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

TL;DR

This paper develops a new framework for quantum groups using quantum divided power algebras and q-derivatives, leading to novel constructions of quantum polynomial algebras and quantum groups without relying on R-matrices.

Contribution

It introduces a braided Hopf algebra structure for quantum divided power algebras and constructs quantum groups via q-derivatives independently of R-matrix methods.

Findings

Quantum divided power algebra forms a braided Hopf algebra in a new category.

Construction of quantum groups from q-derivatives without R-matrix.

Realization of Lusztig's root vectors within the new framework.

Abstract

The discussions in the present paper arise from exploring intrinsically the structure nature of the quantum $n$-space. A kind of braided category $\Cal {GB}$ of $\La$-graded $\th$-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 $\Cal {GB}$ (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 $U_q(\frak {sl}_n)$-module algebra via a certain $q$-differential operators realization. Particularly, one of the four kinds of roots vectors of $U_q(\frak {sl}_n)$ in the sense of Lusztig can be specified precisely under the realization.