Quantum Divided Power Algebra, Q-Derivatives, and Some New Quantum Groups

The discussions in the present paper arise from exploring intrinsically the structural nature of the quantum n-space. A kind of braided category of Λ-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 Uq( n)-module algebra via certain q-differential operatorsʹ realization. Particularly, one of the four kinds of root vectors of Uq( n) in the sense of Lusztig can be specified precisely under the realization.