On a category of $g{l}_{\mathrm{\infty }}$-modules

We study a particular category $\mathcal{C}$ of ${\text{gl}}_{\mathrm{\infty }}$-modules and a subcategory ${\mathcal{C}}_{int}$ of integrable ${\text{gl}}_{\mathrm{\infty }}$-modules. As the main results, we classify the irreducible modules in these two categories and we show that every module in category ${\mathcal{C}}_{int}$ is semi-simple. Furthermore, we determine the decomposition of the tensor products of irreducible modules in category ${\mathcal{C}}_{int}$.