Double extension of flat pseudo-Riemannian \(F\)-Lie algebras (Q6668532)

Frobenius manifolds emerged as a geometric incarnation of the Witten-Dijkgaaf-Vertinde-Verlinde equation within the framework of two-dimensional topological field theories. \textit{B. Dubrovin} [Lect. Notes Math. 1620, 120--348 (1996; Zbl 0841.58065)] pioneered the study of these geometric objects. \textit{C. Hertling} and \textit{Yu. Manin} [Int. Math. Res. Not. 1999, No. 6, 277--286 (1999; Zbl 0960.58003)] introduced the notion of an \(F\)-manifold, a geometric construction relaxing some conditions of a Frobenius manifold. This paper centers on \(F\)-manifolds abiding by an additional weak Frobenius-like identity, introducing a double extension process [\textit{A. Aubert} and \textit{A. Medina}, Tôhoku Math. J. (2) 55, No. 4, 487--506 (2003; Zbl 1058.53055)] tailored to constructing and exploring flat pseudo-Riemannian \(F\)-Lie algebras. The authors leverage concepts from Nijenhuis cohomology for left-symmetric algebras [\textit{A. Nijenhuis}, Enseign. Math. (2) 14, 225--277 (1968; Zbl 0188.08101)] and Hochschild cohomology for associative algebras [\textit{G. Hochschild}, Ann. Math. (2) 46, 58--67 (1945; Zbl 0063.02029)].\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] introduces the concept of a flat pseudo-Riemannian \(F\)-structure on a smooth manifold with a specific focus on Lie groups and Lie algebras, addressing extensions of this structure and its relation to Frobenius manifolds.\N\N\item[\S 3] presents a method for constructing \(\left( n+2\right) \)-dimensional (weakly) flat pseudo-Riemannian \(F\)-Lie algebras from \(n\)-dimensional (weakly) ones, which involves an explicit set of parameters (structure coefficients) abiding by specific algebraic equations. All weakly flat Lorenzian non-abelian bi-nilpotent \(F\)-Lie algebras with 1-dimensional light-cone subspaces that also function as associative two-sided ideals are obtained as a key result. The non-existence of flat Lorentzian non-abelian bi-nilpotent \(F\)-Lie algebras with certain properties is additionally established. These results are extended to nilpotent Lie algebras rigged with flat scalar products of signature \(\left( 2,n-2\right) \)\ for \(n\geq4\).\N\N\item[\S 4] describes how the core construction is to be adapted to generate two additional double extension processes applicable to Lie algebras of \(F\)-strong symmetric type and Frobenius-like Poisson algebras.\N\N\item[\S 5] provides concrete examples.\N\end{itemize}