nLab inner product abelian group

Contents

Context

Algebra

Group theory

Contents

Definition

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

  • for all aGa \in G, 0,a=0 \langle 0, a \rangle = 0 and a,0=0 \langle a, 0 \rangle = 0 ;
  • for all a,b,cGa, b, c \in G, a+b,c=a,c+b,c \langle a + b, c \rangle = \langle a, c \rangle + \langle b, c \rangle and a,b+c=a,b+a,c \langle a, b + c \rangle = \langle a, b \rangle + \langle a, c \rangle
  • for all a,bGa, b \in G and a,b=b,a¯ \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,bGa, b \in G and cc \in \mathbb{Z}, ca,b=ca,b \langle c a, b \rangle = c \langle a, b \rangle and a,c¯b=a,bc¯\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)x,xq(x) \coloneqq \langle x, x \rangle.

See also

Created on May 11, 2022 at 08:27:29. See the history of this page for a list of all contributions to it.