Zhu algebra of a CFT-type vertex operator algebra

Let $V={⨁}_{r\ge 0}{V}_r$ be a conformal vertex algebra of CFT type over $\mathbb{C}$, with state–field convention $Y(a,z)={\sum }_{m\in \mathbb{Z}}{a}_{(m)}{z}^{-m-1}$. For homogeneous $a,b\in V$ of weight $r=\mathrm{w}\mathrm{t}(a)$, define $a\circ b={\mathrm{Res}}_z\frac{(1+z{)}^r}{z^2}Y(a,z)b,O(V)={\mathrm{span}}_{\mathbb{C}}\{a\circ b\},a\ast b={\mathrm{Res}}_z\frac{(1+z{)}^r}{z}Y(a,z)b\mathrm{.}$ Here $(1+z{)}^r={\sum }_{j=0}^r\left(\genfrac{}{}{0px}{}{r}{j}\right)z^j$ is the finite polynomial expansion and residue means coefficient of $z^{-1}$. Extend both operations to bilinear maps $V\times V\to V$: for the unique finite homogeneous decompositions $a={\sum }_r{a}_r$ and $b={\sum }_s{b}_s$, set $a\circ b:={\sum }_{r,s}{a}_r\circ b_s$ and $a\ast b:={\sum }_{r,s}{a}_r\ast b_s$, with empty sums equal to zero. Thus $O(V)$ is equivalently the span of the circle products of arbitrary vectors. Write $A(V)=V\mathrm{/}O(V)$ for the vector-space quotient and call $a\ast b$ the candidate product on representatives. The assertion that this product descends to an associative unital algebra is a separate theorem; this definition alone does not assert it.

Zhu quotient algebra and admissible top-level action

This is the algebraic construction as a vector-space quotient with a candidate product. Associativity, the unit, and the admissible top-level action are not built into the definition.