On Derivation Deviations in an Abstract Pre-Operad

Abstract

We consider basic algebraic constructions associated with an abstract pre-operad, such as a -algebra, total composition •, precoboundary operator δ and tribraces {h, f, g}. A derivation deviation of the pre-coboundary operator over the tribraces is calculated in terms of the -multiplication and total composition. * ORDER REPRINTS 1610KLUGE AND PAAL over the tribraces {·, ·, ·}, dev {·,·,·} δ (h ⊗ f ⊗ g)

The Operadic Modeling of Gauge Systems of the Yang-Mills Type

“…As a matter of fact, these deformations can be described by using a deformed boundary operator∂ ↤ ∂, see [1,10,11,12] for more detailed exposition. Theorem 6.2 (operadic Stokes law [10,11,12], cf [1]). The variations of the operadic flows result as superpositions over the corresponding deformed boundaries, δ =δ ∂ i.eδ ⟨⋅ ⋅⟩ =δ ∂ ⟨⋅⟩ ⟨⋅ ⋅⟩ ,δ ⟨⋅ ⋅ ⋅⟩ =δ ∂ ⟨⋅⋅⟩ ⟨⋅ ⋅ ⋅⟩ ,δ ⟨⋅ ⋯⟩ = ⟨⋅ ⋯⟩ , .…”

“…By performing computations, the result reads as follows. Corollary 6.4 (operadic Stokes law [10,11], cf [1]). In an operad C, the low-order variations of flows read…”

“…We can now state the Leibniz rule in tangent cohomology. Theorem 6.7 (Leibniz rule in tangent cohomology [11], cf. [1]).…”

Note on operadic harmonic oscillator

“…As a variation of this one can pose the following question [7]: how to describe the time evolution of the linear algebraic operations (multiplications) V ⊗n → V . The algebraic operations (multiplications) can be seen as an example of the operadic variables [2,3,4,5].…”

Note on 2d binary operadic harmonic oscillator

“…The algebraic operations (multiplications) can be seen as an example of the operadic variables [2,3,4,5]. If an operadic system depends on time one can speak about operadic dynamics [7].…”