Every monomorphism of the Lie algebra of triangular polynomial derivations is an automorphism (Q447873)

Let \(K\) be a field of characteristic zero and \(P_n=K[x_1,\cdots ,x_n]\) be the polynomial ring. Let \(u_n\) be the Lie algebra such that NEWLINE\[NEWLINEu_n=K\partial_1+P_1\partial_2+\cdots +P_{n-1}\partial_nNEWLINE\]NEWLINE and it is called the Lie algebra of unitriangular polynomial derivations where \(\partial_i\), \(1\leq i\leq n\), is the partial derivative of \(P_n\). \(u_n\) is a subalgebra of the well-known Witt type Lie algebra \(W_n\). Under this review, the author shows that every monomorphism of \(u_n\) is an automorphism.NEWLINENEWLINENEWLINENEWLINE Reviewer's comments on the results: Since the Lie algebra \(u_n\) is not simple, for any monomorphism \(\theta\) of \(u_n\), \(\partial_1\) and \(\partial_n\) are in a monomorphism invariant set, i.e., \(\theta (\partial_1)=c_1\partial_1\) and \(\theta (\partial_n)=c_n\partial_n\) where \(c_1\) and \(c_n\) are non-zero elements of \(K\). So the main results of the paper can also be derived by those properties.