Associated Lie algebra of a positive definite even lattice

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 #m/def/lie

With a nice choice of section

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 identity

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 ,

#state/develop | #lang/en | #SemBr

Footnotes