Center and Lie algebra of outer derivations for algebras of differential operators associated to hyperplane arrangements

TL;DR

This paper computes the center and outer derivations of differential operator algebras linked to hyperplane arrangements, using homological algebra and Lie-Rinehart tools, with specific results for braid and reflection arrangements.

Contribution

It provides explicit descriptions of the center and outer derivations for these algebras, extending understanding of their algebraic structure in geometric contexts.

Findings

Center computed for specific arrangements

Outer derivations characterized for braid and reflection arrangements

Results utilize homological algebra and Lie-Rinehart frameworks

Abstract

We compute the center and the Lie algebra of outer derivations of a familiy of algebras of differential operators associated to hyperplane arrangements of the affine space A 3. The results are completed for 4-braid arrangements and for reflection arrangements associated to the wreath product of a cyclic group with the symmetric group S 3. To achieve this we use tools from homological algebra and Lie-Rinehart algebras of differential operators.