Affine and Lattice Structures in Regular ℕ-Graded Vertex Operator Algebras with Gorenstein V0
Abstract
We study regular ℕ-graded vertex operator algebras V=⨁n≥ 0Vn whose weight-zero algebra V0 is a nontrivial finite-dimensional local Gorenstein algebra, asking which structural features of strongly rational vertex operator algebras persist without the CFT-type condition V0=ℂbf 1. For semisimple Lie subalgebras of the left Leibniz algebra V1, assuming ker(.L(−1)|V0)=ℂ1, the product u1v induces an invariant symmetric bilinear form. Under C2-cofiniteness and a nondegeneracy condition, each simple component with nonzero form generates an affine vertex operator algebra at positive integral level and acts integrably on V. When V1 is solvable, the Frobenius structure of V0 yields a distinguished nondegenerate subspace M⊂ V1. Under a quasi-primary condition, M is abelian and generates a Heisenberg vertex operator algebra. With additional semisimplicity, full-rank integrality, and cocycle compatibility assumptions, V contains a conformally embedded lattice vertex operator algebra VK, where K is positive-definite and even, with rank dimℂM and minimum norm at least $4$. Conformally shifted lattice vertex operator algebras provide explicit models illustrating the distinction among weight-one Lie, mode-generated Lie, and lattice structures. They also show that regularity alone does not ensure semisimplicity of arbitrary Heisenberg zero-mode actions, so the additional lattice-theorem hypotheses represent genuine structural obstructions.
Related
- Locally Gorenstein algebras, a generalization of Gorenstein algebras
- Virtually Gorenstein Artin algebras are weakly Gorenstein
- Endomorphism algebras of Gorenstein-projective modules over string algebras
- Binomial Artin-Schelter Regular Algebras and Twisted Monoid Algebras
- Slice regular functions on alternative *-algebras : prescribing zeroes and values on discrete sets and related extension problems
BibTeX
@misc{yamskulna2026affine,
title = {{Affine and Lattice Structures in Regular $\mathbbN$-Graded Vertex Operator Algebras with Gorenstein $V_0$}},
author = {Gaywalee Yamskulna},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22780},
eprint = {2609.22780},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22780},
}
RIS
TY - UNPB
TI - Affine and Lattice Structures in Regular ℕ-Graded Vertex Operator Algebras with Gorenstein V_0
AU - Gaywalee Yamskulna
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22780
UR - https://arxiv.org/abs/2609.22780
AB - We study regular ℕ-graded vertex operator algebras V=⨁_n≥ 0V_n whose weight-zero algebra V_0 is a nontrivial finite-dimensional local Gorenstein algebra, asking which structural features of strongly rational vertex operator algebras persist without the CFT-type condition V_0=ℂbf 1. For semisimple Lie subalgebras of the left Leibniz algebra V_1, assuming ker(.L(−1)|_V_0)=ℂ1, the product u_1v induces an invariant symmetric bilinear form. Under C_2-cofiniteness and a nondegeneracy condition, each simple component with nonzero form generates an affine vertex operator algebra at positive integral level and acts integrably on V. When V_1 is solvable, the Frobenius structure of V_0 yields a distinguished nondegenerate subspace M⊂ V_1. Under a quasi-primary condition, M is abelian and generates a Heisenberg vertex operator algebra. With additional semisimplicity, full-rank integrality, and cocycle compatibility assumptions, V contains a conformally embedded lattice vertex operator algebra V_K, where K is positive-definite and even, with rank dim_ℂM and minimum norm at least $4$. Conformally shifted lattice vertex operator algebras provide explicit models illustrating the distinction among weight-one Lie, mode-generated Lie, and lattice structures. They also show that regularity alone does not ensure semisimplicity of arbitrary Heisenberg zero-mode actions, so the additional lattice-theorem hypotheses represent genuine structural obstructions.
ER -