Nuclear elements of degree 6 in the free alternative algebra. (English) Zbl 1208.17024

Hentzel, I. R.; Peresi, L. A.

Exp. Math. 17, No. 2, 245-255 (2008).

Authors’ summary: “We construct five new elements of degree 6 in the nucleus of the free alternative algebra. We use the representation theory of the symmetric group to locate the elements. We use the computer algebra system ALBERT and an extension of ALBERT to express the elements in compact form and to show that these new elements are not a consequence of the known degree-5 elements in the nucleus. We prove that these five new elements and four known elements form a basis for the subspace of nuclear elements of degree 6. Our calculations are done using modular arithmetic to save memory and time. The calculations can be done in characteristic zero or any prime greater than 6, and similar results are expected. We generated the nuclear elements using prime 103. We check our answer using five other primes.”
This article together with the authors’ previous ones [Exp. Math. 15, No. 4, 445–454 (2006; Zbl 1147.17006), Non-associative algebra and its applications, Lect. Notes Pure Appl. Math. 246, 195–204 (2006)] gives a partial solution to Problem 2.121 posed by I. P. Shestakov in the Dniester Notebook [Non-associative algebra and its applications, Lect. Notes Pure Appl. Math. 246, 461–516 (2006)] on the description of the center and the associative center (= nucleus in the authors’ terminology) of a free alternative algebra as completely characteristic subalgebras. Are they finitely generated?

MSC:

17D05	Alternative rings
17-04	Software, source code, etc. for problems pertaining to nonassociative rings and algebras
17-08	Computational methods for problems pertaining to nonassociative rings and algebras
68W30	Symbolic computation and algebraic computation

Keywords:

free alternative algebras; nucleus; polynomial identities; computational algebra

Citations:

Zbl 1147.17006

Software:

Albert