Certain associative algebras similar to $U(sl_{2})$ and Zhu's algebra   $A(V_{L})$
  It is proved that Zhu's algebra for vertex operator algebra associated to a positive-definite even lattice of rank one is a finite-dimensional semiprimitive quotient algebra of certain associative algebra introduced by Smith. Zhu's algebra for vertex operator algebra associated to any positive-definite even lattice is also calculated and is related to a generalization of Smith's algebra. 
Chongying Dong, Haisheng Li and Geoffrey Mason