Lagrangian extensions and left symmetric structures on the four-dimensional real Lie superalgebras

Over real numbers, Backhouse classified all four-dimensional Lie superalgebras. From this list, we will investigate those Lie superalgebras that can be obtained as Lagrangian extensions. Moreover, we investigate left-symmetric structures on these Lie superalgebras. Furthermore, except for two of them, they are all Novikov superalgebras.

💡 Research Summary

The paper investigates four‑dimensional real Lie superalgebras, focusing on two intertwined topics: Lagrangian (or “Lagrangian”) extensions and left‑symmetric (LSS) structures. Starting from the complete classification of such superalgebras by Backhouse, the authors aim to determine which of them can be realized as T*‑extensions (or ΠT*‑extensions) of a smaller Lie superalgebra h, a construction originally introduced by Bordemann for Lie algebras and later generalized to the super setting.

A T*‑extension is built on the vector space g = h ⊕ h* (or g = h ⊕ Π(h*)) equipped with a non‑degenerate 2‑cocycle ω that is either even (symmetric) or odd (antisymmetric) depending on the parity of the form. The subspace h* (or Π(h*)) must be Lagrangian, i.e., equal to its ω‑orthogonal complement. The existence of such an ω requires a flat, torsion‑free connection ∇ on h; the connection yields a representation of h on h* (and on Π(h*)) via the dual action. The authors review these constructions, emphasizing the necessity of a quasi‑Frobenius structure (a non‑degenerate closed 2‑cocycle) on the original superalgebra.

Using the tables from Backhouse’s classification, the authors reorganize the 4‑dimensional superalgebras into six families (Tables 1‑6) according to their super‑dimension and whether the odd part brackets non‑trivially. For each family they explicitly exhibit a suitable h, a flat connection, and an ω, thereby proving that every indecomposable 4‑dimensional real Lie superalgebra admits a Lagrangian extension. A notable correction concerns the algebras denoted (D₁₀₀)₁ and (D₁₀₀)₂: previous work claimed only non‑homogeneous (i.e., odd) symplectic forms exist, but the present paper constructs both even and odd non‑degenerate closed forms, showing that both algebras can be obtained as Lagrangian extensions.

The second major contribution is the systematic construction of left‑symmetric superalgebra (LSSA) structures on all these superalgebras. Given a flat connection ∇ on a Lie superalgebra g, one defines a product x·y = ∇ₓ(y); this product satisfies the left‑symmetry condition (the associator is supersymmetric in the first two arguments) and thus endows g with an LSSA structure. The authors provide explicit formulas for the product in each case, thereby demonstrating that every 4‑dimensional real Lie superalgebra admits at least one LSSA.

Most of the constructed LSSAs satisfy the additional identities defining a Novikov superalgebra (right‑multiplication commutes and the product is left‑symmetric). Only the two algebras (D₁₀₀)₁ and (D₁₀₀)₂ fail to be Novikov; nevertheless, all of them satisfy the broader Balinsky‑Novikov (BN) conditions, which combine left‑symmetry, commutativity on the odd part, and several compatibility relations. Consequently, every algebra in the list is a BN‑superalgebra, and all but two are genuine Novikov superalgebras.

The paper also discusses the relationship between the quasi‑Frobenius form ω and the left‑symmetric product: for a quasi‑Frobenius superalgebra (g, ω) one has ω(x·y, z) = (−1)^{|x||y|} ω(y,