Differential Calculus on the Logarithmic Extension of the Quantum 3d Space and Weyl Algebra

Abstract

Noncommutative derivative operators acting on the quantum 3D space in the sense of Manin are introduced. Furthermore, the quantum 3D space is extended by the series expansion of the logarithm of the grouplike generator in the quantum 3D space. We give its differential calculus and the corresponding Weyl algebra. We also obtain algebra of Cartan-Maurer forms on this extension and the corresponding Lie algebra of vector fields. All noncommutative results are found to reduce to those of the standard commutative algebra when the deformation parameter of the quantum 3D space is set to one.

Duality on the quantum space(3) with two parameters

“…Note that we also use T ( ) = ( + + ) obtained from (35)(36)(37). >From the axioms (4-5), we also obtain the counit and antipode…”

“…In this direction, many efforts have been accomplished in order to develop noncommutative differential structures on quantum spaces(groups) [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Among them, as a fundamental work, the noncommutative differential calculus on quantum groups is introduced by Woronowicz [18].…”