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.

The three most important examples of unimodular lattices are:

A lattice is unimodular if and only if its dual lattice is integral. Unimodular lattices are equal to their dual lattices, and for this reason, unimodular lattices are also known as self-dual.

Given a pair (m,n) of nonnegative integers, an even unimodular lattice of signature (m,n) exists if and only if m-n is divisible by 8, but an odd unimodular lattice of signature (m,n) always exists. In particular, even unimodular definite lattices only exist in dimension divisible by 8. Examples in all admissible signatures are given by the II_m,n and I_m,n constructions, respectively.

The theta function of a unimodular positive definite lattice is a modular form whose weight is one half the rank. If the lattice is even, the form has level 1, and if the lattice is odd the form has Γ₀(4) structure (i.e., it is a modular form of level 4). Due to the dimension bound on spaces of modular forms, the minimum norm of a nonzero vector of an even unimodular lattice is no greater than ⎣n/24⎦ + 1. An even unimodular lattice that achieves this bound is called extremal. Extremal even unimodular lattices are known in relevant dimensions up to 80, and their non-existence has been proven for dimensions above 163,264.

For indefinite lattices, the classification is easy to describe. Write R^m,n for the m + n dimensional vector space R^m+n with the inner product of (a₁, ..., a_m+n) and (b₁, ..., b_m+n) given by

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