Example of a non-associative algebra

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This page presents and discusses an example of a non-associative division algebra over the complex numbers.

The multiplication is defined by taking the complex conjugate of the usual multiplication: a*b=\overline{ab}. This is a commutative, non-associative division algebra of dimension 2 over the reals, and has no unit element.

[edit] Proof that (\mathbb{C},*) is a division algebra

For a proof that \mathbb{C} is a field, see complex number. Then, the complex numbers themselves trivially form a vector space.

It remains to prove that the binary operation given above satisfies the requirements of a division algebra

  • (x + y)z = x'z + y'z;
  • x(y + z) = x'y + x'z;
  • (ax')y = a(x'y); and
  • x'(by) = b(x'y);

for all scalars a and b in \mathbb{C} and all vectors x, y, and z (also in \mathbb{C}).

For distributivity:

x*(y+z)=\overline{x(y+z)}=\overline{xy+xz}=\overline{xy}+\overline{xz}=x*y+x*z.

(similarly for right distributivity); and for the third and fourth requirements

(ax)*y=\overline{(ax)y}=\overline{a(xy)}=\overline{a}\cdot\overline{xy}=\overline{a}(x*y).

[edit] Non associativity of (\mathbb{C},*)

[edit] Discussion

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