Algebraic representation

In mathematics, an algebraic representation of a group "G" on a "k"-algebra "A" is a linear representation formula_1 such that, for each "g" in "G", formula_2 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 formula_3 is an algebraic representation of "G".
If "A" is a commutative "G"-algebra, then formula_4 is an affine "G"-scheme.