Date:	Mon, September 5, 2016
Time:	15:00
Place:	Seminar Room (120), Research I

Abstract: During the course of this talk, I shall provide a definition of integrable modules over a Lie algebra and focus on the category \(\text{Int}_{\mathfrak{sl}_\infty(\mathbb{C})}\) of integrable \(\mathfrak{sl}_\infty(\mathbb{C})\)-modules. For a fixed splitting Cartan subalgebra \(\mathfrak{h}\) of \(\mathfrak{sl}_\infty(\mathbb{C})\), there are two important subcategories of \(\text{Int}_{\mathfrak{sl}_\infty(\mathbb{C})}\): \(\text{Int}^{\text{wt}}_{\mathfrak{sl}_\infty(\mathbb{C}),\mathfrak{h}}\) and \(\text{Int}^{\text{fin}}_{\mathfrak{sl}_\infty(\mathbb{C}),\mathfrak{h}}\). The subcategory \(\text{Int}^{\text{wt}}_{\mathfrak{sl}_\infty(\mathbb{C}),\mathfrak{h}}\) is the full subcategory of \(\text{Int}_{\mathfrak{sl}_\infty(\mathbb{C}),\mathfrak{h}}\) consisting of \(\mathfrak{h}\)-weight modules, whereas the subcategory \(\text{Int}^{\text{fin}}_{\mathfrak{sl}_\infty(\mathbb{C}),\mathfrak{h}}\) is the full subcategory of \(\text{Int}^{\text{wt}}_{\mathfrak{sl}_\infty(\mathbb{C}),\mathfrak{h}}\) composed by modules with finite-dimensional weight spaces. The aim of this talk is to show that \(\text{Int}^{\text{fin}}_{\mathfrak{sl}_\infty(\mathbb{C}),\mathfrak{h}}\) is a semisimple category. If time permits, we shall dwell further into other properties of integrable \(\mathfrak{sl}_\infty(\mathbb{C})\)-modules.