可度量化李三系

研究域K上可定义非退化不变对称双线性形的李三系■(称这样的李三系■为可度量化的).对一个李三系■,我们定义了该李三系的T_ω~*-扩张,给出了一个度量化李三系(■,φ)同构于某李三系的T~*-扩张的充分必要条件,并且讨论了什么时候两个T~*-扩张是等价或度量等价的.最后证明当基域的特征为零时,偶数维幂零的度量化李三系与某李三系的T~*-扩张同构,而奇数维幂零的度量化李三系则同构于某李三系的T~*-扩张的余维数为1的非退化理想.

Metrisable Lie Triple Systems

We study the Lie triple system (L.t.s.) J over a field K admitting a nondenerate invariant symmetric bilinear form (call such a J metrisable).We give the definition of T_ω~*-extension of an L.t.s.J,prove a necessary and sufficient condition for a metrised L.t.s.(J,φ) to be isometric to a T~*-extension of some L.t.s.,and determine when two T~*-extension of an L.t.s.are"same",i.e.they are equivalent or isometrically equivalent.In the end,we prove that any nilpotent metrised L.t.s.is either isometric to some T~*-extension or isometric to a nondegenerate ideal of codimension one in some T~*-extension according to the dimension of the algebra is even or odd when the characteristic of the ground field is equal to 0.