Geometry MOC

Rational lattice

A rational lattice 𝐿 of rank 𝑛 is the ℤ-span of a basis of an 𝑛-dimensional quadratic space 𝐿ℚ over Rational numbers. geo Equivalently, a rational lattice 𝐿 is a rank 𝑛 ℤ-module with a symmetric ℤ-bilinear map

⟨⋅,⋅⟩:𝐿×𝐿→ℚ

where for any field 𝐾 with [[Characteristic|char⁡𝐾 =0]] we identify 𝐿𝐾 =𝐾 ⊗ℤ𝐿,¹ which is made a quadratic space under the extension of ⟨ ⋅, ⋅⟩. The following notation is also useful for subsets of a given quadrance

𝐿𝑚={𝛼∈𝐿:⟨𝛼,𝛼⟩=𝑚}

Further terminology

• Given a basis {𝛼𝑖}𝑛𝑖=1 of 𝐿, the Gram matrix is given by 𝐺𝑖𝑗 =⟨𝛼𝑖,𝛼𝑗⟩.
• 𝐿 is nondegenerate iff 𝐿ℚ is ^nondegenerate iff det𝐺 ≠0.
• 𝐿 is integral iff ⟨𝛼,𝛽⟩ ∈ℤ for all 𝛼 ∈𝐿 iff 𝐺 is integral.
• 𝐿 is even iff ⟨𝛼,𝛼⟩ ∈2ℤ for all 𝛼 ∈𝐿, which implies integral by polarization.
• 𝐿 is positive definite iff ⟨𝛼,𝛼⟩ >0 for all nonzero 𝛼 ∈𝐿.
• 𝐿 is unimodular iff |det𝐺| =1.
• Dual of a rational lattice
• Self-dual rational lattice
• Theta function of a positive definite lattice

Properties

• A discrete Z-submodule of a quadratic space is a rational lattice

See also

• Lattice from a binary linear code
• Lattice subgroup
• Associated Lie algebra of a positive definite even lattice

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §6.1, pp. 122–123 ↩