nLab
orthogonal group

Definition

For $n\in \mathbb{N}$ the orthogonal group is the group of isometries of a real $n$-dimensional Hilbert space.

This is canonically isomorphic to the group of $n\times n$ orthogonal matrices.

The analog for complex Hilbert spaces is the unitary group.

In a lined topos

Not just to make this entry interesting for the $n$Lab, but also because I might actually need this for an application, I’d like to give a discussion of the orthogonal group and of the general linear group inside an arbitrary lined topos. What can one say?

Let $(𝒯,R)$ be a lined topos.

Then for $n\in \mathbb{N}$ the orthogonal group $O(n)$ is the subgroup of the automorphism group ${\mathrm{Aut}}_{𝒯}(R^n)$ of the $n$-fold product $R^n$ of the line $R$ in $𝒯$.

Whitehead tower and higher orientation structures

The Whitehead tower of the orthogonal group plays an important role in applications related to quantum physics.

The first steps are

$$\cdots \to \mathrm{Fivebrane}(n)\to \mathrm{String}(n)\to \mathrm{Spin}(n)\to \mathrm{SO}(n)\to \mathrm{O}(n).$$\cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,.

Fivebrane group to String group to Spin group to special orthogonal group to orthogonal group.

Given a manifold $X$, lifts of the structure map $X\to ℬO(n)$ of the $O(n)$-principal bundle to which the tangent bundle is associated through this tower define, respectively

• orientation

• Spin structure

• String structure

• Fivebrane structure

on $X$.