nLab inner product abelian group

Definition

An inner product abelian group is an abelian group $G$ with a binary function $q: G \times G \to \mathbb{Z}$ called the inner product such that the following properties hold:

• for all $a \in G$, $\langle 0, a \rangle = 0$ and $\langle a, 0 \rangle = 0$;
• for all $a, b, c \in G$, $\langle a + b, c \rangle = \langle a, c \rangle + \langle b, c \rangle$ and $\langle a, b + c \rangle = \langle a, b \rangle + \langle a, c \rangle$
• for all $a, b \in G$ and $\langle a, b \rangle = \overline{\langle b, a \rangle}$.

where $\overline{(-)}:\mathbb{Z} \to \mathbb{Z}$ is an involution on the integers.

Typically, the inner product is defined with another axiom

• for all $a, b \in G$ and $c \in \mathbb{Z}$, $\langle c a, b \rangle = c \langle a, b \rangle$ and $\langle a, \overline{c} b \rangle = \langle a, b \rangle \overline{c}$

but this is provable for any binary function which is left and right distributive over the abelian group operations.

Properties

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

See also

• inner product

• abelian group

• ring

• quadratic abelian group

• Clifford ring