A complete description of solvable symplectic Lie algebras

Technical Summary: A Complete Description of Solvable Symplectic Lie Algebras

Problem Statement
The classification of symplectic Lie algebras remains a challenging open problem, even in low dimensions. While the symplectic double extension process (introduced in [15] and refined in [7]) offers a method for constructing symplectic Lie algebras from simpler components, its application has been limited. Specifically, existing generalizations rely on the existence of normal isotropic ideals or utilize left-symmetric products, which become difficult to apply when the isotropic ideal has a dimension greater than or equal to two, or even when it is one-dimensional but not normal. Consequently, a complete inductive description of solvable symplectic Lie algebras via this process has been lacking, particularly for cases involving non-normal isotropic ideals.

Methodology
The authors employ a structural approach based on the symplectic double extension process, generalized to handle isotropic ideals of dimension one and two. The methodology proceeds as follows:

1. Generalized Matched Pairs: The paper introduces the concept of a generalized matched pair (Definition 2.3), a quintuple $(B, H, L_B, L_H, \Phi)$ involving two Lie algebras and specific linear maps. This framework allows for the construction of a Lie algebra $g = B \oplus H$ where $H$ is a subalgebra but $B$ is not necessarily one, extending the standard matched pair notion.
2. Symplectic Double Extension Construction:
   • By a Line (Section 3): The authors define a symplectic double extension of a symplectic Lie algebra $(a, [\cdot, \cdot]_a, \omega_a)$ by a line $\mathbb{R}$. This involves constructing $g = \mathbb{R}d_0 \oplus a \oplus \mathbb{R}e_0$ with a specific bracket structure determined by an admissible element $(a_a, b_a, D, \epsilon)$. The conditions for this to form a symplectic Lie algebra are derived using Chevalley-Eilenberg cohomology and representation theory.
   • By a Plane (Section 4): The process is extended to a plane $\mathbb{R}^2$, constructing $g = \mathbb{R}^2 \oplus a \oplus (\mathbb{R}^2)^*$. This involves a generalized semi-direct product followed by a generalized bicrossed product, governed by a more complex set of admissible parameters.
3. Inductive Reduction: The paper establishes that any solvable symplectic Lie algebra possessing a nonzero isotropic ideal can be decomposed. By applying Lie's theorem to the representation of the algebra on the isotropic ideal, the authors prove the existence of an isotropic ideal of dimension 1 or 2. This allows the algebra to be viewed as a symplectic double extension of a lower-dimensional symplectic Lie algebra (the symplectic reduction).
4. Structural Analysis of Irreducible Cases: For algebras without nonzero isotropic ideals (symplectically irreducible), the paper provides a new purely algebraic proof of a structural theorem characterizing these algebras, specifically showing they are 2-step solvable with a nondegenerate derived ideal.

Key Contributions and Results

• Complete Characterization of Solvable Symplectic Lie Algebras: The primary result (Theorem 5.1) states that any solvable symplectic Lie algebra with a nonzero isotropic ideal is a symplectic double extension by a line or a plane of another solvable symplectic Lie algebra. Consequently, any such algebra can be constructed via a finite sequence of these extensions starting from a symplectically irreducible algebra (Corollary 5.1).
• Unimodularity and Solvability: The paper demonstrates that if a symplectic Lie algebra has a nondegenerate derived ideal, it is necessarily unimodular and solvable (Theorem 5.3). It also provides necessary and sufficient conditions for a symplectic double extension to preserve unimodularity (Corollary 3.2, Proposition 4.2).
• New Algebraic Proof of Structural Theorem: The authors provide a new, completely algebraic proof (Theorem 6.2) for the structure of symplectically irreducible symplectic Lie algebras. They prove that such algebras are 2-step solvable, possess a zero center, and decompose into an orthogonal direct sum of a nondegenerate abelian ideal $[g, g]$ and a nondegenerate abelian subalgebra $[g, g]^\perp$, with a faithful, skew-symmetric action of the latter on the former.
• Classification in Low Dimensions:
  • The paper proves that no nonzero symplectically irreducible symplectic Lie algebras exist in dimensions 2 and 4 (Corollary 6.3).
  • It classifies all 6-dimensional symplectically irreducible symplectic Lie algebras, showing they are isomorphic to a specific algebra defined by two independent 2-dimensional symplectic ideals (Proposition 6.2).
• Limitations of the Process: The paper presents an example (Example 6.2) of an 8-dimensional solvable symplectic Lie algebra constructed from a 6-dimensional irreducible one. It highlights that this specific 8-dimensional algebra possesses no isotropic ideal of dimension 1 or 2 that would allow it to be constructed via the double extension process starting from the zero algebra, illustrating the necessity of the "symplectically irreducible" base case.

Significance
The paper claims to resolve the gap in the inductive description of solvable symplectic Lie algebras. By extending the symplectic double extension process to handle isotropic ideals of dimension one and two (even when not normal), the authors provide a constructive framework that covers all solvable symplectic Lie algebras. This framework reduces the classification problem to the identification of symplectically irreducible algebras, for which a complete structural characterization is provided. The work also clarifies the relationship between the nondegeneracy of the derived ideal, unimodularity, and solvability, offering a rigorous algebraic foundation for understanding the structure of these geometric-algebraic objects.