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 real 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.

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

For a proof that \mathbb{R} is a field, see real number. Then, the complex numbers themselves clearly 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;
  • (a x)y = a(x y); and
  • x(b y) = b(x y);

for all scalars a and b in \mathbb{R} 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).

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

  • a * (b * c) = a * \overline{b c} = \overline{a \overline{b c}} = \overline{a} b c
    (a * b) * c = \overline{a b} * c = \overline{\overline{a b} c} = a b \overline{c}

So, if a, b, and c are all non-zero, and if a and c do not differ by a real multiple, a * (b * c) \neq (a * b) * c.