Presentations of Feigin-Stoyanovsky's type subspaces of standard modules for affine Lie algebras of type $C_\ell^{(1)}$

TL;DR

This paper provides a presentation of Feigin-Stoyanovsky's type subspaces of standard modules for affine Lie algebras of type C, describing their structure via generators and relations based on basis descriptions.

Contribution

It introduces a new presentation of these subspaces in terms of generators and relations, expanding understanding of their algebraic structure.

Findings

Explicit basis description for the subspaces.

Presentation of subspaces as quotients of universal enveloping algebras.

Enhanced understanding of the algebraic structure of these subspaces.

Abstract

Feigin-Stoyanovsky's type subspace $W(\Lambda)$ of a standard $\tilde{{\mathfrak g}}$-module $L(\Lambda)$ is a $\tilde{{\mathfrak g}}_1$-submodule of $L(\Lambda)$ generated by the highest-weight vector $v_\Lambda$, where $\tilde{{\mathfrak g}}_1$ is a certain commutative subalgebra of $\tilde{{\mathfrak g}}$. Based on the description of basis of $W(\Lambda)$ for $\tilde{{\mathfrak g}}$ of type $C_\ell^{(1)}$, we give a presentation of this subspace in terms of generators and relations $$W(\Lambda)\simeq U(\tilde{{\mathfrak g}}_1^-)/J.$$