Editing Vertex operator algebra (section)

=== Heisenberg algebra modules ===

Modules of the Heisenberg algebra can be constructed as [[Fock space]]s <math>\pi_\lambda</math> for <math>\lambda \in \mathbb{C}</math> which are induced representations of the [[Heisenberg Lie algebra]], given by a vacuum vector <math>v_\lambda</math> satisfying <math>b_nv_\lambda = 0</math> for <math>n > 0</math>, <math>b_0v_\lambda = 0</math>, and being acted on freely by the negative modes <math>b_{-n}</math> for <math>n>0</math>.  The space can be written as <math>\mathbb{C}[b_{-1}, b_{-2}, \cdots]v_\lambda</math>. Every irreducible, <math>\mathbb{Z}</math>-graded Heisenberg algebra module with gradation bounded below is of this form.

These are used to construct lattice vertex algebras, which as vector spaces are direct sums of Heisenberg modules, when the image of <math>Y</math> is extended appropriately to module elements.

The module category is not semisimple, since one may induce a representation of the abelian Lie algebra where ''b''<sub>0</sub> acts by a nontrivial [[Jordan block]].  For the rank ''n'' free boson, one has an irreducible module ''V''<sub>λ</sub> for each vector λ in complex ''n''-dimensional space.  Each vector ''b'' ∈ '''C'''<sup>n</sup> yields the operator ''b''<sub>0</sub>, and the Fock space ''V''<sub>λ</sub> is distinguished by the property that each such ''b''<sub>0</sub> acts as scalar multiplication by the inner product (''b'', λ).