Derived operations satisfy standard identities

TL;DR

This paper proves that derived operations in certain algebraic structures inherently satisfy standard identities, with implications for modular forms and deformation quantization.

Contribution

It demonstrates that operators of derived operations always satisfy standard identities, linking algebraic structures to modular forms and deformation quantization.

Findings

Operators of derived operations satisfy standard identities.

Rankin-Cohen brackets of modular forms satisfy standard identities.

Higher brackets in deformation quantization satisfy standard identities.

Abstract

A derived operation is a bilinear operation on a commutative associative algebra $A$ defined intrinsically out of its product and several derivations of the product. We show that operators of left (or right) multiplications of a derived operation always satisfy a "standard identity" of certain order. In particular, it implies that each Rankin-Cohen bracket of modular forms, as well as each higher bracket of Kontsevich's universal deformation quantization formula for Poisson structures on $\mathbb{R}^n$, satisfies standard identities.