Combinatorial relations among relations for level 2 standard $C_{n}\sp{(1)}$-modules

TL;DR

This paper explores the combinatorial structure of relations among vertex operator coefficients that annihilate level 2 standard modules for affine Lie algebras of type C, aiming to aid in constructing a Groebner basis for the associated vertex algebra.

Contribution

It introduces a combinatorial framework for relations among relations in level 2 modules of affine Lie algebras of type C, advancing the understanding of their algebraic structure.

Findings

Constructed combinatorially parameterized relations among annihilating field coefficients.

Provided insights into the structure of relations among relations for level 2 modules.

Suggested potential applications in building Groebner-like bases for vertex operator algebras.

Abstract

For an affine Lie algebra $\hat{\mathfrak g}$ the coefficients of certain vertex operators which annihilate level $k$ standard $\hat{\mathfrak g}$-modules are the defining relations for level $k$ standard modules. In this paper we study a combinatorial structure of the leading terms of these relations for level $k=2$ standard $\hat{\mathfrak g}$-modules for affine Lie algebras of type $C_{n}\sp{(1)}$ and the main result is a construction of combinatorially parameterized relations among the coefficients of annihilating fields. It is believed that the constructed relations among relations will play a key role in a construction of Groebner-like basis of the maximal ideal of the universal vertex operator algebra $V^ k_{\mathfrak g}$ for $k=2$.