# Algebraic representation

In mathematics, an algebraic representation of a group G on a k-algebra A is a linear representation ${\displaystyle \pi :G\to GL(A)}$ such that, for each g in G, ${\displaystyle \pi (g)}$ is an algebra automorphism. Equipped with such a representation, the algebra A is then called a G-algebra.

For example, if V is a linear representation of a group G, then the representation put on the tensor algebra ${\displaystyle T(A)}$ is an algebraic representation of G.

If A is a commutative G-algebra, then ${\displaystyle \operatorname {Spec} (A)}$ is an affine G-scheme.

## References

• Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402.