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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.


References [Enhancements On Off] (What's this?)

  • [1] Wenxin Ma and Michel L. Racine, Minimal identities of symmetric matrices, Trans. Amer. Math. Soc. 320 (1990), 171-192. MR 961598 (90k:16018)
  • [2] J. Marshall Osborn, Identities of nonassociative algebras, Canad. J. Math. 17 (1963), 78-92. MR 0179221 (31:3470)
  • [3] K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative, Academic Press, New York, 1982. MR 668355 (83i:17001)

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

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