Category $\mathcal{O}$ for the Lie algebra of vector fields on the line

TL;DR

This paper studies the category alor the Lie algebra of vector fields on the line, revealing its complex block structure, wild representation type, and establishing connections to modules over an associative algebra.

Contribution

It provides the block decomposition of alor ll simple modules, shows the wild representation type of each block, and constructs new simple modules from Weyl modules and Borel subalgebra modules.

Findings

Each block has infinitely many simple objects.

The representation type of each block is wild.

An exact functor relates alnd modules over a subalgebra H_1.

Abstract

Let $\mathfrak{W}$ be the Lie algebra of vector fields on the line. Via computing extensions between all simple modules in the category $\mathcal{O}$, we give the block decomposition of $\mathcal{O}$, and show that the representation type of each block of $\mathcal{O}$ is wild using the Ext-quiver. Each block of $\mathcal{O}$ has infinite simple objects. This result is very different from that of $\mathcal{O}$ for complex semisimple Lie algebras. To find a connection between $\mathcal{O}$ and the module category over some associative algebra, we define a subalgebra $H_1$ of $U(\mathfrak{b})$. We give an exact functor from $\mathcal{O}$ to the category $\Omega$ of finite dimensional modules over $H_1$. We also construct new simple $\mathfrak{W}$-modules from Weyl modules and modules over the Borel subalgebra $\mathfrak{b}$ of $\mathfrak{W}$.