Evaluations of noncommutative polynomials on finite dimensional algebras. Lvov-Kaplansky Conjecture

Авторы: Louis Rowen, Roman Yavich, Sergey Malev, Alexei Kanel-Belov
Сборник: ArXiv.org.e-Print
Серия: Series arxive "math"
Том: 1904
Год издания: 2019
Издательство: Cornell University Press
Местоположение издательства: United States
Аннотация: Let p be a polynomial in several non-commuting variables with coefficients in an algebraically closed field K of arbitrary characteristic. It has been conjectured that for any n, for p multilinear, the image of p evaluated on the set Mn(K) of n by n matrices is either zero, or the set of scalar matrices, or the set sln(K) of matrices of trace 0, or all of Mn(K). This expository paper describes research on this problem and related areas. We settle this conjecture for n=2 and obtain decisive results for n=3, and partial information for n≥3, also for non-multilinear polynomials. In addition we consider the case of K not algebraically closed, and polynomials evaluated on other finite dimensional simple algebras (in particular the algebra of the quaternions).
Добавил в систему: Белов Алексей Яковлевич

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