Inner form

In mathematics, an inner form of an algebraic group $G$ over a field $K$ is another algebraic group $H$ such that there exists an isomorphism $\phi$ between $G$ and $H$ defined over ${\overline {K}}$ (this means that $H$ is a $K$ -form of $G$ ) and in addition, for every Galois automorphism $\sigma \in \mathrm {Gal} ({\overline {K}}/K)$ the automorphism $\phi ^{-1}\circ \phi ^{\sigma }$ is an inner automorphism of $G({\overline {K}})$ (i.e. conjugation by an element of $G({\overline {K}})$ ).

Through the correspondance between $K$ -forms and the Galois cohomology $H^{1}(\mathrm {Gal} ({\overline {K}}/K),\mathrm {Inn} (G))$ this means that $H$ is associated to an element of the subet $H^{1}(\mathrm {Gal} ({\overline {K}}/K),\mathrm {Inn} (G))$ where $\mathrm {Inn} (G)$ is the subgroup of inner automorphisms of $G$ .

Being inner forms of each other is an equivalence relation on the set of $K$ -forms of a given algebraic group.

A form which is not inner is called an outer form.

For example, the $\mathbb {R}$ -forms of $\mathrm {SL} _{3}(\mathbb {R} )$ are itself and the unitary groups $\mathrm {SU} (2,1)$ and $\mathrm {SU} (3)$ . The latter two are outer forms of $\mathrm {SL} _{3}(\mathbb {R} )$ , and they are inner forms of each other.