Augmentation (algebra)

In algebra, an augmentation of an associative algebra A over a field k is an algebra homomorphism $A \to k$ , typically denoted by ε. An algebra together with an augmentation is called an augmented algebra.

For example, if $A =k[G]$ is the group algebra of a group G, then

A \to k, \, \sum a_i x_i \mapsto \sum a_i

is an augmentation. The kernel is called the augmentation ideal of A.

References

Loday, Jean-Louis; Vallette, Bruno (2012). Algebraic operads. Grundlehren der Mathematischen Wissenschaften 346. Berlin: Springer-Verlag. p. 2. ISBN 978-3-642-30361-6. Zbl 1260.18001.