Quasi-lisse vertex algebras were introduced by Arakawa and Kawasetsu in Quasi-lisse vertex algebras and modular linear differential equations . They satisfy the property that the normalized character of an ordinary representtion of a quasi-lisse vertex operator algebra has a modular invariance property, in the sense that it satisfies a modular linear differential equation. I am wondering how much is known about the classification of these algebras?

1. Is a complete classification possible?
2. What are some examples of quasi-lisse vertex algebras apart from the one's discussed in the paper (affine vertex algebras associated with the Deligne exceptional series at non-permissible levels) and traditional ones (such as affine VOAs at permissible levels and their W-algebras through work of Kac and Wakimoto)?
3. A special case if the classification of rational and $C_2$-cofinite vertex algebras? Can these be classified?
4. What are some examples where the quasi-modular forms of the characters have been explicitly calculated?

My interest is mainly in the modular invariance of vertex algebras and I am looking for lots of different examples where such explicit computations of characters have been made or could be made.

Quasi-lisse vertex algebras were introduced by Arakawa and Kawasetsu in Quasi-lisse vertex algebras and modular linear differential equations . They satisfy the property that the normalized character of an ordinary representation of a quasi-lisse vertex operator algebra has a modular invariance property, in the sense that it satisfies a modular linear differential equation. I am wondering how much is known about the classification of these algebras?

1. Is a complete classification possible?
2. What are some examples of quasi-lisse vertex algebras apart from the one's discussed in the paper (affine vertex algebras associated with the Deligne exceptional series at non-admissible levels) and traditional ones (such as affine VOAs at admissible levels and their W-algebras through work of Kac and Wakimoto)?
3. A special case is the classification of rational and $C_2$-cofinite vertex algebras? Can these be classified?
4. What are some examples where the quasi-modular forms of the characters have been explicitly calculated?

My interest is mainly in the modular invariance of vertex algebras and I am looking for lots of different examples where such explicit computations of characters have been made or could be made.