Zbilinear

Zbilinear describes a class of bilinear maps in algebra that are defined over the ring of integers. Specifically, if M and N are Z-modules (that is, abelian groups) a function B: M × N → Z is Z-bilinear when it is linear in each argument with respect to the integers. This means B(x + x', y) = B(x, y) + B(x', y), B(x, y + y') = B(x, y) + B(x, y'), and B(kx, y) = k B(x, y), B(x, ky) = k B(x, y) for all x, x' in M, y, y' in N and all integers k.

When M and N are free abelian groups of finite rank, a Z-bilinear map corresponds to an integer matrix A such that B(x, y) = x^T A y for x in Z^m and y in Z^n. If A is symmetric, the form is symmetric; if A is alternating, B(x, x) = 0 for all x. A bilinear form that induces an injective map from M to Hom(N, Z) is called nondegenerate; if the induced map is an isomorphism, the form is unimodular and related to lattices with determinant ±1.

Zbilinear forms play a central role in number theory and geometry, where they are used to study lattices, dual lattices, and discriminants. They give rise to associated quadratic forms Q(x) = B(x, x) (often with a factor of 1/2 to match conventions). The study of Zbilinear forms connects to topics such as classification of bilinear and quadratic forms, automorphisms preserving the form, and applications in topology and representation theory. See also bilinear form, quadratic form, lattice, and dual lattice.