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摘要翻译：
本文系统地研究了2D变量自由代数的项链李代数n的李代数结构。我们首先把n描述为一个sp(2d)-模。针对d=1，我们将n分解为sl2的最高权模的直和，其系数由一个封闭的公式给出。其次，我们观察到n有一个非平凡中心，它通过一般2x2矩阵偶的迹环的中心C链接到S(Sl2)的泊松中心。n的李代数结构在C上诱导出一个泊松结构，它的辛叶可以用海森堡李代数h描述为半直积SL2\r×h的李群的伴随轨道。最后，我们给出了二重Poisson代数与Poisson序之间的联系，证明了二重Poisson代数的所有迹环在其中心上都是Poisson序。
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英文标题：
《On the structure of the necklace Lie algebra》
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作者：
Jacques Alev and Geert Van de Weyer
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Rings and Algebras 环与代数
分类描述：Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups
非交换环与代数，非结合代数，泛代数与格论，线性代数，半群
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
In this note, we initiate the systematic study of the Lie algebra structure of the necklace Lie algebra n of a free algebra in 2d variables. We begin by giving a description of n as an sp(2d)-module. Specializing to d = 1, we decompose n into a direct sum of highest weight modules for sl_2, the coefficients of which are given by a closed formula. Next, we observe that n has a nontrivial center, which we link through the center C of the trace ring of couples of generic 2x2 matrices to the Poisson center of S(sl_2). The Lie algebra structure of n induces a Poisson structure on C, the symplectic leaves of which we are able to describe as coadjoint orbits for the Lie group of the semidirect product sl_2\rtimes h of sl_2 with the Heisenberg Lie algebra h. Finally, we provide a link between double Poisson algebras on one hand and Poisson orders on the other hand, showing that all trace rings of a double Poisson algebra are Poisson orders over their center.
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PDF链接：
https://arxiv.org/pdf/0801.1621
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