Noncommutative geometry on the universal envelopping algebra of the Borel subgroup U[sb(2)]

TL;DR

This paper explores the noncommutative geometric structure of the Borel algebra as a manifold, deriving differential relations, cohomology, and solutions to the wave equation within this framework.

Contribution

It introduces a detailed noncommutative geometric analysis of the Borel algebra, including differential forms, derivatives, cohomology, and wave solutions.

Findings

Derived noncommutative differential form relations.

Calculated the de Rham cohomology of the algebra.

Obtained the wave operator and magnetic solutions.

Abstract

We study the Borel algebra de ne by [x a ; x b ] = 2 a;1 x b as a noncommutative manifold R 3 . We calculate its noncommutative di erential form relations. We deduce its partial derivative relations and the derivative of a plane wave. After calculating its de Rham cohomology, we deduce the wave operator and its corresponding magnetic solution.