
  • Lie algebra representation
• Quiver representation

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