Abstract
The level one Zhu algebra for the Heisenberg vertex operator algebra is calculated, and implications for the use of Zhu algebras of higher level for vertex operator algebras are discussed. In particular, we show the Heisenberg vertex operator algebra gives an example of when the level one Zhu algebra, and in fact all its higher level Zhu algebras, do not provide new indecomposable non simple modules for the vertex operator algebra beyond those detected by the level zero Zhu algebra.
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