A classification of nilpotent compatible Lie algebras

Technical Summary: A Classification of Nilpotent Compatible Lie Algebras

Problem Statement
The paper addresses the classification problem for nilpotent compatible Lie algebras over an arbitrary field of characteristic different from 2. A compatible Lie algebra is defined as a vector space $\mathfrak{g}$ equipped with two Lie brackets, $[-,-]$ and $\{-,-\}$, such that any linear combination $\lambda[-,-] + \mu\{-,-\}$ remains a Lie bracket. This condition is equivalent to satisfying a "mixed Jacobi identity." While compatible Lie algebras arise in various contexts, including the classical Yang-Baxter equation, integrable systems, and deformation theory, a complete structural classification, particularly for the nilpotent case, had not been established up to dimension 4. The authors aim to fill this gap by extending existing classification techniques to this specific algebraic variety.

Methodology
The authors generalize the Skjelbred-Sund method, originally developed for classifying nilpotent Lie algebras, to the setting of algebras with two products. The core of their approach relies on the following theoretical framework:

1. Cohomology Theory: Utilizing the cohomology theory for compatible Lie algebras introduced in [15], the authors define central extensions via 2-cocycles. They establish that any finite-dimensional nilpotent compatible Lie algebra can be constructed as a sequence of central extensions starting from the zero-dimensional algebra.
2. Central Extensions and Admissibility: A central extension $\mathfrak{g}_\omega$ of a compatible Lie algebra $\mathfrak{g}$ by a vector space $V$ is constructed using a cocycle $\omega \in Z^2(\mathfrak{g}, V)$. The authors focus on "full" central extensions where the center of the extension is exactly $V$. They introduce the concept of "admissible" cocycles, which ensure the extension is full and does not possess "central components" (direct summands of the form $\mathfrak{i} \oplus Kx$ where $x$ is central).
3. Isomorphism Criterion: The paper proves that two full central extensions $\mathfrak{g}_\omega$ and $\mathfrak{g}_\theta$ are isomorphic if and only if their corresponding cocycles lie in the same orbit under the action of the automorphism group $\text{Aut}(\mathfrak{g})$ and the general linear group $\text{GL}(V)$. Specifically, they reduce the classification problem to determining the orbits of subspaces in the second cohomology group $H^2(\mathfrak{g}, K)$ under the action of $\text{Aut}(\mathfrak{g})$.
4. Algorithmic Classification: An inductive algorithm is proposed:
   • Start with the classification of nilpotent compatible Lie algebras up to dimension $n-1$.
   • For each algebra $\mathfrak{g}$ in the list, compute its center, automorphism group, and second cohomology group.
   • Identify admissible cocycles (those with trivial annihilator intersection with the center).
   • Compute the orbits of the action of $\text{Aut}(\mathfrak{g})$ on the relevant subspaces of $H^2(\mathfrak{g}, K)$.
   • Construct extensions for each orbit and add algebras with central components (direct sums with 1-dimensional centers).

Key Contributions

• Extension of Skjelbred-Sund: The paper successfully adapts the Skjelbred-Sund method to the variety of compatible Lie algebras, proving the necessary theorems regarding central extensions and isomorphism criteria for algebras with two products.
• Handling of Skew-Isomorphism: The authors explicitly define and distinguish between isomorphism ($\cong$) and skew-isomorphism ($\cong$), where the two brackets are swapped. They provide a complete list of skew-isomorphisms among the classified algebras, noting that some algebras are self-skew-isomorphic while others form pairs.
• Explicit Classification: The authors provide a complete, explicit classification of all nilpotent compatible Lie algebras up to dimension 4.

Results
The classification is presented for a base field $K$ of characteristic $\neq 2$. The results are summarized as follows:

• Dimensions 1 and 2: There is only one isomorphism class in each dimension, both being abelian.
• Dimension 3:
  • There are 3 isomorphism classes and 1 one-parameter family.
  • The classes include the abelian algebra, two algebras where only one bracket is non-trivial (isomorphic to the Heisenberg algebra in one bracket and abelian in the other), and a family where both brackets are non-trivial and proportional.
• Dimension 4:
  • There are 12 isomorphism classes, 6 one-parameter families, and 1 two-parameter family.
  • The classification includes algebras with central components (direct sums of lower-dimensional algebras) and non-trivial extensions.
  • The isomorphism classes depend on the field properties. Specifically, for families involving parameters $\beta$, the algebras $NCL_{4,9}^\beta$ and $NCL_{4,14}^\beta$ are isomorphic if and only if the ratio of their parameters has a cube root in $K$.
  • If the base field is cubically closed (every element has a cube root), the number of distinct isomorphism classes reduces: the one-parameter families collapse to single classes, and the two-parameter family reduces to a one-parameter family.

The paper provides a comprehensive table (Section 4.5) listing all algebras up to dimension 4, their defining relations, and the dimension of their centers. Section 4.6 details the skew-isomorphism relationships, showing that many algebras in the list are skew-isomorphic to others in the list (e.g., $NCL_{3,2} \cong NCL_{3,3}$), while some are self-skew-isomorphic.

Significance
The paper claims significance by providing the first complete classification of nilpotent compatible Lie algebras in low dimensions. This work serves as a foundational step for further structural analysis of these algebras, which are relevant in mathematical physics (e.g., bi-Hamiltonian structures and integrable systems). By establishing the cohomological machinery and the classification algorithm, the authors enable future extensions to higher dimensions and other varieties of compatible algebraic structures. The results also clarify the relationship between the two Lie brackets, demonstrating that the compatibility condition imposes specific structural constraints that distinguish these algebras from simple pairs of Lie algebras.