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Quasi-Twilled Lie Pseudolgebras and Their Deformation Maps

Sania Asif, Zhixiang Wu

TL;DR

The paper develops a unified deformation-theory framework for operator-type structures on Lie $H$-pseudoalgebras by introducing quasi-twilled Lie $H$-pseudoalgebras. It defines two deformation-map types (Type I and Type II) and constructs corresponding controlling curved $ rak{L}_fty$-pseudoalgebras, so that Maurer–Cartan elements encode the deformation data. For each type, it establishes Chevalley–Eilenberg–style cohomologies that unify established operator cohomologies (e.g., modified $r$-matrices, relative and twisted Rota–Baxter operators, Reynolds operators, crossed homomorphisms, and derivations) within the pseudoalgebra setting. The work also classifies rank-two quasi-twilled Lie $H$-pseudoalgebras and provides explicit examples (including matched pairs, semidirect and current pseudoalgebras) to illustrate broad applicability in infinite-dimensional contexts. Overall, it lays a rigorous foundation for deformation theory of operators on Lie pseudoalgebras with potential applications to vertex algebras and conformal-field theoretic structures.

Abstract

In this paper, we present a unified framework for studying cohomology theories of various operators in the context of pseudoalgebras. The central tool in our approach is the notion of a quasi-twilled Lie pseudoalgebra. We introduce two types of deformation maps. Type I unifies modified $r$ matrices, crossed homomorphisms, derivations, and homomorphisms; and Type II provides a uniform treatment of relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators, and deformation maps of matched pairs of Lie conformal algebras. We construct the corresponding controlling algebras and define cohomology theories for both types of deformation maps. These results recover existing cohomological results for known operators and yield new results, including the cohomology theory for modified $r$-matrices and deformation maps of matched pairs of Lie pseudoalgebras.

Quasi-Twilled Lie Pseudolgebras and Their Deformation Maps

TL;DR

The paper develops a unified deformation-theory framework for operator-type structures on Lie -pseudoalgebras by introducing quasi-twilled Lie -pseudoalgebras. It defines two deformation-map types (Type I and Type II) and constructs corresponding controlling curved -pseudoalgebras, so that Maurer–Cartan elements encode the deformation data. For each type, it establishes Chevalley–Eilenberg–style cohomologies that unify established operator cohomologies (e.g., modified -matrices, relative and twisted Rota–Baxter operators, Reynolds operators, crossed homomorphisms, and derivations) within the pseudoalgebra setting. The work also classifies rank-two quasi-twilled Lie -pseudoalgebras and provides explicit examples (including matched pairs, semidirect and current pseudoalgebras) to illustrate broad applicability in infinite-dimensional contexts. Overall, it lays a rigorous foundation for deformation theory of operators on Lie pseudoalgebras with potential applications to vertex algebras and conformal-field theoretic structures.

Abstract

In this paper, we present a unified framework for studying cohomology theories of various operators in the context of pseudoalgebras. The central tool in our approach is the notion of a quasi-twilled Lie pseudoalgebra. We introduce two types of deformation maps. Type I unifies modified matrices, crossed homomorphisms, derivations, and homomorphisms; and Type II provides a uniform treatment of relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators, and deformation maps of matched pairs of Lie conformal algebras. We construct the corresponding controlling algebras and define cohomology theories for both types of deformation maps. These results recover existing cohomological results for known operators and yield new results, including the cohomology theory for modified -matrices and deformation maps of matched pairs of Lie pseudoalgebras.
Paper Structure (11 sections, 34 theorems, 163 equations)

This paper contains 11 sections, 34 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Quasi-twilled Lie $H$-Pseuodalgebras
  3. Preliminaries on Pseudoalgebras
  4. The controlling algebras and cohomologies of deformation maps of type I
  5. Deformation maps of type I of a quasi-twilled Lie pseudoalgebra
  6. The controlling algebra of $\mathfrak{D}$-maps
  7. Cohomology of $\mathfrak{D}$ maps of type I
  8. The Controlling Algebras and Cohomologies of Deformation Maps of Type II
  9. Type II Deformation Maps and Operator Unification
  10. The Controlling Algebra of $\mathcal{D}$-Maps of type II
  11. Cohomologies of $\mathcal{D}$-maps

Key Result

Lemma 2.6

If $\|f\| = k_f | l_f$ and $\|g\| = k_g | l_g$, then the $NR$-bracket $[f,g]_{NR}$ has the bidegree $k_f + k_g | l_f + l_g$.

Theorems & Definitions (99)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.1
  • ...and 89 more