On trace forms of higher degree
Let \(A\) be a central simple algebra over a field \(k\). Consider the trace-form of degree \(d\) on \(A\): \(\Psi_d(X)=\text{tr }X^d\), where tr is the reduced trace. A linear map \(f:A\to A\) is a similarity of \((A,\Psi_d)\) if there exists a \(\lambda\in k^*\) such that \(\Psi_d(f(a))=\lambda\Psi_d(a)\) for all \(a\in A\). The main theorem of the paper shows that if \(d>2\) and the characteristic of \(k\) does not divide \(d!\), then the similarities \(f\) of \((A,\Psi_d)\) are in the group generated by the automorphisms, antiautomorphisms and scalar maps of \(A\). (Linear maps in this group are called standard.) In section 1 some background and motivation for the problem is provided. The theorem can be seen as a generalization of a theorem of \textit{J. Rackusin} and \textit{W. Watkins} [Linear Algebra Appl. 17, 269-276 (1977; Zbl 0371.15004)] which states that if \(d>2\) and \(f\) is a linear map on \(M_n(\mathbb{C})\) preserving \(\text{tr }X^d\) such that \(f(1_n)=1_n\), then \(f\) must be standard. To prove the main theorem of the paper it is enough to prove it over some extension of \(k\) (Lemma 3.3). This reduces the result to \(A=M_n(k)\). The largest part of the proof consists in showing that for a similarity \(f\), \(f(1_n)\) is a scalar matrix. In section 2 it is shown that \((A,\Psi_d)\) as a symmetric \(d\)-linear space is regular and indecomposable.
scientific article; zbMATH DE number 951805
scientific article