Let F be a field of characteristic zero. W_n = F[t_1^(+-1), t_2^(+-1), ...,t_n^(+-1)] (partial deriv)/((partial deriv)t_1) + ... + F[t_1^(+-1), t_2^(+-1), ..., t_n^(+-1)](partial deriv)/((partial deriv)t_n) is the Wit...Let F be a field of characteristic zero. W_n = F[t_1^(+-1), t_2^(+-1), ...,t_n^(+-1)] (partial deriv)/((partial deriv)t_1) + ... + F[t_1^(+-1), t_2^(+-1), ..., t_n^(+-1)](partial deriv)/((partial deriv)t_n) is the Witt algebra over F, W_n^+ = F[t_1, t_2 ..., t_n](partial deriv)/((partial deriv)t_1) + ... + F[t_1, t_2 ..., t_n] (partial deriv)/((partialderiv)t_n) is Lie subalgebra of W_n. It is well known both W_n and W_n^+ are simple infinitedimensional Lie algebra. In Zhao's paper, it was conjectured that End(W_n^+) - {0} = Aut(W_n^+) andit was proved that the validity of this conjecture implies the validity of the well-known Jacobianconjecture. In this short note, we check the conjecture above for n = 1. We show End(W_1^+) - {0} =Aut(W_1^+).展开更多
文摘Let F be a field of characteristic zero. W_n = F[t_1^(+-1), t_2^(+-1), ...,t_n^(+-1)] (partial deriv)/((partial deriv)t_1) + ... + F[t_1^(+-1), t_2^(+-1), ..., t_n^(+-1)](partial deriv)/((partial deriv)t_n) is the Witt algebra over F, W_n^+ = F[t_1, t_2 ..., t_n](partial deriv)/((partial deriv)t_1) + ... + F[t_1, t_2 ..., t_n] (partial deriv)/((partialderiv)t_n) is Lie subalgebra of W_n. It is well known both W_n and W_n^+ are simple infinitedimensional Lie algebra. In Zhao's paper, it was conjectured that End(W_n^+) - {0} = Aut(W_n^+) andit was proved that the validity of this conjecture implies the validity of the well-known Jacobianconjecture. In this short note, we check the conjecture above for n = 1. We show End(W_1^+) - {0} =Aut(W_1^+).
