Modular quantizations of Lie algebras of Cartan type K via Drinfeld twists of Jordanian type

We construct explicit Drinfel'd twists of Jordanian type for the generalized Cartan type K Lie algebras in characteristic 0 and obtain the corresponding quantizations, especially their integral forms. By making modular reductions including modulo p and modulo p-restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra u(K(2n+1;1_)) for the restricted simple Lie algebra of Cartan type K in characteristic p. They are new families of noncommutative and noncocommutative Hopf algebras of dimension pp2n+1+1 (if 2n+4. ≢0. (mod. p)) or pp2n+1 (if 2n+ 4. ≢ 0(mod. p)) over a truncated p-polynomials ring, which also contain the well-known Radford algebras (see [20]) as Hopf subalgebras. Some open questions are proposed.