In this paper, we explore natural connections among trigonometric Lie algebras, (general) affine Lie algebras, and vertex algebras. Among the main results, we obtain a realization of trigonometric Lie algebras as what were called the covariant algebras of the affine Lie algebra $\widehat{\mathcal{A}}$ of Lie algebra $\mathcal{A}=\frak{gl}_{\infty}\oplus\frak{gl}_{\infty}$ with respect to certain automorphism groups. We then prove that restricted modules of level $\ell$ for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebras $V_{\widehat{\mathcal{A}}}(\ell,0)$ (or $V_{\widehat{\mathcal{A}}}(2\ell,0)$). Furthermore, we determine irreducible modules and equivariant quasi modules for simple vertex algebra $L_{\widehat{\mathcal{A}}}(\ell,0)$ with $\ell$ a positive integer. In particular, we prove that every quasi-finite unitary highest weight (irreducible) module of level $\ell$ for type $A$ trigonometric Lie algebra gives rise to an irreducible equivariant quasi $L_{\widehat{\mathcal{A}}}(\ell,0)$-module.

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三角李代数、仿射李代数和顶点代数

在本文中，我们探索了三角李代数、（一般）仿射李代数和顶点代数之间的自然联系。在主要结果中，我们获得了三角李代数的实现，即李代数的仿射李代数 $\widehat{\mathcal{A}}$ 的协变代数 $\mathcal{A}=\frak{gl }_{\infty}\oplus\frak{gl}_{\infty}$ 关于某些自同构群。然后，我们证明三角李代数的 $\ell$ 级受限模自然对应于仿射顶点代数 $V_{\widehat{\mathcal{A}}}(\ell,0)$（或 $ V_{\widehat{\mathcal{A}}}(2\ell,0)$)。此外，我们确定了简单顶点代数 $L_{\widehat{\mathcal{A}}}(\ell,0)$ 的不可约模和等变准模，其中 $\ell$ 是一个正整数。特别是，