Augmentation (algebra) explained

A \to k , typically denoted by ε. An algebra together with an augmentation is called an augmented algebra. The kernel of the augmentation is a two-sided ideal called the augmentation ideal of A.

For example, if

A = k[G] is the group algebra of a finite group G, thenA \to k, \sum a _{i} x _{i} \mapsto \sum a _{i} is an augmentation.

If A is a graded algebra which is connected, i.e.

A _{0=k} , then the homomorphism A\to k which maps an element to its homogeneous component of degree 0 is an augmentation. For example,k[x]\to k, \sum a ^{i} _{ix} \mapsto a _{0} is an augmentation on the polynomial ring k[x] .

References

• Book: Loday . Jean-Louis . Jean-Louis Loday . Vallette . Bruno . Algebraic operads . 1260.18001 . Grundlehren der Mathematischen Wissenschaften . 346 . Berlin . . 978-3-642-30361-6 . 2012 . 2 .