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Constructing the polynomial identities and central identities of degree \(<9\) of \(3\times 3\) matrices. (English) Zbl 0884.15009

Let \(R\) be an algebra over a field \(\phi\) and \(X=\{x_1,x_2,\dots\}\) be a countable set of symbols. A polynomial \(f(x_1,\dots,x_n)\) from the free associative algebra \(\phi\langle X\rangle\) is said to be a polynomial identity of \(R\) if \(f(x_1,\dots,x_n)=0\) for all \(x_1,\dots,x_n\in R\). It is called a polynomial central identity of \(R\) if \(f\) is not an identity and \(f(x_1,\dots,x_n)\in C(R)\) for all \(x_1,\dots,x_n\in R\), where \(C(R)\) is the center of \(R\).
The author presents an algorithm for computing an independent generating set for the multilinear identities and the multilinear central identities of the \(m\times m\) matrices over a field \(\phi\) of characteristic zero or a large enough prime. Then the author uses it to construct all the multilinear identities and all the multilinear central identities of degree \(<9\) for \(M_3(\phi)\).

MSC:

15A24 Matrix equations and identities
15A69 Multilinear algebra, tensor calculus
65F30 Other matrix algorithms (MSC2010)
Full Text: DOI

References:

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