Subalgebra-preserving deformations of the Lie algebras of Witt type (Q1296182)

Algebraic geometrical aspects of deformations of Lie algebra \(L\) over the field \(k\) with a fixed maximal subalgebra \(L_0\) are investigated. In connection with this a Lie algebra \(\mathcal L\) over commutative ring \(K\) is called a deformation of \(L\) if \(\mathcal L\) is a projective \(K\)-module and \(\mathcal L/\mathfrak n\mathcal L\cong L\) over \(K/\mathfrak n\cong k\) for some maximal ideal \(\mathfrak n\) of \(K\). \(\mathcal L\) is supposed to contain a subalgebra \(\mathcal L_0\) which is a direct summand of \(\mathcal L\) as a \(K\)-module and \(\mathcal L_0/\mathfrak n\mathcal L_0\) coincides with the fixed subalgebra \(L_0\). Such a deformation is called by the author a distinguished deformation. The description of distinguished deformations \(\mathcal W\) of the Lie algebra \(W(n:m)\) of general Cartan type over a field of characteristic \(p>0\) is given. It is proved that there exists \(s\notin \mathfrak n\) such that \(\mathcal W[s^{-1}]=\mathcal W\otimes K[s^{-1}]\) may be described in terms of deformations of commuting derivations \(\partial /\partial x_i\) [see \textit{A. S. Dzhumadil'daev}, Mat. Sb. 180, No. 2, 168-186 (1989; Zbl 0691.17009)]. Under certain restrictions for a transitive Lie algebra \(L\), \(L\subset W(n:m),\) with the maximal subalgebra \(L_0\) the existence of \(s\notin \mathfrak n\) such that \((\mathcal L[s^{-1}]\), \(\mathcal L_0[s^{-1}])\) is a transitive \(K[s^{-1}]\)-subalgebra in \(\mathcal W[s^{-1}]\) by using the construction of the minimal embedding is proved.