Kvadriline

Kvadriline is a term used in some mathematical literatures to denote a quadrilinear object: a map that is linear in each of four vector arguments. Formally, given vector spaces V1, V2, V3, and V4 over a field F, a kvadriline form is a function f: V1 × V2 × V3 × V4 → F that is linear in each argument separately. If all Vi are equal to a common space V, one speaks of a 4-linear form on V.

The components of a kvadriline form relative to chosen bases give a 4th-order tensor T with components Tijkl. The space of kvadriline forms is isomorphic to the tensor product V1* ⊗ V2* ⊗ V3* ⊗ V4*. Contraction of indices yields lower-order forms, and imposing symmetry or antisymmetry selects subspaces of forms with particular invariance properties.

Symmetry types include fully symmetric forms, which satisfy f(vσ(1), vσ(2), vσ(3), vσ(4)) = f(v1, v2, v3, v4) for all permutations σ, and alternating (antisymmetric) forms, where f changes sign according to the permutation. The Levi-Civita symbol εijkl on R4 is a classic example of a fully antisymmetric 4-linear form.

Relations and applications: kvadriline forms arise as higher-order tensors in differential geometry, continuum mechanics, and theoretical physics. They generalize bilinear and trilinear forms and underpin multilinear polarization identities and tensor representation theory.

Terminology: kvadriline is less common in English-language texts, where quadrilinear form or 4-linear form is standard. Some languages or older writings preserve kvadriline as the usual name for the concept.