On classification and construction of algebraic Frobenius manifolds

We develop the theory of generalized bi-Hamiltonian reduction. Applying this theory to a suitable loop algebra we recover a generalized Drinfeld-Sokolov reduction. This gives a way to construct new examples of algebraic Frobenius manifolds.

💡 Research Summary

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The paper develops a comprehensive framework for constructing algebraic Frobenius manifolds by exploiting a generalized bi‑Hamiltonian reduction and its equivalence with a generalized Drinfeld‑Sokolov reduction. The authors begin by recalling the Dubrovin conjecture, which predicts a one‑to‑one correspondence between massive, irreducible algebraic Frobenius manifolds with positive degrees and primitive regular conjugacy classes in Coxeter groups. While the conjecture has been verified for polynomial Frobenius manifolds (which correspond to Coxeter conjugacy classes), a systematic method for producing algebraic examples associated with arbitrary primitive conjugacy classes has been lacking.

To fill this gap, the authors introduce a bi‑Hamiltonian structure on the loop space (L(\mathfrak g)) of a simple Lie algebra (\mathfrak g). Two compatible Poisson tensors are defined: \