Modules of a Heisenberg algebra

Heisenberg module

Let . Given a -graded Heisenberg algebra over with centre and a basis of homogenous elements satisfying the Heisenberg commutation relations

one may construct the Heisenberg module , a certain -graded irreducible -module linearly isomorphic to the symmetric algebra of polynomials in indeterminate so that for

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

Construction

Let Let be the -graded -module defined by

and let be the induced module lie

The claimed linear isomorphism follows from the Poincaré-Birkhoff-Witt theorem and the fact that .

Examples

• Standard Heisenberg algebra for QM

---

tidy | en | sembr

Footnotes

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