Innenform

Innenform, literally “inner form,” is a term used in German mathematics to denote a form of an algebraic object that becomes isomorphic to the original object over an algebraic closure through an inner automorphism rather than through an outer twist.

In the context of algebraic groups, an inner form of a connected reductive group G over a field F is another F-group G' such that, after base change to an algebraic closure, Ḡ and Ḡ' are isomorphic via a conjugation (an inner automorphism of Ḡ). Inner forms are obtained by twisting G with a cocycle taking values in the adjoint group Ad(G). They are distinguished from outer forms, which arise from twists by automorphisms outside the inner automorphism group.

The set of inner forms is classified by Galois cohomology. Concretely, inner forms of G correspond to elements of the pointed set H^1(F, Ad(G)) (or, sometimes, H^1(F, G/Z(G))). This framework explains how different but related group structures can share common algebraic properties after extension to a larger field.

A standard example is found with the projective general linear group. All inner forms of PGL_n over F are of the form PGL_1(D), where D is a central division algebra over F of degree n. When D is the matrix algebra Mn(F), one recovers the split form PGL_n(F).

The term Innenform is primarily used in this algebraic sense; in other disciplines, phrases like “inner form” may appear with broader or different meanings. See also outer form, Galois cohomology, and inner twists in algebraic groups.