Relation Subspaces in Vertex Operator Algebras: Residue Generators for O_n^\circ(V) and Intersections with (L(−1) +L(0))V

Authors

• Junghyun Kim, University of Nebraska at KearneyFollow

Abstract

We study the relation subspace 𝑂◦𝑛 (𝑉) that appears in the definition o f the level-𝑛 Zhu algebra 𝐴𝑛 (𝑉) = 𝑉/𝑂𝑛 (𝑉), where 𝑂𝑛 (𝑉) = 𝑂𝐿 (𝑉) + 𝑂◦𝑛 (𝑉) and 𝑂𝐿 (𝑉) =(𝐿(−1) + 𝐿(0))𝑉. Using residue calculus, we introduce operators 𝑅𝑛,𝑘 that encode the circle products 𝑢◦𝑛 𝑣 and prove explicit change-of-generators formulas between the standard generators (𝑢−𝑚1)◦𝑛 𝑣 and the residue generators 𝑢 𝑅𝑛,0𝑣, together with a binomial inversion. These identities provide a practical framework for computing 𝑂◦𝑛 (𝑉), especially in strongly generated VOAs. As progress toward understanding the overlap 𝑂◦𝑛 (𝑉) ∩ 𝑂𝐿 (𝑉), we show that the 𝑘 = 0 level collapses modulo the 𝑘 ≥ 1 subspace and we derive the induced action of 𝐿(−1) + 𝐿(0) on the resulting circle-layer quotient. This leads to a conjecture describing 𝑂◦𝑛 (𝑢, 𝑉) ∩ 𝑂𝐿 (𝑉) as (𝐿(−1) + 𝐿(0)) 𝑂◦𝑛 (𝑢, 𝑉).

Recommended Citation

Kim, Junghyun (2026) "Relation Subspaces in Vertex Operator Algebras: Residue Generators for O_n^\circ(V) and Intersections with (L(−1) +L(0))V," Undergraduate Research Journal: Vol. 27, Article 3.
Available at: https://openspaces.unk.edu/undergraduate-research-journal/vol27/iss1/3