A decomposition of elements of the free algebra
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- by Wen Xin Ma
- Proc. Amer. Math. Soc. 118 (1993), 37-45
- DOI: https://doi.org/10.1090/S0002-9939-1993-1126198-4
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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
- Wen Xin Ma and Michel L. Racine, Minimal identities of symmetric matrices, Trans. Amer. Math. Soc. 320 (1990), no. 1, 171–192. MR 961598, DOI 10.1090/S0002-9947-1990-0961598-6
- J. Marshall Osborn, Identities of non-associative algebras, Canadian J. Math. 17 (1965), 78–92. MR 179221, DOI 10.4153/CJM-1965-008-3
- K. A. Zhevlakov, A. M. Slin′ko, I. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative, Pure and Applied Mathematics, vol. 104, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Translated from the Russian by Harry F. Smith. MR 668355
Bibliographic Information
- © Copyright 1993 American Mathematical Society
- 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