Invitation to operadic dynamics

Abstract

Based on the Gerstenhaber Theory, clarification is made of how operadic dynamics may be introduced. Operadic observables satisfy the Gerstenhaber algebra identities and their time evolution is governed by operadic evolution equation. The notion of an operadic Lax pair is also introduced. As an example, an operadic (representation of) harmonic oscillator is proposed. MSC: 18D50, 70G60

Dynamical deformations of three-dimensional Lie algebras in Bianchi classification over the harmonic oscillator

“…Dynamics in operadic systems (operadic dynamics) may be introduced by Definition 3.1 (operadic Lax pair [6]). Allow a classical dynamical system to be described by the Hamiltonian system (1.1).…”

“…If an operadic system depends on time one can speak about operadic dynamics [6]. The latter may be introduced by simple and natural analogy with the Hamiltonian dynamics.…”

“…Thus, from the algebraic point of view, mechanical systems can be described by linear operators, i.e by linear maps V → V of a vector space V . As a generalization of this one can pose the following question [6]: how can the time evolution of the linear operations (multiplications) V ⊗n → V be described?…”

Quantum counterparts of three-dimensional real Lie algebras over harmonic oscillator

“…If an operadic system depends on time, one can speak about operadic dynamics [7]. The latter may be introduced by simple and natural analogy with the Hamiltonian dynamics by using the Gerstenhaber brackets instead of the commutator bracketing in the Lax equation (2).…”

“…by linear maps V → V of a vector space V . As a generalization of this one can pose the following question [7]: how can the time evolution of the linear operations (multiplications) V ⊗ → V be described?…”