arXiv 1407.04282014-07math.RA

李半直积与偏序集代数的上同调

Cohomology of Lie semidirect products and poset algebras

Vincent E. Coll, Murray Gerstenhaber

备注：: Revised, errors corrected, 16 pages

摘要：

当$\mathfrak h$是域$\mathbf k$上的李代数$\mathfrak g$的一个环面子代数，$M$是一个$\mathfrak g$ -模，$\mathfrak h$也在其上环面作用时，Chevalley-Eilenberg上链复形的Hochschild-Serre滤子允许比任意子代数更强的形式。对于半直积$\mathfrak g = \mathfrak h \ltimes \mathfrak k$和$\mathfrak h$ toral，有$H^*(\mathfrak g, M) \cong \bigwedge\mathfrak h^{\vee} \bigotimes H^*(\mathfrak k,M)^{\mathfrak h} = H^*(\mathfrak h, \mathbf k)\bigotimes H^*(\mathfrak k,M)^{\mathfrak h}$，对于李偏序集代数$\mathfrak g$，控制$\mathfrak g$变形的$H^*(\mathfrak g, \mathfrak g)$可以从基础偏序集的神经计算。李偏序集代数的变形理论，类似于复解析流形，它是一个小模型，举例说明。

英文摘要：

When $\mathfrak h$ is a toral subalgebra of a Lie algebra $\mathfrak g$ over a field $\mathbf k$, and $M$ a $\mathfrak g$-module on which $\mathfrak h$ also acts torally, the Hochschild-Serre filtration of the Chevalley-Eilenberg cochain complex admits a stronger form than for an arbitrary subalgebra. For a semidirect product $\mathfrak g = \mathfrak h \ltimes \mathfrak k$ with $\mathfrak h$ toral one has $H^*(\mathfrak g, M) \cong \bigwedge\mathfrak h^{\vee} \bigotimes H^*(\mathfrak k,M)^{\mathfrak h} = H^*(\mathfrak h, \mathbf k)\bigotimes H^*(\mathfrak k,M)^{\mathfrak h}$, and for a Lie poset algebra $\mathfrak g$, that $H^*(\mathfrak g, \mathfrak g)$, which controls the deformations of $\mathfrak g$, can be computed from the nerve of the underlying poset. The deformation theory of Lie poset algebras, analogous to that of complex analytic manifolds for which it is a small model, is illustrated by examples.

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摘要：

英文摘要：

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