Nondegenerate Invariant Bilinear Forms And The Centroid On Lie Triple Systems

A bilinear form f on a nonassociative triple system T is said to be invariant if f((abc~,d) = f(a, (dcb)) f(c, (bad~) for every a,b,c,d € T. Finite-dimensional complex semisimple lie triple system (with their killing form) carry s~ch a structure. In addition, (T, f) is called a pseudo-metnsed triple system if f is nondegenerate and invariant. In the first part of prest paper, we mainly investigate the decomposition theory of triple systems and pseudo-metrised triple systems. We also characterize the finite-dimensional metrised lie triple system by the structure of nondegenerate invariant and symmetric bilinear form on it.In the second part of this paper, we give the definition of the centroid of triple systems and the basic results of the centroid of simple triple systems .In the progress of this paper, the definition that the centroid of lie triple system is small is given and a sufficient condition that the composable lie triple system is small is discussed. In addition, we put forth the relationship between the centroid of a lie triple system T and the centroid of the standard embedding lie algebra of T. Finally, we disct~ss the properties of the centroid of metrised lie triple systems.