nLab quadratic abelian group

Definition

A quadratic abelian group is an abelian group $G$ with a function $q: G \to \mathbb{Z}$ such that the following properties hold:

• (cube relation) For any $x,y,z \in G$,
  $$q(x+y+z) - q(x+y) - q(x+z) - q(y+z) + q(x) + q(y) + q(z) = 0$$
• (homogeneous of degree 2) For any $x \in G$ and any $r \in \mathbb{Z}$,
  $$q(r x) = r^2 q(x)$$

Examples

• Every ring is a quadratic abelian group.

• Every inner product abelian group is a quadratic abelian group with $q(x) \coloneqq \langle x, x \rangle$.

 See also

• quadratic function

• abelian group

• ring

• Clifford ring

• inner product abelian group