nLab nonassociative algebra

Definition

Let $k$ be a commutative unital ring, usually a field (but conceivably even a commutative rig).

A nonassociative $k$-algebra is a $k$-module $V$ equipped with a bilinear product $V\otimes V\to V$.

This product is typically neither associative nor unital, although it can be (an example of the red herring principle).

Mathematicians working in the field of nonassociative algebras often say simply ‘algebra’ meaning a nonassociative algebra.

Examples

Some interesting subclasses are Lie algebra, Jordan algebra, Leibniz algebra, alternative algebra, associative unital algebra, composition algebra…

The octonions are a (slightly) non-associative real normed division algebra.

Related concepts

• nonassociative ring

• noncommutative algebra, nonunital algebra

References

• Richard D. Schafer, Introduction to Non-Associative Algebras, Dover, New York, 1995. (pdf)