Lie algebras MOC

Associated Lie algebra of a positive definite even lattice

Let be a positive definite even rational lattice. Let for [[Characteristic|]], and set

which is alternating -bilinear. Then determines a unique 2 central extension of a free abelian group

such that ¹. Let , , and define

where is a free module, whence follows . Then is a quadratic Lie algebra under the bilinear bracket defined by

and the nonsingular bilinear form extending that of by

for and .² lie

With a nice choice of section

Let be a bilinear map such that (cf. associated elementary 2-group of an even lattice)

then

and by the Correspondence between 2-cocycles and central extensions we have a central extension of the above form with a -section such that

and in particular . Denote

We then have

and are free to define for , and the commutation relations become

for and the nonsingular bilinear form is given by

Proof of quadratic Lie algebra

It is clear that the bracket is alternating on . For some , we have so . Thus the bracket is alternating.

To prove that is a Lie algebra, it is sufficient to prove the ^Jacobi

for . Clearly if the identity holds. If and

If , , and

where in case ,

in case , and thus

anf case , we have and thus . Finally consider the case

where in case ,

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Footnotes

1. where we denote . ↩

2. 1988. Vertex operator algebras and the Monster, §6.2, 126 ↩