For q generic or q=a primitive l-th root of 1, q-Witt algebras are described by means of q-divided power algebras. The structure of the universal q-central extension of the q-Witt algebra, the q-Virasoro algebra Virq, is also determined. q-Lie algebras are investigated and the q-PBW theorem for the universal enveloping algebras of q-Lie algebras is proved. A realization of a class of representations of the q-Witt algebras is given. Based on it, the q-holomorph structure for the q-Witt algebras is constructed, which interprets the realization in the context of epresentation theory.

The discussions in the present paper arise from exploring intrinsically the structure nature of the quantum n-space. A kind of braided category GB of-graded-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 GB (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 of our discussion, the quantum divided power algebra is equipped with a structure of Uq(sl n)-module algebra via a certain q-differential operators realization. Particularly, one of the four kinds of roots vectors of Uq(sl n) in the sense of Lusztig can be specified precisely under the realization.

In this paper, all one-dimensional Leibniz central extensions on the algebras of differential operators over C[t; t¡1] and C((t)), as well as on the quantum 2-torus, the Virasoro-like algebra and its q-analog are studied. We determine all nontrivial Leibniz 2-cocycles on these infinite dimensional Lie algebras.

For any Virasoro-toroidal Lie algebra of type X (1) l (X=A; B; C; D; E; F;G), we give some explicit irreducible representations fromthe vertex operators and some oscillator representations of Virasoro algebra.

The purpose of this work is to construct a class of homogeneous vertex representations for the extended toroidal Lie algebra of type G2, which are showed to be completely reducible.

In this paper, we mainly study the Steinberg unitary Leibniz algebra stul(n,D) defined over a unital dialgebra D. We consider its universal central extension and obtain some results as in the Steinberg unitary Lie algebra case.

The universal central extensions and their extension kernels of the matrix Lie superalgebra sl (m, n,A), the Steinberg Lie superalgebra st(m, n,A) in category SLeib of Leibniz superalgebras are determined under a weak assumption (compared with [MP]) using the first Hochschild homology and the first cyclic homology group.

We find the defining structures of two-parameter quantum groups Ur,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 (Ur,s(g), h,i) and its associated object (Us−1,r−1 (g), h | i).

We investigate the finite-dimensional representation theory of twoparameter quantum orthogonal and symplectic groups that we found in [BGH] under the assumption that rs−1 is not a root of unity and extend some results [BW1, BW2] obtained for type A to types B, C and D. We construct the corresponding R-matrices and the quantum Casimir operators, by which we prove that the complete reducibility Theorem also holds for the categories of finite-dimensional weight modules for types B, C, D.