On generators and relations for $O_n^\circ(V)$ in higher-level Zhu algebras, Part I

Abstract

We study the circle-relation subspace $O_n^\circ(V)$ in the defining relations of the level-$n$ Zhu algebra of a vertex operator algebra $V$. We assemble the residue relations generating $O_n^\circ(V)$ into generating functions and obtain an explicit recursion that replaces every sufficiently deep mode by retained modes with indices at least $-2n-1$, modulo $O_n^\circ(V)$. This recursion is encoded in a left truncated field $Y_L$. Since such mode replacements are valid only in the leftmost position of an operator product, we prove the replacement, derivative, commutator, and one-step iterate identities for $Y_L$ needed for finite-length calculations. For a finite homogeneous strong generating set $S$, we define the finite-length circle subspaces $O_n^\circ(N,S,V)$ and their plus subspaces. We formulate a finite-length conjecture under an explicit auxiliary-degree hypothesis and a separate termwise conjecture for the correction residues of one-sided iterates. We prove reductions to the $k=0$ strip and to last vacuum mode $-1$, recover the length-one relation spaces, and give explicit Virasoro computations at word length two for $n=1$ and $n=2$. The $Y_L$ identities are the input for the companion paper (Part II).