Evaluations of nonassociative polynomials on finite dimensional algebras. Sergey Malev (Ariel University of Samaria, Israel) and Alexei Kanel-Belov (Bar-Ilan University, Israel) 05/Aug/2021 - 14:00 GMT-3 (São Paulo time)
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 \(M_n(K)\) of \(n\) by \(n\) matrices is either zero, or the set of scalar matrices, or the set \(sl_n(K)\) of matrices of trace 0, or all of \(M_n(K)\).
In this talk we will discuss the generalization of this result for non-associative algebras such as Cayley-Dickson algebra (i.e. algebra of octonions), pure (scalar free) octonion Malcev algebra and basic low rank Jordan algebras.