Clifford系统、Clifford结构及其正则微分形式

IF 0.4 4区 数学 Q4 MATHEMATICS
Kai Brynne M. Boydon, Paolo Piccinni
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引用次数: 0

摘要

在\(\mathbb {H}^2 \cong {\mathbb {R}}^8\)四元数4-形式\(\Phi _{\text {Sp}(2)\text {Sp}(1)}\)和Cayley校准\(\Phi _{\text {Spin}(7)}\)的不同结构之间的比较表明,可以从相同的“Kähler 2-形式”集合开始,同时输入四元数Kähler和\(\text {Spin}(7)\)几何。这种比较涉及到连克利福德结构和克利福德系统的概念。转到维度16,类似的结构允许在\(\mathbb {R}^{16}\)中为规范4-form \(\Phi _{\text {Spin}(8)}\)和\(\Phi _{\text {Spin}(7)\text {U}(1)}\)编写显式公式,它们与与\(\text {SO}(16)\)的子组\(\text {Spin}(8)\)和\(\text {Spin}(7)\text {U}(1)\)相关的Clifford系统相关联。我们描述了4-形式\(\Phi _{\text {Spin}(8)}\)和\(\Phi _{\text {Spin}(7)\text {U}(1)}\)的校准4-平面,以两种不同的方式将Cayley 4-平面的概念扩展到16维。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Clifford systems, Clifford structures, and their canonical differential forms

A comparison among different constructions in \(\mathbb {H}^2 \cong {\mathbb {R}}^8\) of the quaternionic 4-form \(\Phi _{\text {Sp}(2)\text {Sp}(1)}\) and of the Cayley calibration \(\Phi _{\text {Spin}(7)}\) shows that one can start for them from the same collections of “Kähler 2-forms”, entering both in quaternion Kähler and in \(\text {Spin}(7)\) geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension 16, similar constructions allow to write explicit formulas in \(\mathbb {R}^{16}\) for the canonical 4-forms \(\Phi _{\text {Spin}(8)}\) and \(\Phi _{\text {Spin}(7)\text {U}(1)}\), associated with Clifford systems related with the subgroups \(\text {Spin}(8)\) and \(\text {Spin}(7)\text {U}(1)\) of \(\text {SO}(16)\). We characterize the calibrated 4-planes of the 4-forms \(\Phi _{\text {Spin}(8)}\) and \(\Phi _{\text {Spin}(7)\text {U}(1)}\), extending in two different ways the notion of Cayley 4-plane to dimension 16.

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来源期刊
CiteScore
0.80
自引率
0.00%
发文量
7
审稿时长
>12 weeks
期刊介绍: The first issue of the "Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg" was published in the year 1921. This international mathematical journal has since then provided a forum for significant research contributions. The journal covers all central areas of pure mathematics, such as algebra, complex analysis and geometry, differential geometry and global analysis, graph theory and discrete mathematics, Lie theory, number theory, and algebraic topology.
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