Deformation maps on quasi-twilled Lie conformal algebras

Technical Summary: Deformation Maps on Quasi-Twilled Lie Conformal Algebras

Problem Statement
The paper addresses the need for a unified algebraic framework to study various operator-theoretic structures on Lie conformal algebras. While the theory of Lie conformal algebras (introduced by Kac and developed by Bakalov, D'Andrea, and Kac) provides the algebraic language for the operator product expansion in two-dimensional conformal field theory, the study of specific operators—such as Rota-Baxter operators, derivations, crossed homomorphisms, and modified $r$-matrices—has often been treated in isolation. Recent developments in the theory of Lie algebras (specifically by Jiang, Sheng, and Tang) introduced the concept of "quasi-twilled" structures and "deformation maps" to unify these operators. However, this unification had not yet been extended to the conformal setting. The authors aim to generalize the theory of deformation maps to quasi-twilled Lie conformal algebras, establishing a coherent cohomological and homotopical framework that encompasses both classical and conformal operators.

Methodology
The authors employ a unified approach based on the following mathematical tools:

1. Quasi-Twilled Lie Conformal Algebras: They utilize a decomposition $E = \mathfrak{a} \oplus \mathfrak{b}$ where $\mathfrak{b}$ is a subalgebra, but $\mathfrak{a}$ is not necessarily one. The structure is encoded by a 2-cochain $\Omega$ which admits a bihomogeneous decomposition with respect to the Nijenhuis-Richardson (NR) bigrading.
2. Nijenhuis-Richardson Bracket: The NR bracket and its bigrading properties are used to decompose the Maurer-Cartan equation $[\Omega, \Omega]_{NR} = 0$ into a hierarchical system of compatibility conditions.
3. Derived Bracket Construction: Following Voronov's method, the authors construct controlling homotopy algebras. They define projection maps on the space of cochains to generate $L_\infty$-algebras (or curved $L_\infty$-algebras) where the specific operators of interest appear as Maurer-Cartan elements.
4. Deformation Theory: By twisting the controlling $L_\infty$-algebras with a given deformation map, the authors derive the governing equations for infinitesimal deformations ($D + D'$ or $B + B'$).
5. Cohomology: They define Chevalley-Eilenberg-type cochain complexes associated with the induced structures on the summands to define the cohomology of the deformation maps.

Key Contributions and Results

The paper introduces two dual families of operators on a quasi-twilled Lie conformal algebra $(E, \mathfrak{a}, \mathfrak{b})$:

1. Right Deformation Maps ($D: \mathfrak{a} \to \mathfrak{b}$)

• Definition: A $C[\partial]$-module homomorphism $D$ is a right deformation map if its graph $\text{Gr}(D)$ is a Lie conformal subalgebra of $E$.
• Unification: This single definition subsumes:
  • Conformal modified $r$-matrices.
  • Conformal crossed homomorphisms.
  • Conformal derivations.
  • Conformal Lie algebra homomorphisms.
• Induced Structures: $D$ induces a new Lie conformal algebra structure on $\mathfrak{a}$ (denoted $\mathfrak{a}_D$) and a module structure on $\mathfrak{b}$.
• Controlling Algebra: The authors prove that right deformation maps are precisely the Maurer-Cartan elements of a curved $L_\infty$-algebra on $\bigoplus_{n \geq 0} C^{n+1}(\mathfrak{a}, \mathfrak{b})$. The structure maps are explicitly given by iterated NR brackets involving the components of $\Omega$.
• Cohomology: A cohomology complex is defined for $D$. The second cohomology group $H^2(D)$ classifies infinitesimal deformations of $D$.

2. Left Deformation Maps ($B: \mathfrak{b} \to \mathfrak{a}$)

• Definition: A $C[\partial]$-module homomorphism $B$ is a left deformation map if its graph $\text{Gr}(B)$ is a Lie conformal subalgebra of $E$.
• Unification: This definition subsumes:
  • Conformal relative Rota-Baxter operators (including O-operators).
  • Conformal twisted Rota-Baxter operators.
  • Conformal Reynolds operators.
  • Deformation maps of matched pairs of Lie conformal algebras.
• Induced Structures: $B$ induces a new Lie conformal algebra structure on $\mathfrak{b}$ (denoted $\mathfrak{b}_B$) and a module structure on $\mathfrak{a}$.
• Controlling Algebra: Left deformation maps are shown to be Maurer-Cartan elements of an (uncurved) $L_\infty$-algebra on $\bigoplus_{n \geq 0} C^{n+1}(\mathfrak{b}, \mathfrak{a})$. The structure maps involve up to ternary brackets ($l_3$) derived from the NR bracket.
• Cohomology: A cohomology complex is defined for $B$. The second cohomology group $H^2(B)$ classifies infinitesimal deformations.

Significance and Claims
The paper claims to provide a "unified approach" that recovers and extends existing theories. Specifically:

• Generalization: It extends the work of Jiang, Sheng, and Tang (who worked on quasi-twilled Lie algebras) to the conformal setting.
• Unification: It demonstrates that diverse operators (Rota-Baxter, Reynolds, derivations, etc.) are not isolated phenomena but specific instances of deformation maps within a single quasi-twilled framework.
• New Results: The authors note that several of the resulting controlling algebras and deformation cohomologies appear to be new even within the conformal setting, particularly for conformal modified $r$-matrices and deformation maps of matched pairs.
• Theoretical Framework: By establishing the $L_\infty$-algebraic control, the paper provides the precise obstruction-theoretic description for the deformation of these operators, linking the algebraic structure directly to deformation theory via the Maurer-Cartan formalism.

The paper concludes that this framework successfully integrates the cohomological and deformation theories of all the mentioned classical and conformal operators under the umbrella of quasi-twilled Lie conformal algebras.