Lie algebras of type $D_4$ over number fields.
Lie algebras of type $D_4$ over number fields.
复制标题
数域上 $D_4$ 类型的李代数。
DOI：
10.2140/pjm.1992.156.209
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发表时间：
1992
影响因子：
0.6
通讯作者：
B. Allison
中科院分区：
文献类型：
--
作者：
B. Allison
本文证明了如何构造代数数域上所有的D4型中心单李代数。我们使用的构式是由于G.B.塞利格曼而修改的构式的特例。构造的起点是一个具有对合CD(^，μ)的8维非结合代数，它是由一个4维可分交换结合代数38和一个非零标量μ通过Cayley-Dickson加倍过程得到的。利用代数CD(^，μ)作为李代数JF(CΌ(&，μ)，γ)的系数代数，JF(Cβ，μ)可以粗略地描述为关于对合X-·y~x Xιy的3×3-斜Hermite矩阵的李代数，其中y是具有标量项的可逆对角矩阵.我们证明了数域上的任何D4型李代数都可以构造为c/fCDfJ‘，μ)，γ)，对于38，μ和γ中的某一个选择。我们还给出了这样构造的两个李代数的同构条件。
In this paper we show how to construct all central simple Lie algebras of type D 4 over an algebraic number field. The construction that we use is a special case of a modified version of a construction due to G. B. Seligman. The starting point for the construction is an 8-dimensional nonassociative algebra with involution C D ( ^, μ) that is obtained by the Cayley-Dickson doubling process from a 4-dimensional separable commutative associative algebra 38 and a nonzero scalar μ . The algebra C D ( ^, μ) is used as the coefficient algebra for a Lie algebra Jf(CΌ(& , μ), γ) that can be roughly described as the Lie algebra of 3 x 3-skew hermitian matrices with entries from CD{β , μ) relative to the involution X —• y~ x X ι y , where y is an invertible diagonal matrix with scalar entries. We show that any Lie algebra of type D 4 over a number field can be constructed as c/fCDfJ', μ), γ) for some choice of 38 , μ and γ . We also give isomorphism conditions for two Lie algebras constructed in this way.
DOI：
10.1090/s0002-9947-1984-0735416-2
发表时间：
1984
影响因子：
1.3
作者：
B. Allison;J. Faulkner
通讯作者：
B. Allison;J. Faulkner