Poisson-commutative subalgebras and the Feigin-Frenkel centre: relations, interactions, and new perspectives.

• Applicant: Professorin Dr. Oksana Yakimova
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 575190730

Let L be a Lie algebra. We are interested in commutative subalgebras of the enveloping algebra U(L) and Poisson-commutative subalgebras of the symmetric algebra S(L). These two types of objects are closely related and there is a fruitful interplay between them. Let 𝜽 be an automorphism of finite order of a reductive Lie algebra g. Then there is a Poisson-commutative subalgebra Z(g,𝜽) of S(g) associated to 𝜽. We aim to show that for many automorphism, Z(g,𝜽) has nice geometric and algebraic properties. Another goal is to lift Z(g,𝜽) to a commutative subalgebra of U(g). The Feigin--Frenkel (FF-) centre is a remarkable commutative subalgebra of U(L), where L is the current algebra associated to a simple Lie algebra g. It has many applications to Hamiltonian and quantum integrable systems. Explicit formulas for generators of the FF-centre are known, if g is a classical Lie algebra. We intend to find explicit formulas for the generators of the FF-centre for exceptional g. We plan to find generalisations of the FF-centre in the context of non-reductive finite-dimensional Lie algebras and to a 𝜽-twisted current algebra.

• DFG Programme: Research Grants