Algebraic representation

Short description: Group representation via algebra automorphisms

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

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

See also

• Algebraic character

References

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