Lie trusses and heaps of Lie affebras

Abstract

A frame-independent formulation of Lie brackets on affine spaces or Lie affebras introduced in [K. Grabowska, J. Grabowski & P. Urbański, Lie brackets on affine bundles, Ann. Global Anal. Geom. 24 (2003), 101-130] is given.

Heaps of Linear Connections and Their Endomorphism Truss

“…If we have both left and right distributivity, then we speak of a truss. We divert the reader to the original literature on trusses for more details (see [7][8][9][10][11]). A video lecture outlining the theory of trusses, including motivation and a historical perspective, is available in [17].…”

“…An interesting observation is that the set of endomorphisms of linear connections forms a truss. The latter structures are ring-like algebraic structures in which the binary addition is replaced with a heap operation, together with some natural distributivity axioms (see [7][8][9][10][11]). We explicitly construct the endomorphism truss of linear connections on a Lie algebroid (or just an anchored vector bundle) in this note.…”

Affine Nijenhuis operators and Hochschild cohomology of trusses

“…Let A be an affine space over an F-vector space → A. As explained for example in [3,Section 4] or [7], the action + of (5.4)…”

On Lie semiheaps and ternary principal bundles

“…Even in high school physics, energy, voltage and position are not 'absolutely measurable', but one needs to fix a zero point and measure things relative to this chosen point -mathematically one should be thinking in terms of torsors. A gauge or frame-independent formulation of analytical dynamics requires affine bundles and, because of this, Grabowska, Grabowski and Urbański (see [12]) defined Lie brackets on sections of affine bundles -this too has been reformulated by Brzeziński using ternary operations (see [8]). Semiheaps and ternary algebras have been applied to quantum mechanics (see [4,16,17]).…”