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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 T(A) is an algebraic representation of G.

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


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

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