II25,1

In mathematics, II25,1 is the even 26-dimensional Lorentzian unimodular lattice. It has several unusual properties, arising from Conway's discovery that it has a norm zero Weyl vector. In particular it is closely related to the Leech lattice Λ, and has the Conway group Co1 at the top of its automorphism group.
Construction.
Write "R"m,n for the "m+n" dimensional vector space
"R"m+n with the inner product of 
("a"1...,"a"m+n) and ("b"1...,"b"m+n) given by
The lattice "II25,1" is given by all vectors ("a"1...,"a"26)
in "R"25,1 such that either all the "ai" are integers or they are all integers
plus 1/2, and their sum is even.
Reflection group.
The lattice II25,1 is isomorphic to Λ⊕H where:
and the two summands are orthogonal. So we can write vectors of II25,1 as (λ,"m", "n") = λ+"mz"+"nw" with λ in Λ and "m","n" integers, where (λ,"m", "n") has norm λ2 –2"mn". To give explicitly the isomorphism, let formula_1, and formula_2, so that the subspace formula_3 generated by formula_4 and formula_5 is the 2-dimensional even Lorentzian lattice. Then formula_6 is isomorphic to formula_7 and we recover one of the definitions of Λ.
Conway showed that the roots (norm 2 vectors) having inner product –1 with "w"=(0,0,1) are the simple roots of the reflection group. These are the vectors (λ,1,λ2/2–1) for λ in the Leech lattice. In other words, the simple roots can be identified with the points of the Leech lattice, and moreover this is an isometry from the set of simple roots to the Leech lattice.
The reflection group is a hyperbolic reflection group acting on 25-dimensional hyperbolic space.
The fundamental domain of the reflection group has 1+23+284 orbits of vertices as follows:
Automorphism group.
 described the automorphism group Aut(II25,1) of II25,1 as follows. 
Vectors.
Every non-zero vector of II25,1 can be written uniquely as a positive integer multiple of a primitive vector, so to classify all vectors it is sufficient to classify the primitive vectors.
Positive norm vectors.
Any two positive norm primitive vectors with the same norm are conjugate under the automorphism group.
Norm zero vectors.
There are 24 orbits of primitive norm 0 vectors, corresponding to the 24 Niemeier lattices. The correspondence is given as follows: if "z" is a norm 0 vector, then the lattice "z"⊥/"z" is a 24-dimensional even unimodular lattice and is therefore one of the Niemeier lattices.
The Niemeier lattice corresponding to the norm 0 Weyl vector of the reflection group of II25,1 is the Leech lattice.
Norm –2 vectors.
There are 121 orbits of vectors "v" of norm –2, corresponding to the 121 isomorphism classes of 25-dimensional even lattices "L" of determinant 2. In this correspondence, the lattice "L" is isomorphic to the orthogonal complement of the vector "v".
Norm –4 vectors.
There are 665 orbits of vectors "v" of norm –4, corresponding to the 665 isomorphism classes of 25-dimensional unimodular lattices "L". In this correspondence, the index 2 sublattice of the even vectors of the lattice "L" is isomorphic to the orthogonal complement of the vector "v".
Other vectors.
There are similar but increasingly complicated descriptions of the vectors of norm –2"n" for "n"=3, 4, 5, ..., and the number of orbits of such vectors increases quite rapidly.