We find the defining structures of two-parameter quantum groups U r,s (g) corresponding to the orthogonal and the symplectic Lie algebras, which are realized as Drinfel'd doubles. We further investigate the environment conditions upon which the Lusztig's symmetries exist between (U r,s (g), , ) and its associated object (U s −1 ,r −1 (g), | ).

The discussions in the present paper arise from exploring intrinsically the structural nature of the quantum n-space. A kind of braided category of -graded θ-commutative associative algebras over a field k is established. The quantum divided power algebra over k related to the quantum n-space is introduced and described as a braided Hopf algebra in (in terms of its 2-cocycle structure), over which the so-called special q-derivatives are defined so that several new interesting quantum groups, especially the quantized polynomial algebra in n variables (as the quantized universal enveloping algebra of the abelian Lie algebra of dimension n) and the quantum group associated to the quantum n-space, are derived from our approach independently of using the R-matrix. As a verification of its validity for our discussion, the quantum divided power algebra is equipped with the structure of a U q n -module algebra via certain q-differential operators' realization. Particularly, one of the four kinds of root vectors of U q n in the sense of Lusztig can be specified precisely under the realization.

We quantize the generalized-Witt algebra in characteristic 0 with its Lie bialgebra structures discovered by Song-Su [G. Song, Y. Su, Lie bialgebras of generalized-Witt type, arXiv: math.QA/0504168, Sci. China Ser. A 49 (4) (2006) 533-544]. Via a modulo p reduction and a modulo "p-restrictedness" reduction process, we get 2 n − 1 families of truncated p-polynomial noncocommutative deformations of the restricted universal enveloping algebra of the Jacobson-Witt algebra W(n; 1 ) (for the Cartan type simple modular restricted Lie algebra of W type). They are new families of noncommutative and noncocommutative Hopf algebras of dimension p 1+np n in characteristic p. Our results generalize a work of Grunspan [C. Grunspan, Quantizations of the Witt algebra and of simple Lie algebras in characteristic p, J. Algebra 280 (2004) 145-161] in rank n = 1 case in characteristic 0. In the modular case, the argument for a refined version follows from the modular reduction approach (different from [C. Grunspan, Quantizations of the Witt algebra and of simple Lie algebras in characteristic p, J. Algebra 280 (2004) 145-161]) with some techniques from the modular Lie algebra theory.

Abstract. We further define two-parameter quantum affine algebra Ur,s( c sln) (n > 2) after the work on the finite cases (see, which turns out to be a Drinfel'd double. Of importance for the quantum affine cases is that we can work out the compatible two-parameter version of the Drinfel'd realization as a quantum affinization of Ur,s(sln) and establish the Drinfel'd isomorphism Theorem in the two-parameter setting, via developing a new combinatorial approach (quantum calculation) to the quantum affine Lyndon basis we present (with an explicit valid algorithm based on the use of Drinfel'd generators).

Mainly motivated by Pirashvili's spectral sequences on a Leibniz algebra, a cohomological characterization of Leibniz central extensions of Lie algebras is given. In particular, as applications, we obtain the cohomological version of Gao's main theorem for Kac-Moody algebras and answer a question in an earlier paper by Liu and Hu (2004).