In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24, which were classified by Niemeier. The best known example is the Leech lattice. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... Generally, rank is a system of hierarchy used to classify like things. ... In mathematics, the Leech lattice is a lattice Λ in R24 discovered John Leech ( 16 (1964), 657--682). ...

Classification

Niemeier lattices are usually labeled by the Dynkin diagram of their root systems. These Dynkin diagrams have rank either 0 or 24, and all of their components have the same Coxeter number. (The Coxeter number, at least in these cases, is the number of roots divided by the dimension.) There are exactly 24 Dynkin diagrams with these properties, and there turns out to be a unique Niemeier lattice for each of these Dynkin diagrams. See also Simple Lie group. ... See also Simple Lie group. ...

The complete list of Niemeier lattices is

• The Leech lattice (empty root system), Coxeter number 0.
• A₁²⁴, Coxeter number 2.
• A₂¹², Coxeter number 3.
• A₃⁸, Coxeter number 4.
• A₄⁶, Coxeter number 5.
• A₅⁴D₄, Coxeter number 6.
• D₄⁶, Coxeter number 6.
• A₆⁴, Coxeter number 7.
• A₇²D₅², Coxeter number 8.
• A₈³, Coxeter number 9.
• A₉²D₆, Coxeter number 10.
• D₆⁴, Coxeter number 10.
• E₆⁴, Coxeter number 12.
• A₁₁D₇E₆, Coxeter number 12.
• A₁₂², Coxeter number 13.
• D₈³, Coxeter number 14.
• A₁₅D₉, Coxeter number 16.
• A₁₇E₇, Coxeter number 18.
• D₁₀E₇², Coxeter number 18.
• D₁₂², Coxeter number 22.
• A₂₄, Coxeter number 25.
• D₁₆E₈, Coxeter number 30.
• E₈³, Coxeter number 30.
• D₂₄, Coxeter number 46.

Properties

Some of the Niemeier lattices are related to sporadic simple groups. The Leech lattice is acted on by a double cover of the Conway group, and the lattices A₁²⁴ and A₂¹² are acted on by the Mathieu groups M₂₄ and M₁₂. The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway. ... In mathematics, the Mathieu groups are five finite simple groups discovered by the French mathematician Emile Léonard Mathieu. ...

The Niemeier lattices, other than the Leech lattice, correspond to the deep holes of the Leech lattice. This implies that the affine Dynkin diagrams of the Niemeier lattices can be seen inside the Leech lattice, when two points of the Leech lattice are joined by no lines when they have distance , by 1 line if they have distance , and by a double line if they have distance .

Further reading

• Conway and Sloane, Sphere packings, lattices, and groups, ISBN 0387985859
• Sloane's lattice catalogue