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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A decomposition of elements of the free algebra


Author: Wen Xin Ma
Journal: Proc. Amer. Math. Soc. 118 (1993), 37-45
MSC: Primary 16S10
DOI: https://doi.org/10.1090/S0002-9939-1993-1126198-4
MathSciNet review: 1126198
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Abstract: Let $ f$ be an element of $ F[X]$, the free associative algebra over a field $ F$ and $ n$ the maximum of the degrees of the variables and the multiplicities of the degrees in $ f$. A partial ordering on the homogeneous elements of $ F[X]$ is defined such that if $ f$ is homogeneous and $ F\nmid n!$, then $ f$ can be decomposed into a sum of two polynomials $ {f_0}$ and $ {f_1}$ such that for $ 0 < m \leqslant n,\;{f_0}$ is symmetric or skew symmetric in all its arguments of degree $ m$ depending on whether $ m$ is even or odd and $ {f_1}$ is a consequence of polynomials of lower type than $ f$. Osborn's Theorem about the symmetry of the absolutely irreducible polynomial identities is obtained as a corollary. The same holds in the free nonassociative algebra. The proofs are combinatorial.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1126198-4
Article copyright: © Copyright 1993 American Mathematical Society