Algebraic representation

In mathematics, an algebraic representation of a group G on a k-algebra A is a linear representation $$\pi: G \to GL(A)$$ such that, for each g in G, $$\pi(g)$$ 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 $$T(A)$$ is an algebraic representation of G.

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