[논문 리뷰] Equivariant Nijenhuis Lie Algebras: Extensions to Classical Lie-Theoretic Structures

Shuai Hou, Zohreh Ravanpak외 1명|arXiv (Cornell University)|2026. 01. 27.

Advanced Topics in Algebra인용 수 0

한 줄 요약

본 논문은 equivariant Nijenhuis Lie algebras (ENL algebras) 이론을 전개하여 ENL bialgebras, matched pairs, Manin triples, Drinfel’d doubles를 확립하고, ENL r-matrices와 relative Rota-Baxter operators를 고전적인 Yang–Baxter 이론과 연결한다. 또한 pre-ENL 구조를 통해 pre-ENL 프레임워크로 확장한다.

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.

연구 동기 및 목표

• Introduce ENL algebras: Lie algebras with equivariant Nijenhuis operators.
• Develop ENL bialgebra theory and duality via matched pairs and Manin triples.
• Extend ENL framework to Rota-Baxter Lie algebras and EN-relative Rota-Baxter operators.
• Study EN-r matrices and the equivariant classical Yang–Baxter equation.
• Extend to pre-Lie algebras and define pre-ENL structures.

제안 방법

• Define equivariant Nijenhuis operators and ENL algebras with adjoint-equivariance E∘ad_x = ad_x∘E.
• Establish representations compatible with E (EN-representations) and duals.
• Develop matched pairs of ENL algebras and prove the bicrossed product yields a ENL double.
• Introduce ENL-RB algebras and EN-relative Rota-Baxter operators to realize ENL r-matrices.
• Characterize coboundary ENL bialgebras via EN r-matrices and study doubles.
• Extend the framework to pre-Lie algebras with weak/strong equivariance (pre-ENL).

실험 결과

연구 질문

• RQ1 How can the classical Lie bialgebra correspondences (matched pairs, Manin triples, Drinfel’d doubles) be extended to the category of ENL algebras?
• RQ2 What are the operator-theoretic realizations (Rota-Baxter, relative Rota-Baxter) of ENL r-matrices and their role in coboundary ENL bialgebras?
• RQ3 How does equivariance interact with duality and doubles in ENL structures?
• RQ4 How can ENL theory be extended to pre-Lie algebras and what new structures arise (pre-ENL)?

주요 결과

• ENL algebras with equivariant Nijenhuis operators yield ENL brackets and a hierarchy of ENL structures.
• Matched pairs of ENL algebras correspond to ENL doubles via a bicrossed product, preserving ENL structure.
• Quadratic ENL algebras admit an S-sharp isomorphism linking adjoint and coadjoint EN representations.
• EN-relatively Rota-Baxter operators realize ENL r-matrices and induce coboundary ENL bialgebras.
• The ENL framework extends to pre-Lie algebras, producing pre-ENL algebras compatible with ENL constructions.