non-associative algebra
In mathematics, non-associative algebra is a subfield of abstract algebra, in which are studied algebraic structures endowed with a binary operation that is not assumed to obey the associative law.
The associative law is the rule of ordinary algebra allowing one to dispense with parentheses in dealing with addition, or multiplication: it makes good sense to write down 3 + 5 + 7, and there is no requirement to specify (3 + 5) + 7, or 3 + (5 + 7). In group theory and many other branches of algebra the associative law is also assumed. A reason lying behind this is that composition of functions is associative; so that abstraction away from function composition may still permit the associative law to be assumed.
The study of non-associative structures therefore arises from reasons somewhat different from the mainstream of classical algebra. One area within non-associative algebra that has grown very large is that of Lie algebras. There the associative law is replaced by the Jacobi identity. Lie algebras abstract the essential nature of infinitesimal transformations, and have become ubiquitous in mathematics.
There are other specific types of non-associative structures that have been studied in depth. They tend to come from some specific applications. Some of these arise in combinatorial mathematics. See Category:Nonassociative algebra for further examples.
Non-associative algebras over a field
A non-associative algebra (or distributive algebra) over a field K is a K-vector space A
equipped with a K-bilinear map A×A→A. There are left and right multiplication maps 
and 
. The enveloping algebra
of A is the subalgebra of all K-endomorphisms of A generated by the multiplication maps.
An algebra is unital or unitary if it has a unit or identity element I with Ix = x = xI for all x in the algebra.
An algebra is power associative if xn is well defined for all x in the algebra and all positive integer n: equivalently the subalgebra generated by any one element is associative.
An algebra is alternative if (xx)y = x(xy) and y(xx) = (yx)x for all x and y: equivalently the subalgebra generated by any two elements is associative.
A Jordan algebra is commutative and satisfies the Jordan property (xy)(xx) = x(y(xx)) for all x and y.
A Lie algebra is anticommutative and satisfies xx = 0 and the Jacobi identity x(yz) + y(zx) + z(xy) = 0.
These properties are related by associative implies alternative implies power associative; commutative and associative implies Jordan implies power associative. None of the converse implications hold.
References
- Richard D. Schafer, An Introduction to Nonassociative Algebras (1996) ISBN 0486688135
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