The Fourth-Linear <inline-formula><math display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi mathvariant="bold-fraktur">aff</mi><mo>(</mo><mn>1</mn><mo>)</mo></mstyle></mrow></semantics></math></inline-formula>-Invariant Differential Operators and the First Cohomology of the Lie Algebra of Vector Fields on <inline-formula><math display="inline"><semantics><mstyle mathvariant="bold"><msup><mi mathvariant="double-struck">RP</mi><mn>1</mn></msup></mstyle></semantics></math></inline-formula>

Abstract

In this paper, we denote the Lie algebra of smooth vector fields on RP1 by V(RP1). This paper focuses on two parts. In the beginning, we determine the cohomology space of aff(1) with coefficients in Dτ,λ,μ;ν. Afterward, we classify aff(1)-invariant fourth-linear differential operators from V(RP1) to Dτ,λ,μ;ν vanishing on aff(1). This result enables us to compute the aff(1)-relative cohomology of V(RP1) with coefficients in Dτ,λ,μ;ν.

Keywords

• invariant differential operators
• Lie algebra of vector fields
• cohomology