Free Akivis Algebras, Primitive Elements, and Hyperalgebras
Free Akivis Algebras, Primitive Elements, and Hyperalgebras
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自由 Akivis 代数、原元和超代数
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
U. Umirbaev
中科院分区：
文献类型：
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作者：
I. Shestakov;U. Umirbaev
研究自由Akivis代数及其泛包络代数中的本原元。证明了自由Akivis代数的子代数是自由的，并且它的生成子代数是自由剩余的.也证明了Akivis代数的各种字问题的可判定性。
K. H. Hofmann和K. Strambach（问题6.15在[拓扑和分析循环，在“群和循环理论和应用，”系列纯数学（O。Chein，H. O. Pflugfelder和J. D. H. Smith，Eds.），Vol. 8，pp. 205-262，Heldermann Verlag，柏林，1990]）关于本原元结构的结论是不成立的，并构造了自由非结合代数中的一个本原元全系.
最后，证明了每个代数B都可以看作是超代数，即具有一系列多重线性运算的系统对于局部解析环起着切代数的作用，其中B上的超代数运算由一定的本原元解释。
Free Akivis algebras and primitive elements in their universal enveloping algebras are investigated. It is proved that subalgebras of free Akivis algebras are free and that finitely generated subalgebras are finitely residual. Decidability of the word problem for the variety of Akivis algebras is also proved.
The conjecture of K. H. Hofmann and K. Strambach (Problem 6.15 in [Topological and analytic loops, in “Quasigroups and Loops Theory and Applications,” Series in Pure Mathematics (O. Chein, H. O. Pflugfelder, and J. D. H. Smith, Eds.), Vol. 8, pp. 205–262, Heldermann Verlag, Berlin, 1990]) on the structure of primitive elements is proved to be not valid, and a full system of primitive elements in free nonassociative algebra is constructed.
Finally, it is proved that every algebra B can be considered as a hyperalgebra, that is, a system with a series of multilinear operations that plays a role of a tangent algebra for a local analytic loop, where the hyperalgebra operations on B are interpreted by certain primitive elements.