arXiv subjects

Thanasin Nampaisarn

Publications and source records attributed to Thanasin Nampaisarn.

3 recordsLinked to original sources

arXiv2020

Categories $\mathcal{O}$ for Root-Reductive Lie Algebras: II. Translation Functors and Tilting Modules

This is the second paper of a series of papers on a version of categories $\mathcal{O}$ for root-reductive Lie algebras. Let $\mathfrak{g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$ with a splitting Borel subalgebra $\mathfrak{b}$ containing a splitting maximal toral subalgebra $\mathfrak{h}$. For some pairs of blocks $\overline{\mathcal{O}}[λ]$ and $\overline{\mathcal{O}}[μ]$, the subcategories whose objects have finite length are equivalence via functors obtained by the direct limits of translation functors. Tilting objects can also be defined in $\overline{\mathcal{O}}$. There are also universal tilting objects $D(λ)$ in parallel to the finite-dimensional cases.

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arXiv2017

On Categories $\mathcal{O}$ for Root-Reductive Lie Algebras

Let $\mathfrak{g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$ with a splitting Borel subalgebra $\mathfrak{b}$ containing a splitting maximal toral subalgebra $\mathfrak{h}$. We study the category $\bar{\mathcal{O}}$ consisting of all $\mathfrak{h}$-weight $\mathfrak{g}$-modules which are locally $\mathfrak{b}$-finite and have finite-dimensional $\mathfrak{h}$-weight spaces. The focus is on very special Borel subalgebras called the Dynkin Borel subalgebras. This paper serves as an initial passage to the understanding of categories $\mathcal{O}$ for infinite-dimensional root-reductive Lie algebras.

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arXiv2017

Categories O for Dynkin Borel Subalgebras of Root-Reductive Lie Algebras

The purpose of my Ph.D. research is to define and study an analogue of the classical Bernstein-Gelfand-Gelfand (BGG) category $\mathcal{O}$ for the Lie algebra $\mathfrak{g}$, where $\mathfrak{g}$ is one of the finitary, infinite-dimensional Lie algebras $\mathfrak{gl}_\infty(\mathbb{K})$, $\mathfrak{sl}_\infty(\mathbb{K})$, $\mathfrak{so}_\infty(\mathbb{K})$, and $\mathfrak{sp}_\infty(\mathbb{K})$. Here, $\mathbb{K}$ is an algebraically closed field of characteristic $0$. We call these categories "extended categories $\mathcal{O}$" and use the notation $\bar{\mathcal{O}}$. While the categories $\bar{\mathcal{O}}$ are defined for all splitting Borel subalgebras of $\mathfrak{g}$, this research focuses on the categories $\bar{\mathcal{O}}$ for very special Borel subalgebras of $\mathfrak{g}$ which we call Dynkin Borel subalgebras. Some results concerning block decomposition and Kazhdan-Lusztig multiplicities carry over from usual categories $\mathcal{O}$ to our categories $\bar{\mathcal{O}}$. There are differences which we shall explore in detail, such as the lack of some injective hulls. In this connection, we study truncated categories $\bar{\mathcal{O}}$ and are able to establish an analogue of BGG reciprocity in the categories $\bar{\mathcal{O}}$.

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