Narrow Results

We formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical Frobenius theorem, which says that an involutive $C^1$ distribution is uniquely integrable.

Generalizing Lie algebras, we consider anti-commutative algebras with skew-symmetric identities of degree > 3. Given a skew-symmetric polynomial f, we call an anti-commutative algebra f-Lie if it satisfies the identity f = 0. If s_n is a standard skew-symmetric polynomial of degree n, then any s₄-Lie algebra is f-Lie if deg f ≥ 4. We describe a free anti-commutative super-algebra with one odd generator. We exhibit various constructions of generalized Lie algebras, for example: given any derivations D, F of an associative commutative algebra U, the algebras (U, D ∧ F) and (U, id ∧ D²) are s₄-Lie. An algebra (U, id ∧ D³ - 2D ∧ D²) is s'₅-Lie, where s'₅ is a non-standard skew-symmetric polynomial of degree 5.

This paper aims to present a general idea for description of finite physical objects with consistent internal dynamical structure and evolution as a whole making use of the mathematical concepts and structures connected with the Frobenius integrability/nonintegrability theorems and to present an example. The idea consists in to consider some distribution Δ_o of vector fields on a manifold, then to separate some integrable subdistribution Δ ⊂ Δ_o, representing the integrity of the object considered. The curvatures of all nonintegrable subdistributions of Δ will be interpreted as generators of processes of internal energy-momentum exchange, i.e. of the internal dynamics of the object. The curvatures of distributions including vector fields from Δ_o and Δ will be interpreted as generators of interaction of the physical object with the outside world. Example of photon-like objects is considered in detail.