A construction of Lie algebras and Lie superalgebras by Freudenthal-Kantor triple systems. I
A Freudenthal-Kantor triple system (FKS) \(U(\varepsilon,\delta)\) is a finite-dimensional vector space over a field of characteristic not 2 or 3 with a triple product \(\{xyz\}\) on \(U(\varepsilon,\delta)\) satisfying \[ [L(x,y),L(u,v)]=L(L(x,y)u,v)+\varepsilon L(u,L(y,x)v), \] \[ K(K(x,y)u,v)=L(v,u)K(x,y)-\varepsilon K(x,y)L(u,v) \] for all \(x,y,u,v\) in \(U(\varepsilon,\delta)\), where \(L(x,y)u=\{xyu\}\), \(K(x,y)u=\{xuy\}-\delta \{yux\}\), \(\varepsilon =\pm 1\), and \(\delta =\pm 1\). If \(U(\varepsilon,\delta)\) is an FKS, then there is a two-dimensional associative triple system \(W\) such that \(W\otimes U(\varepsilon,\delta)\) is an FKS. A vector space direct sum of two copies of \(W\otimes U(\epsilon,\delta)\) can be made into a Lie triple system \(V\). \(G(\varepsilon,\delta)=D\oplus V\) is a Lie algebra (when \(\delta =1)\) or a Lie superalgebra (when \(\delta =-1)\), where \(D\) is the Lie algebra of inner derivations of the Lie triple system \(V\). \(G(\varepsilon,\delta)\) is a graded Lie algebra or Lie superalgebra of second order.
scientific article; zbMATH DE number 3920735