A quantum space and some associated quantum groups

TL;DR

This paper introduces a quantum n-space with a Hopf algebra structure, constructs a related derivation algebra as a quantum group, and develops a bicovariant differential calculus on it.

Contribution

It constructs a new quantum n-space with a Hopf algebra structure and establishes its associated derivation algebra as a noncommutative quantum group, along with a differential calculus.

Findings

Quantum n-space with Hopf algebra structure introduced.

Derivation algebra shown to be a noncommutative quantum group.

Bicovariant differential calculus constructed for the quantum n-space.

Abstract

In this paper, we first introduce a quantum $n$-space with a cocommutative Hopf algebra structure. Then it is shown that to this quantum $n$-space there corresponds a derivation algebra of $\sigma$-twisted derivations related to some algebra automorphisms $\sigma$ on the quantum $n$-space. Furthermore, we show that this derivation algebra is a noncommutative and non-cocommutative Hopf algebra, namely, a quantum group. Morever, for this the quantum $n$-space, we show how to construct a bicovariant differential calculus related to the derivation algebra.