# Thanasin Nampaisarn

## "Introduction to the Theory of Integrable 𝔰𝔩∞(ℂ)-Modules"

 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.