Nonassociative coefficient algebras for Steinberg unitary Lie algebras
Given a unital associative algebra with involution \((A,-)\) over a scalar ring \(\Phi\), \(n\geq 3\) and \(\gamma_1, \ldots, \gamma_n\) invertible elements in \(\Phi\), the unitary Lie algebra \(u_ n (A,-)\) is the Lie algebra of the skew elements in \(\operatorname{Mat}_n(A)\) relative to the involution \(x^*= \gamma^{-1} \overline{x}^t \gamma\), where \(\gamma= \text{diag} (\gamma_ 1, \dots, \gamma_ n)\). The Lie subalgebra generated by the elements \(ae_{ij}- \gamma_i \gamma_j^{-1} \overline {a} e_{ji}\) for \(a\in A\) and \(i\neq j\), where the \(e_{ij}\)'s are the usual matrix units, is called the elementary unitary Lie algebra. The authors consider a more general situation by taking \((A,-)\) to be any unital nonassociative algebra with involution over \(\Phi\) and the Lie algebra generated by the symbols \(u_{ij} (a)\), \(1\leq i\neq j\leq n\), \(a\in A\), subject to the natural relations verified in the elementary unitary Lie algebras by their generators. Then they show that it is enough to deal with algebras \((A,-)\) for which the linear maps \(a\mapsto u_{ij} (a)\) are one-to-one (termed \(n\)-faithful algebras) and that this occurs for \(n\geq 4\) if and only if \(A\) is associative. However, for \(n=3\) the situation is more complicated and thoroughly studied by the authors. The main result of the paper shows that \((A,-)\) is 3-faithful if and only if \((A,-)\) is a structurable algebra [\textit{B. N. Allison}, Jordan algebras, Proc. Conf. Oberwolfach 1992, 1--8 (1994; Zbl 0812.17003)]. For these algebras, a related Lie algebra \(K(A,-, \gamma)\) is constructed which is simple or central if and only if so is \((A,-)\) and which is isomorphic to the quotient by its center of the Steinberg unitary Lie algebra of order three.
scientific article; zbMATH DE number 554259