unimodulär

Unimodular is a mathematical term used to describe certain objects that preserve volume or have balanced measure properties. The term appears in several closely related contexts, including linear algebra over the integers, lattice theory, and harmonic analysis on locally compact groups.

Unimodular matrices are integer matrices with determinant ±1. Such matrices belong to the group GL(n, Z) and act as invertible changes of basis that preserve the Z^n lattice. Their inverses are also integer matrices, and they preserve volume in the sense that the absolute value of the determinant is 1. Unimodular transformations are fundamental in algorithms for integer linear algebra, lattice reduction, and the computation of Smith normal forms.

Unimodular lattices are lattices in Euclidean space with determinant (volume of a fundamental parallelepiped) equal to 1. For integral lattices, a common equivalent condition is that the lattice equals its dual lattice. Even unimodular lattices are unimodular lattices in which all vector norms are even; notable examples include the E8 lattice in eight dimensions and the Leech lattice in 24 dimensions. Unimodular lattices arise in number theory, coding theory, and string theory.

Unimodular groups are locally compact groups for which the left and right Haar measures coincide, equivalently the modular function is identically 1. Most familiar groups, such as abelian Lie groups, compact groups, and SL(n, R), are unimodular. This property simplifies harmonic analysis on the group and influences representations and integration.

In summary, unimodular describes a class of structures with volume-preserving or measure-symmetric properties, with distinct but related meanings in matrices, lattices, and groups.