Quantum supergroup structure of the 1+1-dimensional quantum superplane, its dual and its differential calculus

TL;DR

This paper demonstrates that the 1+1-dimensional quantum superplane is a quantum supergroup, constructs its dual and differential calculus, and reveals its quantum Lie superalgebra structure with a non-cocommutative Hopf algebra.

Contribution

It introduces a new perspective by identifying the quantum superplane as a quantum supergroup and constructs its dual and differential calculus within this framework.

Findings

The quantum superplane is a quantum supergroup with explicit supermatrix and R-matrix.

A dual Hopf superalgebra is realized from initial pairings.

A right-invariant differential calculus and associated quantum Lie superalgebra are constructed.

Abstract

We show that the 1+1-dimensional quantum superplane introduced by Manin is a quantum supergroup according to the Faddeev-Reshetikhin-Takhtajan approach. We give its supermatrix element, its corresponding R-matrix and its Hopf structure. This new point of view allows us, first, to realize its dual Hopf superalgebra starting from postulated initial pairings. Second, we construct a right-invariant differential calculus on it and then deduce the corresponding quantum Lie superalgebra which as a commutation super-algebra appears classical, and as Hopf structure is a non-cocommutative q-deformed one. An isomorphism between the latter and the dual one obtained in the first method is given.