Triangular Lie algebra

Contravariant form on a triangular module

Let 𝔀 =π”«βˆ’ βŠ•π”₯ βŠ•π‘›+ be a triangular Lie algebra over 𝕂 with an involutive antiautomorphism πœ” :𝔀𝐨𝐩 →𝔀 so that

πœ”([π‘₯,𝑦])=[πœ”(𝑦),πœ”(π‘₯)]πœ”π”₯=π”₯πœ”π”«Β±=π”«Β±πœ”2=1

and let πœ† :π”₯ →𝕂 be an πœ”-invariant1 linear form. Then the corresponding Triangular module 𝑀(πœ†) has a symmetric unique contravariant form, a bilinear form 𝑏 :𝑀(πœ†) ×𝑀(πœ†) →𝕂 satisfying #m/thm/lie

  1. 𝑏(π‘₯ βŠ™π‘£,𝑀) =𝑏(𝑣,πœ”(π‘₯) βŠ™π‘€) for all π‘₯ βˆˆπ”€ and 𝑣,𝑀 βˆˆπ‘€(πœ†)
  2. 𝑏(π‘£πœ†,π‘£πœ†) =1
Proof

Note that πœ” extends to an involutive antiautomorphism of the universal enveloping algebra πœ” :π‘ˆ(𝔀)𝐨𝐩 β†’π‘ˆ(𝔀) so that

πœ”(π‘₯𝑦)=πœ”(𝑦)πœ”(π‘₯)

for π‘₯,𝑦 βˆˆπ‘ˆ(𝔀).

First we prove uniqueness of 𝑏. Clearly ^F1 extends to

𝑏(π‘₯βŠ™π‘£,𝑀)=𝑏(𝑣,πœ”(π‘₯)𝑀)

for π‘₯ βˆˆπ‘ˆ(𝔀) and 𝑣,𝑀 βˆˆπ‘€(πœ†).

Note by the PoincarΓ©-Birkhoff-Witt theorem

π‘ˆ(𝔀)=π–΅π–Ύπ–Όπ—π•‚π‘ˆ(π”«βˆ’)βŠ—π‘ˆ(π”₯)βŠ—π‘ˆ(𝔫+)=𝖡𝖾𝖼𝗍𝕂(π•‚βŠ•π”«βˆ’π‘ˆ(π”«βˆ’))βŠ—π‘ˆ(π”₯)βŠ—(π•‚βŠ•π‘ˆ(𝔫+)𝔫+)=π–΅π–Ύπ–Όπ—π•‚π”«βˆ’π‘ˆ(π”«βˆ’)βŠ—π‘ˆ(π”₯)βŠ—π‘ˆ(𝔫+)𝔫+βŠ•π”«βˆ’π‘ˆ(π”«βˆ’)βŠ—π‘ˆ(π”₯)βŠ•π‘ˆ(π”₯)βŠ—π‘ˆ(𝔫+)𝔫+βŠ•π‘ˆ(π”₯)=𝖡𝖾𝖼𝗍𝕂(π”«βˆ’π‘ˆ(π”«βˆ’)π‘ˆ(π”₯)π‘ˆ(𝔫+)+π”«βˆ’π‘ˆ(π”«βˆ’)π‘ˆ(π”₯)+π‘ˆ(π”₯)π‘ˆ(𝔫+)𝔫+)βŠ•π‘ˆ(π”₯)=𝖡𝖾𝖼𝗍𝕂(π”«βˆ’π‘ˆ(𝔀)+π‘ˆ(𝔀)𝔫+)βŠ•π‘ˆ(π”₯)

so we may define the projection operator

𝑃:π‘ˆ(𝔀)β†’π‘ˆ(π”₯)=π–΄π– π—Œπ– π—…π—€π•‚π‘†βˆ™(π”₯)

Given any 𝑣,𝑀 βˆˆπ‘€(πœ†) by irreducibility

𝑣=π‘₯βŠ™π‘£πœ†π‘€=π‘¦βŠ™π‘£πœ†

for some π‘₯,𝑦 βˆˆπ‘ˆ(𝔀) so

𝑏(𝑣,𝑀)=𝑏(π‘₯βŠ™π‘£πœ†,π‘¦βŠ™π‘£πœ†)=𝑏(π‘£πœ†,πœ”(π‘₯)π‘¦βŠ™π‘£πœ†)

Since π‘£πœ† is a vacuum vector and

𝑏(π‘£πœ†,π”«βˆ’π‘ˆ(𝔀)βŠ™π‘£πœ†)=𝑏(𝔫+βŠ™π‘£πœ†,π‘ˆ(𝔀)βŠ™π‘£πœ†)=0

it follows

𝑏(𝑣,𝑀)=𝑏(π‘£πœ†,πœ”(π‘₯)π‘¦βŠ™π‘£πœ†)=𝑏(π‘£πœ†,𝑃(πœ”(π‘₯)𝑦)βŠ™π‘£πœ†)=𝑏(π‘£πœ†,πœ†(𝑃(πœ”(π‘₯)𝑦))π‘£πœ†)=πœ†(𝑃(πœ”(π‘₯)𝑦))

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

To prove existence, consider the annihilator of π‘£πœ†

𝔍=π‘ˆ(𝔀)(𝔫++βˆ‘β„Žβˆˆπ”₯𝕂(β„Žβˆ’πœ†(β„Ž)1))

which is a left-ideal π‘ˆ(𝔀). Thus

πœ‘:π‘ˆ(𝔀)/𝔍→𝑀(πœ†)π‘₯+ℑ↦π‘₯βŠ™π‘£πœ†

is a 𝔀-module isomorphism. Now

𝑃(π‘₯𝑦)=𝑃(π‘₯)𝑦𝑃(𝑦π‘₯)=𝑦𝑃(π‘₯)

for π‘₯ βˆˆπ‘ˆ(𝔀) and 𝑦 βˆˆπ‘ˆ(π”₯), whence

πœ†(𝑃(𝔍))=πœ†(𝑃(πœ”(𝔍)))=0

Thus the above formula

𝑏(π‘₯βŠ™π‘£πœ†,π‘¦βŠ™π‘£πœ†)=πœ†(𝑃(πœ”(π‘₯)𝑦))

for π‘₯,𝑦 βˆˆπ‘ˆ(𝔀) is well-defined, since

πœ”(π‘₯𝑦)𝑧=πœ”(𝑦)πœ”(π‘₯)π‘§πœ†(𝑃(πœ”(1)1))=1

for π‘₯,𝑦,𝑧 βˆˆπ‘ˆ(𝔀). Therewithal since

π‘ƒβˆ˜πœ”=πœ”βˆ˜π‘ƒ

it follows

πœ†(𝑃(πœ”(𝑦)π‘₯))=πœ†(𝑃(πœ”(πœ”(π‘₯)𝑦)))=πœ†(πœ”(𝑃(πœ”(π‘₯)𝑦)))=πœ†(𝑃(πœ”(π‘₯)𝑦))

for π‘₯,𝑦 βˆˆπ‘ˆ(𝔀), so 𝑏 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. πœ†πœ”β„Ž =πœ†β„Ž for any β„Ž ∈π”₯. ↩