In mathematics, an algebraic representation of a group G on a k-algebra A is a linear representation π : G → G L ( A ) {\displaystyle \pi :G\to GL(A)} such that, for each g in G, π ( 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 T ( A ) {\displaystyle T(A)} is an algebraic representation of G. If A is a commutative G-algebra, then Spec ( A ) {\displaystyle \operatorname {Spec} (A)} is an affine G-scheme.
See also Algebraic character
References Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402.