Unimodular

Unimodular is a term used in several areas of mathematics to describe objects that preserve a canonical notion of size or lattice structure under integral transformations. The most common usage applies to square matrices with integer entries: a matrix A is unimodular if det(A) = ±1. Such matrices are invertible with integer inverses and define automorphisms of the integer lattice Z^n. They preserve volume up to a sign and can be decomposed into elementary row- and column- operations with integer coefficients. The subgroup with determinant +1, det(A) = 1, is SL(n, Z); together with det = −1 they form GL(n, Z).

A related notion concerns lattices: a lattice L in R^n is unimodular if it is integral and its dual lattice L* equals L, which is equivalent to having determinant 1 (the covolume of L is 1). Unimodular lattices are classified by properties such as evenness. Notable examples include the E8 lattice in dimension 8 and the Leech lattice in dimension 24, both even unimodular. Odd unimodular lattices exist in other dimensions.

In harmonic analysis, a locally compact group is unimodular if its left and right Haar measures coincide, ensuring that integration is invariant under both left and right translations. The term is also used in algebraic geometry in related contexts involving volume forms and dualities, though the core idea is invariance under a canonical measure or lattice structure.