Examples of simple vertex operator algebras (VOAs) A vertex operator algebra $V$ is called simple if $V$ is a simple $V$-module. What are some examples of simple VOAs? Are there lots of examples or this is a very strong condition? Is there a classification? In particular I am interested if the following VOAs are simple or not and under what conditions:


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*The rank $d$ Heisenberg (free field) VOA;

*A lattice VOA of some non-degenerate lattice;

*Affine Lie algebra at level $k$ for a semisimple Lie algebra $\mathfrak{g}$;

*$\mathcal{W}_k(\mathfrak{g},f)$, $\mathcal{W}$ algebras associated to a semisimple Lie algebras at level $k$ and nilpotent element $f$. What if $\mathcal{W}_k(\mathfrak{g},f)$ is the principal $\mathcal{W}$-algebra?

 A: I expect there will never be a classification of simple VOAs, unless perhaps one is only sorting according to very rough criteria.  This is because there are too many of them - even the rational case is wide open.  For your examples, we have the following:


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*The irreducible modules of the rank $d$ free boson are naturally parametrized by points in $d$-dimensional space.  The VOA corresponds to the zero vector, hence is simple.

*The irreducible modules of a lattice VOA are naturally parametrized by cosets of the positive definite even lattice in its dual lattice.  The lattice VOA corresponds to the zero coset, hence is simple.

*You need to specify a vertex operator algebra here.  We often see $V^k(\mathfrak{g})$ used for the vacuum module, and $V_k(\mathfrak{g})$ for its unique simple quotient.  They differ when $k$ is a positive integer (and perhaps other cases that I don't recall right now).

*The W-algebra $W^k(\mathfrak{g},f)$ is given by the functor $H^0_f$ applied to $V^k(\mathfrak{g})$, and ideals are taken to ideals.  In particular, the quotient by $H^0_f$ applied to the unique maximal submodule is the unique simple quotient $W_k(\mathfrak{g},f)$.
