Example of a non-associative algebra

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 = a b ¯ {\displaystyle 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 (C,*) is a division algebra

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For a proof that  R {\displaystyle \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  R {\displaystyle \mathbb {R} } and all vectors x, y, and z (also in  C {\displaystyle \mathbb {C} } ).

For distributivity:

x ∗ ( y + z ) = x ( y + z ) ¯ = x y + x z ¯ = x y ¯ + x z ¯ = x ∗ y + x ∗ z , {\displaystyle 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

( a x ) ∗ y = ( a x ) y ¯ = a ( x y ) ¯ = a ¯ ⋅ x y ¯ = a ¯ ( x ∗ y ) . {\displaystyle (ax)*y={\overline {(ax)y}}={\overline {a(xy)}}={\overline {a}}\cdot {\overline {xy}}={\overline {a}}(x*y).}

## Non associativity of (C,*)

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*      a ∗ ( b ∗ c ) = a ∗ b c ¯ = a b c ¯ ¯ = a ¯ b c {\displaystyle a*(b*c)=a*{\overline {bc}}={\overline {a{\overline {bc}}}}={\overline {a}}bc}
( a ∗ b ) ∗ c = a b ¯ ∗ c = a b ¯ c ¯ = a b c ¯ {\displaystyle (a*b)*c={\overline {ab}}*c={\overline {{\overline {ab}}c}}=ab{\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 ) ≠ ( a ∗ b ) ∗ c {\displaystyle a*(b*c)\neq (a*b)*c} .