An expansion formula for the norm of a Jordan algebra
A generalisation of the known Leibniz formula for the determinant of an \(n \times n\) matrix for certain types of polynomial functions on Jordan algebras is obtained. Let \(B\) be a class of Jordan algebras over an infinite field \(k\) (of characteristic \(\not = 2\)) that is closed under taking Peirce subalgebras, and \(L\) be a fixed extension field of \(k\). Assume that for every Jordan algebra \(J \in B\) we are given a polynomial \(N_J : J \to L\). The author defines for the set \(\{N_J \mid J \in B\}\) a norm-like system for \(B\) over \(L\) and calculates the reduced generic norm for it. In theorem 1 he proves that if \(\{N_J\}\) is a norm-like system for \(B\), \(J \in B\) and \(\{c_1,\dots,c_n\}\) is an orthogonal system of absolutely primitive idempotents of \(J\), then for all \(x \in J\) and all \(\lambda_i \in k\) the following formula holds: \[ N_J(x + \sum^r_{i=1} \lambda_i c_i) = \sum_{T \subset \{1,\dots ,r\}} N_{J_0 (c_T)} (U (e - c_T)x)\lambda_T \] where \[ c_T = \sum_{i \in T} c_i, \quad \lambda_T = \prod_{i \in T} \lambda_i. \] In section 2 of the article the author shows that the class of finite-dimensional Jordan algebras with reduced generic norm is a norm-like system.
scientific article; zbMATH DE number 1145401