Associative Algebra

An **associative algebra** is a vector space V with a new binary operator called multiplication written by juxtaposing two vector expressions.
The operator is required to be associative and bilinear.
If x, y and z denote arbitrary vectors then associativity means

x(yz) = (xy)z

and bilinearity means that (for any x, if Fy = xy then F is a linear operator) and conversely.

In formulae bilinearity means:

x(y+z) = xy + xz

(x+y)z = xz + yz

(αx)y = x(αy) = α(xy)

where α denotes any element of the field of the vector space.

Commutivity (xy = yx) is not required.

There may be a vector 1 such that for all x in the algebra 1x = x1 = x. This is called an algebra with one. All of the algebras at this site have one.

If the vector space is n-dimensional and e_{i} are n basis vectors, then it suffices to know the n^{2} e_{i}e_{j}’s to know all vector products.
Indeed for some set of n^{3} field elements c^{k}_{ij},
e_{i}e_{j}
= Σ_{k}c^{k}_{ij}e_{k}.
For any such n^{3} field elements there is an algebra defined by this equation, but it may not be associative.
The octonions are a non associative algebra.
The c’s determine the algebra up to isomorphism.
Of course the c’s depend on choice of basis vectors.

An arbitrary element x of V can be expressed

x = Σ_{k}x^{k}e_{k}

where x^{k} are n field values.

xy = Σ_{k}x^{k}e_{k}Σ_{j}y^{j}e_{j}
= Σ_{k}Σ_{j}(x^{k}e_{k})(y^{j}e_{j})
= Σ_{k}Σ_{j}(x^{k}y^{j})(e_{k}e_{j})
= Σ_{i}(Σ_{k}Σ_{j}c^{i}_{kj}x^{k}y^{j})e_{i}

and thus the c’s determine multiplication for all vector pairs.

An associative algebra is a division algebra whenever
for all x, if x ≠ 0 (the zero vector) then (for all z there exists a y such that xy = z).
Here x, y and z range over the associative algebra (vector space).
xy = Σ_{i}(Σ_{k}Σ_{j}c^{i}_{kj}x^{k}y^{j})e_{i}.
y will exist just in case the determinant of this linear equation is not 0 for each choice of non-zero x and z.
z = Σ_{k}z^{k}e_{k}.
When x = Σ_{k}x^{k}e_{k}
and z = Σ_{k}z^{k}e_{k}

xy = z =>
Σ_{i}(Σ_{k}Σ_{j}c^{i}_{kj}x^{k}y^{j})e_{i} =
Σ_{i}z^{i}e_{i}

Which by the nature of basis sets such as e_{i} means that

Σ_{k}Σ_{j}c^{i}_{kj}x^{k}y^{j} = z^{i} for each i.

If we take d_{ij} = Σ_{k}c^{i}_{kj}x^{k}

the determinant of d must not be 0.
For some associative algebras where will be non zero vectors x for which the determinant is zero and then it is not a division algebra.