Geometry MOC

Rational lattice

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

where for any field with [[Characteristic|]] we identify ,¹ which is made a quadratic space under the extension of . The following notation is also useful for subsets of a given quadrance

Further terminology

• Given a basis of , the Gram matrix is given by .
• is nondegenerate iff is ^nondegenerate iff .
• is integral iff for all iff is integral.
• is even iff for all , which implies integral by polarization.
• is positive definite iff for all nonzero .
• is unimodular iff .
• 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 ↩