Rational lattice

Even lattice

A rational lattice is said to be even iff for all . geo It immediately follows from polarization that is ^integral.

Properties

• Associated Lie algebra of a positive definite even lattice

Associated elementary 2-group

An even lattice of rank has an associated elementary abelian 2-group of dimension , where we write . We have a canonical alternating -bilinear map

which induces the alternating -bilinear form

Similarly we have a canonical map

which induces the quadratic form

so that is the polar form of . Now or equivalently is ^nondegenerate iff the Gram matrix has odd determinant, in particular if is a unimodular lattice.

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