An Approach to Finite-Dimensional Real Division Composition Algebras through Reflections
An Approach to Finite-Dimensional Real Division Composition Algebras through Reflections
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通过反射求解有限维实除复合代数
DOI：
10.1016/j.bulsci.2014.10.001
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发表时间：
2015
影响因子：
1.3
通讯作者：
Seidon Alsaody
中科院分区：
文献类型：
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作者：
Seidon Alsaody
研究了有限维真实的复合代数的除代数范畴。这些都是有限维绝对值代数，并且只存在于1、2、4和8维中。我们构建三个分解这一类，每一个决定的数量的反射组成的左和右乘幂等元。因此，我们得到了新的8维全子范畴，其中所有态射都是八元数的自同构。这减少了相当大的部分仍然开放的分类问题在8维的正常形式的问题的行动的自同构群的八元数，这是一个紧凑的李群的类型G 2，对正交映射。我们进一步描述这些子范畴的子群Aut（O）和他们的陪集，我们表示几何。这推广了对有限维真实的可除复合代数的单边单位性的研究。
We consider the category of all finite-dimensional real composition algebras which are division algebras. These are precisely the finite-dimensional absolute valued algebras, and exist only in dimension 1, 2, 4 and 8. We construct three decompositions of this category, each determined by the number of reflections composing left and right multiplication by idempotents. As a consequence, we obtain new full subcategories in dimension 8, in which all morphisms are automorphisms of the octonions. This reduces considerable parts of the still open classification problem in dimension 8 to the normal form problem of an action of the automorphism group of the octonions, which is a compact Lie group of type G 2, on pairs of orthogonal maps. We describe these subcategories further in terms of subgroups of Aut (O) and their cosets, which we express geometrically. This extends the study of finite-dimensional real division composition algebras with a one-sided unity.