  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.
