Standard polynomials in matrix algebras

Author:
Louis H. Rowen

Journal:
Trans. Amer. Math. Soc. **190** (1974), 253-284

MSC:
Primary 15A30

DOI:
https://doi.org/10.1090/S0002-9947-1974-0349715-3

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. Amitsur-Levitzki 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.

**[1]**S. A. Amitsur and J. Levitzki,*Minimal identities for algebras*, Proc. Amer. Math. Soc. 1 (1950), 449-463. MR 12, 155. MR**0036751 (12:155d)****[2]**N. Jacobson,*Structure of rings*, 2nd rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R.I., 1964, Chap. X. MR**36**#5158. MR**0222106 (36:5158)****[3]**B. Kostant,*A theorem of Frobenius, a theorem of Amitsur-Levitzki, and cohomology theory*, J. Math. Mech. 7 (1958), 237-264. MR 19, 1153. MR**0092755 (19:1153e)****[4]**Oystein Ore,*Theory of graphs*, Amer. Math. Soc. Colloq. Publ., vol. 38, Amer. Math. Soc., Providence, R.I., 1962. MR**27**#740. MR**0150753 (27:740)****[5]**F. Owens,*Applications of graph theory to matrix theory*(to appear). MR**0376708 (51:12883)****[6]**L. Rowen, Thesis, Yale University, New Haven, Conn., 1973.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0349715-3

Keywords:
Antisymmetric,
involution,
matrix algebra,
polynomial identity,
standard identity,
symmetric,
transpose

Article copyright:
© Copyright 1974
American Mathematical Society