Unimodular lattice
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1.
The E8 lattice and the Leech lattice are two famous examples.
Definitions
A lattice is a free abelian group of finite rank with a symmetric bilinear form (·,·).
The lattice is integral if (·,·) takes integer values.
The dimension of a lattice is the same as its rank (as a Z-module).
The norm of a lattice element a is (a, a).
A lattice is positive definite if the norm of all nonzero elements is positive.
The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (ai, aj), where the elements ai form a basis for the lattice.
An integral lattice is unimodular if its determinant is 1 or −1.
A unimodular lattice is even or type II if all norms are even, otherwise odd or type I.
The minimum of a positive definite lattice is the lowest nonzero norm.
