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Quasi-identities on matrices and the Cayley-Hamilton polynomial. (English) Zbl 1332.16019
The trace identities of the full matrix algebra of order \(n\) over a field of characteristic 0 were described by Procesi and independently by Razmyslov. These all follow from the Cayley-Hamilton polynomial. Recall that one can write, using Newton’s formulae, the coefficients of the characteristic polynomial of a matrix in terms of traces of powers of the matrix.
The paper under review deals with quasi-identities, so let us recall what this is. Take a set \(X=\{x_k\}\) of noncommuting variables and let \(\{x_{ij}^{(k)}\mid 1\leq i,j\leq n\}\) be commuting ones. Form the free associative algebra \(C(X)\) on the variables \(X\) over the polynomial algebra of all \(x_{ij}^{(k)}\); its elements are the quasi-polynomials. (One may consider the ring of scalars of \(C(X)\) as the algebra of polynomial functions on the matrix algebra of order \(n\).)
The authors define in a natural way a trace in \(C(X)\), and study whether the trace quasi-identities for the \(n\times n\) matrices are generated by the Cayley-Hamilton polynomial. They describe large classes of trace quasi-identities that follow from the Cayley-Hamilton polynomial. Nevertheless there exist classes of (antisymmetric) quasi-identities which do not follow from the Cayley-Hamilton polynomial, and the authors describe one such class.

16R60 Functional identities (associative rings and algebras)
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16R30 Trace rings and invariant theory (associative rings and algebras)
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