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

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 $\mathfrak{ W}$.

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