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The generalized Cartan type $\mathbf{S}$ Lie algebras in char 0 with the Liebialgebra structures involved are quantized, where the Drinfel'd twist we usedis proved to be a variation of the Jordanian twist. As the passage from char 0to char p, their quantization integral forms are given. By the modularreduction and base changes, we obtain certain quantizations of the restricteduniversal enveloping algebra $\mathbf u(\mathbf{S}(n;\underline{1}))$ (for theCartan type simple modular restricted Lie algebra $\mathbf{S}(n;\underline{1})$of $\mathbf{S}$ type). They are new Hopf algebras of truncated $p$-polynomialnoncommutative and noncocommutative deformation of dimension$p^{1+(n-1)(p^n-1)}$, which contain the well-known Radford algebra (\cite{DR})as a Hopf subalgebra. As a by-product, we also get some Jordanian quantizationsfor $\mathfrak {sl}_n$, which are induced from those horizontal quantizationsof $\mathbf S(n;\underline1)$.

Quantizations of generalized Cartan type $S$ Lie algebras and of the special algebra $\mathbf{S}(n;\underline{1})$ in the modular case