Torsor

up:: 021 MOC Algebra

“A torsor is like a group that has forgotten its identity.” (John Baez)

In a $G$-torsor, one cannot add elements ─ but rather, one can think of their differences as being elements of an underlying group $G$.

A $G$-torsor is a set $T$ with a Group Action over some Group $G$

$$+:G\times T\to T$$

which is Free and Transitive. This means that every pair of points in $T$ has only one element in $G$ which “connects” them ─ which can be seen as their difference (or ratio).

Examples

• An Affine Space is an example of an ${\mathbb{R}}^n$-torsor ─ we can only measure real position differences
• Energies lie in an $\mathbb{R}$-torsor ─ we can only measure real energy differences
• Voltages lie in an $\mathbb{R}$-torsor ─ we can only measure real voltage differences

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References

• torsor in nLab
• torsors (John Baez)