Quantizations Of Cartan-type Lie Algebras Of H And K-series

This paper studies the quantization of Cartan type Lie algebras of H and K-series, obtains some new noncommutative and noncocommutative Hopf algebras.In chapter2, we construct explicit Drinfel'd twists for the generalized Car-tan type H Lie algebras and obtain the corresponding quantizations. Via taking modular reductions and base changes, we obtain certain modular quantizations of the restricted universal enveloping algebra u(H(2n;1)) in characteristic p. They are new Hopf algebras of truncated p-polynomial noncommutative and nonco-commutative deformation of prime-power dimension pp2n-1, which contain the well-known Radford algebra as a Hopf subalgebra. As a by-product, we also get some Jordanian quantizations for sp2n.Then we study the quantization of the generalized Contact algebra in charac-teristic0, and using the same techniques we get the quantization of the restricted enveloping algebra u(K(2n+1;1)) in characteristic p. They are new Hopf alge-bras of noncommutative and noncocommutative and with dimension pP2n+1+1(if2n+4≠0(modp)) orpp2n+1(if2n+4≡0(modp)) over a truncated p-polynomials ring, which also contain the well-known Radford algebra as a Hopf subalgebra.Since the classification of the finite-dimensional Hopf algebra seems very far off, it is quite useful to have various means of constructing examples of finite-dimensional Hopf algebras. Our results on quantizations of Cartan-type H and K Lie algebras provide many new examples of finite-dimensional Hopf algebras, which are of prime-power. This work, together with paper [29,30], completes the work of this kinds of quantizations for the Cartan type Lie algebras over a field K with charK＞7.