In the mathematical field of representation theory, a trivial representation is a

representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...

of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial

of an

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

is a (

Lie A lie is an assertion that is believed to be false, typically used with the purpose of deceiving or misleading someone. The practice of communicating lies is called lying. A person who communicates a lie may be termed a liar. Lies can be inter ...

) 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 In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...

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

*. Representation theory {{algebra-stub