可度量化李三系

Metrisable Lie Triple Systems

研究域K上可定义非退化不变对称双线性形的李三系J（称这样的李三系J为可度量化的）.对一个李三系J,我们定义了该李三系的Tω^＊ -扩张,给出了一个度量化李三系（J,Ф）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.

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^＊ -extension of an L.t.s. J, prove a necessary and sufficient condition for a metrised L.t.s. （J, Ф） 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.