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. Lattices are often embedded in a real vector space with a symmetric bilinear form. The lattice is positive definite, Lorentzian, and so on if its vector space is. The signature of a lattice is the signature of the form on the vector space.
Examples The three most important examples of unimodular lattices are:
The lattice Z (the set of all integers), in one dimension. The E8 lattice, an even 8-dimensional lattice. The Leech lattice, the 24-dimensional even unimodular lattice with no roots.
Properties An integral 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 IIm,n and Im,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.
Classification For indefinite lattices, the classification is easy to describe. Write Rm,n for the m + n dimensional vector space Rm+n with the inner product of (a1, ..., am+n) and (b1, ..., bm+n) given by
a 1 b 1 + ⋯ + a m b m − a m + 1 b m + 1 − ⋯ − a m + n b m + n . {\displaystyle a_{1}b_{1}+\cdots +a_{m}b_{m}-a_{m+1}b_{m+1}-\cdots -a_{m+n}b_{m+n}.\,}
In Rm,n there is one odd indefinite unimodular lattice up to isomorphism, denoted by
Im,n, which is given by all vectors (a1,...,am+n) in Rm,n with all the ai integers. There are no indefinite even unimodular lattices unless
m − n is divisible by 8, in which case there is a unique example up to isomorphism, denoted by
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