Modules of a Heisenberg algebra

Heisenberg module

Let 𝑘 ∈𝕂. Given a ℤ-graded Heisenberg algebra 𝔩 over 𝕂 with centre 𝔩0 and a basis of homogenous elements satisfying the Heisenberg commutation relations

[𝑥𝑖,𝑧]=[𝑦𝑖,𝑧]=[𝑥𝑖,𝑥𝑗]=[𝑦𝑖,𝑦𝑗]=0[𝑥𝑖,𝑦𝑗]=𝛿𝑖𝑗𝑧

one may construct the Heisenberg module 𝑀(𝑘), a certain ℤ-graded irreducible 𝔩-module linearly isomorphic to the symmetric algebra 𝑆(𝔩−) ≅𝕂[𝑦𝑖]𝑖∈𝐼 of polynomials in indeterminate {𝑦𝑖}𝑖∈𝐼 so that for 𝑓 ∈𝕂[𝑦𝑖]𝑖∈𝐼 ≅𝑀(𝑘)

𝑧⊙𝑓=𝑘𝑓𝑦𝑖⊙𝑓=𝑦𝑖𝑓𝑥𝑖⊙𝑓=𝑘𝜕𝑓𝜕𝑦𝑖

which is the canonical realization of the Heisenberg commutation relations.¹ These modules form an important parameterized family of simple modules over 𝔩.

Construction

Let 𝔟+ =𝔩0 ⊕𝔩+ Let 𝕂𝑘 be the ℤ-graded 𝔟+-module defined by

𝑧⊙1=𝑘𝔩+⊙1=0deg⁡1=0

and let 𝑀(𝑘) be the induced module lie

𝑀(𝑘)=Ind𝔩𝔟+⁡𝕂𝑘=𝑈(𝔩)⊗𝑈(𝔟+)𝕂𝑘

The claimed linear isomorphism 𝑀(𝑘) ≅𝑆(𝔩−) follows from the Poincaré-Birkhoff-Witt theorem and the fact that 𝑈(𝔩−) =𝑆(𝔩−).

Examples

• Standard Heisenberg algebra for QM

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §1.6, p. 22 ↩