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Derived operations satisfy standard identities

Vladimir Dotsenko

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.

Derived operations satisfy standard identities

Abstract

A derived operation is a bilinear operation on a commutative associative algebra 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 , satisfies standard identities.

Paper Structure

This paper contains 13 sections, 3 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Examples of derived operations
  3. Finite-dimensional Lie algebras of derivations
  4. Derived operations in classical mechanics and the theory of PDEs
  5. Novikov algebras and related examples
  6. Rankin--Cohen algebras
  7. Derived operations for one-dimensional subalgebras of DerA
  8. More general finite sets of derivations
  9. "Almost Poisson" brackets defined by a bivector field
  10. Higher brackets of the universal deformation quantization formulas
  11. An example of Dzhumadildaev
  12. The main results
  13. A generalization for higher number of arguments

Key Result

Theorem 2

Every derived operation $\{-,-\}$ defined using $n>1$ derivations $D_1$, …, $D_n$ satisfies a standard identity: there exists $d>0$ such that we have for all $a_1,\ldots,a_d\in A$. In fact, we can take $d=1+\frac{n^{p+1}-1}{n-1}$ for any $p\ge m+1$, where $m$ is the $D$-order of $\{-,-\}$.

Theorems & Definitions (7)

  • Definition 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 4
  • proof