# Trivial representation

In the mathematical field of representation theory, a **trivial representation** is a representation (*V*, *φ*) of a group *G* on which all elements of *G* act as the identity mapping of *V*. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of *V* to the zero vector.

For any group or Lie algebra, an irreducible trivial representation always exists over any field, and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to unital algebras and unital representations.

Although the trivial representation is constructed in such a way as to make its properties seem tautologous, it is a fundamental object of the theory. A subrepresentation is equivalent to a trivial representation, for example, if it consists of invariant vectors; so that searching for such subrepresentations is the whole topic of invariant theory.

The **trivial character** is the character that takes the value of one for all group elements.

## References

- Fulton, William; Harris, Joe (1991).
*Representation theory. A first course*. Graduate Texts in Mathematics, Readings in Mathematics.**129**. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249, ISBN 978-0-387-97527-6..