On the structure of simple bounded weight modules of $\mathfrak{s}\mathfrak{l}(\mathrm{\infty })$, $\mathfrak{o}(\mathrm{\infty })$, $\mathfrak{s}\mathfrak{p}(\mathrm{\infty })$

We study the structure of bounded simple weight $\mathfrak{s}\mathfrak{l}(\mathrm{\infty })$-, $\mathfrak{o}(\mathrm{\infty })$-, $\mathfrak{s}\mathfrak{p}(\mathrm{\infty })$-modules, which have been recently classified in [6]. Given a splitting parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{s}\mathfrak{l}(\mathrm{\infty })$, $\mathfrak{o}(\mathrm{\infty })$, $\mathfrak{s}\mathfrak{p}(\mathrm{\infty })$, we introduce the concepts of $\mathfrak{p}$-aligned and pseudo $\mathfrak{p}$-aligned $\mathfrak{s}\mathfrak{l}(\mathrm{\infty })$-, $\mathfrak{o}(\mathrm{\infty })$-, $\mathfrak{s}\mathfrak{p}(\mathrm{\infty })$-modules, and give necessary and sufficient conditions for bounded simple weight modules to be $\mathfrak{p}$-aligned or pseudo $\mathfrak{p}$-aligned. The existence of pseudo $\mathfrak{p}$-aligned modules is a consequence of the fact that the Lie algebras considered have infinite rank.