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Frölicher-Nijenhuis bracket and derived bracket associated to a nonsymmetric operad with multiplication

Anusuiya Baishya, Apurba Das

TL;DR

This work develops two graded Lie brackets on a nonsymmetric operad with multiplication, namely the Frölicher-Nijenhuis bracket and a derived bracket, whose Maurer-Cartan elements correspond to Nijenhuis and Rota-Baxter elements, respectively. It establishes differential graded Lie algebra structures, representations, and cohomologies in this operadic setting, and derives a deformation-theoretic framework linking multiplicative structures to operator theory on Loday-type algebras. The authors provide explicit bracket formulas and demonstrate their applicability to a range of associative-like algebras, including dendriform, Hom-associative, and associative conformal algebras, with special cases recovering classical results. Overall, the paper unifies and extends operator theory in a broad operadic context, offering concrete tools for studying Nijenhuis and Rota-Baxter operators across binary quadratic operads.

Abstract

This paper aims to construct two graded Lie algebras associated with a nonsymmetric operad with multiplication. Maurer-Cartan elements of these graded Lie algebras correspond respectively to Nijenhuis elements and Rota-Baxter elements for the given multiplication. Explicit forms of these brackets are given to study Nijenhuis operators and Rota-Baxter operators on some Loday-type algebras.

Frölicher-Nijenhuis bracket and derived bracket associated to a nonsymmetric operad with multiplication

TL;DR

This work develops two graded Lie brackets on a nonsymmetric operad with multiplication, namely the Frölicher-Nijenhuis bracket and a derived bracket, whose Maurer-Cartan elements correspond to Nijenhuis and Rota-Baxter elements, respectively. It establishes differential graded Lie algebra structures, representations, and cohomologies in this operadic setting, and derives a deformation-theoretic framework linking multiplicative structures to operator theory on Loday-type algebras. The authors provide explicit bracket formulas and demonstrate their applicability to a range of associative-like algebras, including dendriform, Hom-associative, and associative conformal algebras, with special cases recovering classical results. Overall, the paper unifies and extends operator theory in a broad operadic context, offering concrete tools for studying Nijenhuis and Rota-Baxter operators across binary quadratic operads.

Abstract

This paper aims to construct two graded Lie algebras associated with a nonsymmetric operad with multiplication. Maurer-Cartan elements of these graded Lie algebras correspond respectively to Nijenhuis elements and Rota-Baxter elements for the given multiplication. Explicit forms of these brackets are given to study Nijenhuis operators and Rota-Baxter operators on some Loday-type algebras.
Paper Structure (7 sections, 21 theorems, 118 equations)

This paper contains 7 sections, 21 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Nonsymmetric operad with multiplication
  3. Cup bracket
  4. Nijenhuis elements and the Frölicher-Nijenhuis bracket
  5. Rota-Baxter elements and the derived bracket
  6. Averaging elements
  7. Frölicher-Nijenhuis bracket and derived bracket for some Loday-type algebras

Key Result

Theorem 2.3

Let $\mathcal{P}$ be a nonsymmetric operad. Then the graded vector space $\mathcal{P}_{\bullet +1}:= \bigoplus_{n \geq 0} \mathcal{P}_{n+1}$ equipped with the bracket for $f \in \mathcal{P}_{m+1}, ~ \! g \in \mathcal{P}_{n+1}$ forms a graded Lie algebra.

Theorems & Definitions (51)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • ...and 41 more