Automorphic Orbits in Free Non-associative Algebras
Automorphic Orbits in Free Non-associative Algebras
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DOI：
10.1006/jabr.2001.8794
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发表时间：
2001-09
影响因子：
0.9
通讯作者：
A. Mikhalev;U. Umirbaev;Jietai Yu
中科院分区：
文献类型：
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作者：
A. Mikhalev;U. Umirbaev;Jietai Yu
摘要考虑有限生成的自由非结合代数、自由交换非结合代数和自由反交换非结合代数。我们研究了这些代数在自同构群作用下元素的轨道。利用自由微分学，我们得到了一个元素系统具有给定秩（或原元）的矩阵准则。它给我们提供了一种构建快速识别元元系统的算法的可能性。我们证明了如果一个自由代数的自同构保持了一个非零元素的自同构轨道，那么它就是这个代数的自同构。特别地，保持元素原性的自同态是自同态。
Abstract We consider finitely generated free non-associative algebras, free commutative non-associative algebras, and free anti-commutative non-associative algebras. We study orbits of elements of these algebras under the action of automorphism groups. Using free differential calculus we obtain matrix criteria for a system of elements to have given rank (or to be primitive). It gives us a possibility to construct fast algorithm to recognize primitive systems of elements. We show that if an endomorphism of a free algebra preserves the automorphic orbit of a nonzero element, then it is an automorphism of this algebra. In particular, endomorphisms preserving primitivity of elements are automorphisms.