素特征的“李定理”

ANALOGUES OF LIE'S THEOREM IN PRIME CHARACTERISTIC

设L为代数闭域F上有限维李代数,著名的李定理说:若char F=0,则L为可解当且仅当L的任一有限维不可约模为1维的.在这里特征为0及模为有限维两个条件都是本质的.(1)若charF=P>0,则L为交换当且仅当L的任一(有限维)不可约模为1维的;(2)若char F=0,则L为交换当且仅当L的任一(有限维或无限维)不可约模为1维的; (3)若char F=P>7,L为李代数(限制李代数),则L为可解当且仅当L的任一不可约模(限制模)的维数为p的幂.

Let L be a finite-dimensional Lie algebra over an algebraically closed field F. If char F = 0, the well-known Lie's Theorem characterizes the solvability of L by the condition that every finite-dimensional irreducible module of L is one-dimensional. Here the conditions that Char F = 0 and that the module is finite-dimensional are both essential. (1) If Char F = p > 0, L is abelian if and only if every (finite-dimensional) irreducible module of L is one-dimensional. (2) If Char F = 0, L is abelian if and only if every (finite or infinite-dimensional) irreducible module of L is one-dimensional. (3) If Char F = p > 7, L is a Lie algebra (resp. restricted Lie algebra), L is solvable if and only if the dimension of every (finite-dimensional) irreducible module (resp. restricted module) of L is a power of p.