Involutive distributions of operator-valued evolutionary vector fields and their affine geometry. II

TL;DR

This paper extends the concept of Lie algebroids over infinite jet bundles by introducing N-tuples of differential operators with collective closure, revealing bi-differential Christoffel symbols in their algebraic structure.

Contribution

It generalizes Lie algebroid structures to include N-tuples of operators with bi-differential constants, advancing the understanding of their algebraic and geometric properties.

Findings

Introduction of bi-differential structural constants

Identification of bi-differential Christoffel symbols

Canonical structure of the operator algebra

Abstract

We generalize the notion of a Lie algebroid over infinite jet bundle by replacing the variational anchor with an N-tuple of differential operators whose images in the Lie algebra of evolutionary vector fields of the jet space are subject to collective commutation closure. The linear space of such operators becomes an algebra with bi-differential structural constants, of which we study the canonical structure. In particular, we show that these constants incorporate bi-differential analogues of Christoffel symbols.