Triangular Lie algebra

Contravariant form on a triangular module

Let be a triangular Lie algebra over with an involutive antiautomorphism so that

and let be an -invariant¹ linear form. Then the corresponding Triangular module has a ^symmetric unique contravariant form, a bilinear form satisfying lie

1. for all and

Proof

Note that extends to an involutive antiautomorphism of the universal enveloping algebra so that

for .

First we prove uniqueness of . Clearly ^F1 extends to

for and .

Note by the Poincaré-Birkhoff-Witt theorem

so we may define the projection operator

Given any by irreducibility

for some so

Since is a vacuum vector and

it follows

Therefore the behaviour of is completely determined by properties ^F1 and ^F2, so if exists it is unique.

To prove existence, consider the annihilator of

which is a left-ideal . Thus

is a -module isomorphism. Now

for and , whence

Thus the above formula

for is well-defined, since

for . Therewithal since

it follows

for , so is ^symmetric.

See also the special case of a Hermitian contravariant form on a complex triangular module.

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Footnotes

1. i.e. for any . ↩