论文标题
类别$ \ MATHCAL {O} $ for vector字段的Lie代数
Category $\mathcal{O}$ for the Lie algebra of vector fields on the line
论文作者
论文摘要
令$ \ mathfrak {w} $为行上向量字段的Lie代数。通过类别$ \ MATHCAL {O} $中所有简单模块之间的计算扩展，我们给出了$ \ Mathcal {O} $的块分解，并证明$ \ Mathcal {O} $的每个块的表示类型是使用Ext-Quiver使用的。 $ \ Mathcal {O} $的每个块都有无限的简单对象。对于复杂的半imple lie代数，此结果与$ \ Mathcal {O} $的结果大不相同。要在某些关联代数上找到$ \ Mathcal {O} $与模块类别之间的连接，我们定义了$ u（\ Mathfrak {b}）$的subalgebra $ h_1 $ h_1 $。我们从$ \ MATHCAL {O} $向$ H_1 $的有限维模块的类别$ω$提供了一个确切的函数。我们还构建了来自borel subalgebra $ \ mathfrak {b} $ of $ \ mathfrak {w} $的新的简单$ \ mathfrak {w} $ - 来自Weyl模块和模块的模块。
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 $Ω$ 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 $\mathfrak{W}$.
