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 n² e_ie_j’s to know all vector products. Indeed for some set of n³ field elements c^k_ij, e_ie_j = Σ_kc^k_ije_k. For any such n³ 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_kx^ke_k
where x^k are n field values.

xy = Σ_kx^ke_kΣ_jy^je_j = Σ_kΣ_j(x^ke_k)(y^je_j) = Σ_kΣ_j(x^ky^j)(e_ke_j) = Σ_i(Σ_kΣ_jcⁱ_kjx^ky^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_kΣ_jcⁱ_kjx^ky^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_kz^ke_k. When x = Σ_kx^ke_k and z = Σ_kz^ke_k
xy = z => Σ_i(Σ_kΣ_jcⁱ_kjx^ky^j)e_i = Σ_izⁱe_i
Which by the nature of basis sets such as e_i means that
Σ_kΣ_jcⁱ_kjx^ky^j = zⁱ for each i.
If we take d_ij = Σ_kcⁱ_kjx^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.