$A$-Generalized Hessian pre-Lie algebras and $A$-Generalized Yang--Baxter Equations

Abstract: Inspired by the problem of constructing ($ω$-)pre-Lie algebra structures on the dual space of a pre-Lie algebra, we introduce the (A)-generalized Yang--Baxter equation as a generalization of the Yang--Baxter equation of pre-Lie algebras. We study its symmetric solutions through (A)-generalized Hessian pre-Lie algebras and split these solutions into two types. We further consider factorizable solutions of this equation and establish a one-to-one correspondence between them and generalized quadratic Rota--Baxter pre-Lie algebras of nonzero weight. By studying the structure of these algebras, we find all factorizable solutions. Finally, we study the structure of (A)-generalized Hessian pre-Lie algebras. In particular, we obtain a structural description via central and double extensions and classify low-dimensional non-trivial (A)-generalized Hessian pre-Lie algebras.

• The paper introduces A-generalized Yang–Baxter equations that extend classical conditions to pre-Lie algebras using right annihilator elements.
• It employs operator-theoretic and geometric approaches to establish a one-to-one correspondence with generalized Hessian pre-Lie algebras.
• It provides explicit low-dimensional classifications and shows that factorizable solutions exist only on two-dimensional abelian structures.

$A$-Generalized Hessian pre-Lie Algebras and $A$-Generalized Yang--Baxter Equations

Introduction and Motivation

The paper develops a comprehensive framework for understanding and classifying generalized algebraic structures arising from the study of Yang--Baxter equations on pre-Lie algebras. Motivated by the question of when a pre-Lie algebra structure can be constructed on the dual of a given pre-Lie algebra, the authors generalize the classical (and pre-Lie) Yang--Baxter equation to what they call the $A$-generalized Yang--Baxter equation (AYBE). This generalization is parameterized by an element $u$ of the right annihilator, $\operatorname{Ann}_{\mathrm R}(A)$, of the pre-Lie algebra $A$.

The analysis unifies and extends known relationships between the classical Yang--Baxter equation, Rota--Baxter operators, Hessian pre-Lie algebras, and their generalizations, while providing structural results and low-dimensional classifications for the associated new algebraic objects.

$A$-Generalized Yang--Baxter Equation and Structures on Duals

The $A$-generalized Yang--Baxter equation is defined for a pre-Lie algebra $(A, \cdot)$ and $u \in \operatorname{Ann}_{\mathrm R}(A)$ by

$A$0

with $A$1. This equation extends the classical case ($A$2) and leads to a direct criterion for when certain tensors $A$3 endow $A$4, the dual, with a multiplicative $A$5-)pre-Lie algebra structure via a derived bilinear product.

Explicitly, $A$6 solves the AYBE if and only if the dual structure $A$7, defined by

$A$8

with $A$9, is a multiplicative $A$0-pre-Lie algebra.

The framework covers both symmetric and factorizable solutions, generalizing the previous rigid dichotomy in the theory of the Yang--Baxter equation (YBE).

Symmetric Solutions and $A$1-Generalized Hessian pre-Lie Algebras

Symmetric solutions ($A$2) to the $A$3-generalized $A$4-equation are encapsulated in the notion of $A$5-generalized Hessian pre-Lie algebras. These are quadruples $A$6, with $A$7 a nondegenerate symmetric bilinear form on $A$8 satisfying a specific $A$9-generalized $u$0-cocycle condition: $u$1 The authors show a one-to-one correspondence between nondegenerate symmetric solutions of the $u$2-generalized $u$3-equation and these $u$4-generalized Hessian pre-Lie algebras.

Symmetric solutions are split into:

• Type 1: $u$5 where $u$6, corresponding to $u$7-generalized Hessian pre-Lie subalgebras of $u$8.
• Type 2: $u$9, associated with pairs $\operatorname{Ann}_{\mathrm R}(A)$0, where $\operatorname{Ann}_{\mathrm R}(A)$1 with a modified product and $\operatorname{Ann}_{\mathrm R}(A)$2 forms a Hessian pre-Lie algebra.

Operator-Theoretic and Geometric Correspondences

Symmetric solutions of the generalized $\operatorname{Ann}_{\mathrm R}(A)$3-equation are further characterized via $\operatorname{Ann}_{\mathrm R}(A)$4-generalized (relative) Rota--Baxter operators. Given a linear map $\operatorname{Ann}_{\mathrm R}(A)$5 symmetric with respect to the dual pairing, $\operatorname{Ann}_{\mathrm R}(A)$6 is a symmetric solution if and only if

$\operatorname{Ann}_{\mathrm R}(A)$7

This operator-theoretic viewpoint both generalizes and includes the classical results for pre-Lie and Lie algebras.

Geometrically, the $\operatorname{Ann}_{\mathrm R}(A)$8-generalized Hessian cocycle condition is related to the total symmetry of $\operatorname{Ann}_{\mathrm R}(A)$9 for a left-invariant affine connection and pseudo-Riemannian metric on the associated Lie group, linking to the theory of locally conformally Hessian manifolds.

Factorizable Solutions and Generalized Rota--Baxter Pre-Lie Algebras

For factorizable AYBE solutions ($A$0 invertible and $A$1 $A$2-invariant), the authors develop a theory of $A$3-generalized quadratic Rota--Baxter pre-Lie algebras of nonzero weight. For such $A$4, the weight-$A$5 operator $A$6 satisfies a specific identity involving both $A$7 and $A$8: $A$9 and $A$0 is quadratic, with compatibility between $A$1 and $A$2. These structures are in bijective correspondence with factorizable solutions.

In a remarkable constraint, all nontrivial $A$3-generalized quadratic Rota--Baxter pre-Lie algebras (and thus all factorizable solutions of the AYBE) are shown to exist only on two-dimensional abelian Lie algebras.

Structural Results and Double/Central Extensions

The paper provides a structural theorem for $A$4-generalized Hessian pre-Lie algebras, showing that every such algebra is obtained:

• By a central (one-dimensional annihilator) extension of a Hessian pre-Lie algebra, when $A$5 is non-isotropic for $A$6.
• As a double extension by combining annihilator and derivation data, when $A$7 is isotropic.

These descriptions give rise to explicit construction algorithms and enable full low-dimensional classification.

Low-Dimensional Classification

The authors carry out a complete classification of three-dimensional $A$8-generalized Hessian pre-Lie algebras over $A$9 with nonzero $A$0, cataloging all isomorphism classes (distinguished by the form $A$1 and the structure of the annihilator) and relating them to commutative, associative, and non-abelian pre-Lie structures as in Bai's work.

Each algebra is specified by its multiplication table and the nonzero entries of the symmetric form $A$2; a precise moduli description for the parameters is given in each case. The classification recovers all symmetric solutions (type 1 and type 2) of the AYBE in these dimensions, confirming the general structural results.

Implications and Outlook

This comprehensive approach to the $A$3-generalized Yang--Baxter equation and its symmetries:

• Unifies operator and tensor formulations for the construction of dual algebraic structures.
• Extends the classical relationships between YBE solutions and bialgebra or bialgebroid structures to the pre-Lie and $A$4-pre-Lie settings.
• Shows intrinsic limitations: for instance, the strong constraint that factorizable solutions exist only in abelian, two-dimensional settings.
• Connects to geometric theories of locally conformal Hessian manifolds, suggesting links with geometry of flat connections and invariant metrics.

Looking forward, this framework could be extended to yet more general settings: higher arity ($A$5-ary) operations, non-associative or superalgebra analogues, or homotopic generalizations. Connections to deformation quantization, integration of pre-Lie bialgebraic structures, and further geometric applications are anticipated directions. The results on double and central extensions provide powerful tools for explicit construction and classification in broader algebraic and geometric contexts.

Conclusion

The systematic development of $A$6-generalized Hessian pre-Lie algebras and their correspondence to solutions of the $A$7-generalized Yang--Baxter equations fosters a deep unification of algebraic, operadic, and geometric themes in the theory of pre-Lie algebras. The explicit structural and classification results open avenues for future research in generalized bialgebraic structures, noncommutative geometry, and their applications in mathematical physics.