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Equivariant Nijenhuis Lie Algebras: Extensions to Classical Lie-Theoretic Structures

Shuai Hou, Zohreh Ravanpak, Yunhe Sheng

TL;DR

This work introduces equivariant Nijenhuis Lie (ENL) algebras, integrating a Nijenhuis operator $E$ that commutes with the adjoint action to yield a simplified $[x,y]_E=[Ex,y]$ bracket and a hierarchical ENL structure. It develops ENL analogues of matched pairs, Manin triples, and Drinfel'd doubles, establishing a robust duality theory and showing that ENL doubles correspond to quadratic ENL algebras. The paper then enlarges the framework to Rota-Baxter theory, proving that ENL structures are preserved under Rota-Baxter descents and that EN $r$-matrices generate coboundary ENL bialgebras, including extensions to semidirect products via EN-relative Rota-Baxter operators. It further generalizes to ENL versions of the classical Yang–Baxter equation and demonstrates how EN-relative RB operators yield descendent ENL algebras and EN $r$-matrices, with a parallel development for pre-Lie algebras leading to pre-ENL algebras. Overall, the framework provides a symmetric, dualizable algebraic setting linking Lie theory, operator theory, and integrability, with potential applications to quantization and geometric mechanics.

Abstract

We develop a structural theory of equivariant Nijenhuis Lie algebras (ENL algebras), namely, Lie algebras equipped with Nijenhuis operators satisfying an equivariance condition with respect to the adjoint representation. This rigidity allows classical Lie bialgebra constructions to extend systematically to the operator-equipped setting. Within this framework, we define ENL bialgebras and establish the associated notions of matched pairs, Manin triples, and Drinfel'd doubles. We show that coboundary ENL bialgebras are characterized by EN $r$-matrices satisfying an equivariant classical Yang-Baxter equation. We further introduce EN-relative Rota-Baxter operators and prove that they provide an operator-theoretic realization of such $r$-matrices, leading to descendent ENL algebras and to solutions of the classical Yang--Baxter equation on semidirect ENL algebras. In the quadratic case, this construction reduces to Rota-Baxter operators of weight zero. Finally, we extend the EN framework to pre-Lie algebras and show that pre-ENL algebras naturally induce associated ENL structures.

Equivariant Nijenhuis Lie Algebras: Extensions to Classical Lie-Theoretic Structures

TL;DR

This work introduces equivariant Nijenhuis Lie (ENL) algebras, integrating a Nijenhuis operator that commutes with the adjoint action to yield a simplified bracket and a hierarchical ENL structure. It develops ENL analogues of matched pairs, Manin triples, and Drinfel'd doubles, establishing a robust duality theory and showing that ENL doubles correspond to quadratic ENL algebras. The paper then enlarges the framework to Rota-Baxter theory, proving that ENL structures are preserved under Rota-Baxter descents and that EN -matrices generate coboundary ENL bialgebras, including extensions to semidirect products via EN-relative Rota-Baxter operators. It further generalizes to ENL versions of the classical Yang–Baxter equation and demonstrates how EN-relative RB operators yield descendent ENL algebras and EN -matrices, with a parallel development for pre-Lie algebras leading to pre-ENL algebras. Overall, the framework provides a symmetric, dualizable algebraic setting linking Lie theory, operator theory, and integrability, with potential applications to quantization and geometric mechanics.

Abstract

We develop a structural theory of equivariant Nijenhuis Lie algebras (ENL algebras), namely, Lie algebras equipped with Nijenhuis operators satisfying an equivariance condition with respect to the adjoint representation. This rigidity allows classical Lie bialgebra constructions to extend systematically to the operator-equipped setting. Within this framework, we define ENL bialgebras and establish the associated notions of matched pairs, Manin triples, and Drinfel'd doubles. We show that coboundary ENL bialgebras are characterized by EN -matrices satisfying an equivariant classical Yang-Baxter equation. We further introduce EN-relative Rota-Baxter operators and prove that they provide an operator-theoretic realization of such -matrices, leading to descendent ENL algebras and to solutions of the classical Yang--Baxter equation on semidirect ENL algebras. In the quadratic case, this construction reduces to Rota-Baxter operators of weight zero. Finally, we extend the EN framework to pre-Lie algebras and show that pre-ENL algebras naturally induce associated ENL structures.
Paper Structure (7 sections, 39 theorems, 147 equations, 1 figure)

This paper contains 7 sections, 39 theorems, 147 equations, 1 figure.

Table of Contents

  1. Equivariant Nijenhuis Lie ( ENL) algebras
  2. Matched pairs and Manin triples of ENL algebras
  3. Structure Theory of ENL Bialgebras
  4. Equivariant Nijenhuis structures on Rota-Baxter Lie algebras: ENL-RB algebras
  5. EN $r$-matrices and the classical Yang-Baxter equation
  6. EN-Relative Rota-Baxter operators on ENL algebras
  7. Equivariant Nijenhuis operators on pre-Lie algebras: pre- ENL algebras

Key Result

Lemma 1.2

Let $(\mathfrak g,[\cdot,\cdot]_\mathfrak g,E)$ be an ENL algebra. Then $E$ is a Nijenhuis operator.

Figures (1)

  • Figure 1: The compatibility condition \ref{['E-rep']} for an $N$-representation. The identity expresses the vanishing of the signed sum of all compositions of the maps $T$, $\rho(x)$, and $\rho(Nx)$ along paths from the lower-left copy of $W$ to the upper-right one. The dotted arrows indicate the weaker intertwining relations associated with an averaging operator; see Remark \ref{['aver-com']}.

Theorems & Definitions (95)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Proposition 1.3
  • proof
  • Corollary 1.4
  • Remark 1.5
  • Proposition 1.6
  • proof
  • Proposition 1.7
  • ...and 85 more