A Koszul category of representations of finitary Lie algebras
A Koszul category of representations of finitary Lie algebras
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有限李代数表示的 Koszul 范畴
DOI：
10.1016/j.aim.2015.10.023
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发表时间：
2011
影响因子：
1.7
通讯作者：
V. Serganova
中科院分区：
文献类型：
--
作者：
Elizabeth Dan;I. Penkov;V. Serganova
我们发现，对于每个简单有限李代数g的一个范畴T g的可积模块中的张量积的副本的自然和conatural模块是内射的。T g中的对象可以定义为有限长绝对权模，这里绝对权模指的是g的每个可裂Cartan子代数的权模.范畴T g是Koszul，因为它与有限维Koszul代数的某个直极限上的局部酉有限维模范畴反等价。我们明确地描述了这些有限维代数。证明了正交有限李代数o（∞）和辛有限李代数sp（∞）分别对应的范畴To（∞）和Tsp（∞）的一个等价性.
We find for each simple finitary Lie algebra g a category T g of integrable modules in which the tensor product of copies of the natural and conatural modules is injective. The objects in T g can be defined as the finite length absolute weight modules, where by absolute weight module we mean a module which is a weight module for every splitting Cartan subalgebra of g. The category T g is Koszul in the sense that it is antiequivalent to the category of locally unitary finite-dimensional modules over a certain direct limit of finite-dimensional Koszul algebras. We describe these finite-dimensional algebras explicitly. We also prove an equivalence of the categories T o (∞) and T sp (∞) corresponding respectively to the orthogonal and symplectic finitary Lie algebras o (∞), sp (∞).