Certain Associative Algebras Similar toU(sl2) and Zhuʹs AlgebraA(VL) Original Research Article

It is proved that Zhuʹs algebra for the vertex operator algebra associated to a positive-definite even lattice of rank one is a finite-dimensional semiprimitive quotient algebra of a certain associative algebra introduced by Smith. Zhuʹs algebra for the vertex operator algebra associated to any positive-definite even lattice is also calculated and is related to a generalization of Smithʹs algebra.