Tensor Representations of Mackey Lie Algebras and Their Dense Subalgebras

Tensor Representations of Mackey Lie Algebras and Their Dense Subalgebras
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麦基李代数及其稠密子代数的张量表示

DOI：
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发表时间：
2014
期刊：
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通讯作者：
V. Serganova
V. Serganova
中科院分区：
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文献类型：
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作者：
I. Penkov;V. Serganova

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在这篇文章中，我们回顾了早期论文[PStyr， PS]和[DPS]的主要结果，并建立了相关的更普遍的新结果。我们引入了一类无限维李代数\(\mathfrak{g}^{M}\)，我们称之为麦基李代数，并定义了张量\(\mathfrak{g}^{M}\) -模的一元范畴\(\mathbb{T}_{\mathfrak{g}^{M}}\)。我们也考虑密子代数\(\mathfrak{a} \subset \mathfrak{g}^{M}\)和相应的范畴\(\mathbb{T}_{\mathfrak{a}}\)。局部有限李代数\(\mathfrak{s}\mathfrak{l}(V,W),\mathfrak{o}(V ),\mathfrak{s}\mathfrak{p}(V )\)是各自麦基李代数的密子代数。我们的主要结果是，如果\(\mathfrak{g}^{M}\)是一个麦基李代数，\(\mathfrak{a} \subset \mathfrak{g}^{M}\)是一个稠密的子代数，那么一元范畴\(\mathbb{T}_{\mathfrak{a}}\)等价于\(\mathbb{T}_{\mathfrak{s}\mathfrak{l}(\infty )}\)或\(\mathbb{T}_{\mathfrak{o}(\infty )}\)；[DPS]中详细研究了后一种单型分类。一个可能的选择\(\mathfrak{a}\)是众所周知的广义雅可比矩阵的李代数。
In this article we review the main results of the earlier papers [PStyr, PS] and [DPS], and establish related new results in considerably greater generality. We introduce a class of infinite-dimensional Lie algebras \(\mathfrak{g}^{M}\), which we call Mackey Lie algebras, and define monoidal categories \(\mathbb{T}_{\mathfrak{g}^{M}}\) of tensor \(\mathfrak{g}^{M}\)-modules. We also consider dense subalgebras \(\mathfrak{a} \subset \mathfrak{g}^{M}\) and corresponding categories \(\mathbb{T}_{\mathfrak{a}}\). The locally finite Lie algebras \(\mathfrak{s}\mathfrak{l}(V,W),\mathfrak{o}(V ),\mathfrak{s}\mathfrak{p}(V )\) are dense subalgebras of respective Mackey Lie algebras. Our main result is that if \(\mathfrak{g}^{M}\) is a Mackey Lie algebra and \(\mathfrak{a} \subset \mathfrak{g}^{M}\) is a dense subalgebra, then the monoidal category \(\mathbb{T}_{\mathfrak{a}}\) is equivalent to \(\mathbb{T}_{\mathfrak{s}\mathfrak{l}(\infty )}\) or \(\mathbb{T}_{\mathfrak{o}(\infty )}\); the latter monoidal categories have been studied in detail in [DPS]. A possible choice of \(\mathfrak{a}\) is the well-known Lie algebra of generalized Jacobi matrices.