Standard polynomials in matrix algebras
Author:
Louis H. Rowen
Journal:
Trans. Amer. Math. Soc. 190 (1974), 253284
MSC:
Primary 15A30
MathSciNet review:
0349715
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Abstract: Let be an matrix ring with entries in the field F, and let be the standard polynomial in k variables. AmitsurLevitzki have shown that vanishes for all specializations of to elements of . Now, with respect to the transpose, let be the set of antisymmetric elements and let be the set of symmetric elements. Kostant has shown using Lie group theory that for n even vanishes for all specializations of to elements of . By strictly elementary methods we have obtained the following strengthening of Kostant's theorem: vanishes for all specializations of to elements of , for all n. vanishes for all specializations of to elements of and of to an element of , for all n. vanishes for all specializations of to elements of and of to an element of , for n odd. These are the best possible results if F has characteristic 0; a complete analysis of the problem is also given if F has characteristic 2.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403497153
PII:
S 00029947(1974)03497153
Keywords:
Antisymmetric,
involution,
matrix algebra,
polynomial identity,
standard identity,
symmetric,
transpose
Article copyright:
© Copyright 1974
American Mathematical Society
