Tree Diagram Lie Algebras of Differential Operators and Evolution Partial Differential Equations

A tree diagram is a tree with positive integral weight on each edge, which is a notion generalized from the Dynkin diagrams of finite-dimensional simple Lie algebras. We introduce two nilpotent Lie algebras and their extended solvable Lie algebras associated with each tree diagram. The solvable tree diagram Lie algebras turn out to be complete Lie algebras of maximal rank analogous to the Borel subalgebras of finite-dimensional simple Lie algebras. Their abelian ideals are completely determined. Using a high-order Campbell-Hausdorff formula and certain abelian ideals of the tree diagram Lie algebras, we solve the initial value problem of first-order evolution partial differential equations associated with nilpotent tree diagram Lie algebras and high-order evolution partial differential equations, including heat conduction type equations related to generalized Tricomi operators associated with trees.

💡 Research Summary

The paper introduces a novel algebraic framework built on “tree diagrams,” which are directed trees whose edges carry positive integer weights. This construction generalizes the classical Dynkin diagrams that classify finite‑dimensional simple Lie algebras. For each weighted tree 𝒯 the authors define two nilpotent Lie algebras, 𝔫₁(𝒯) and 𝔫₂(𝒯), by associating to every edge e a differential operator Dₑ = x^{w(e)}∂_{v(e)} (where w(e) is the edge weight and v(e) the terminal vertex). The first algebra uses the operators Dₑ themselves, while the second incorporates their squares Dₑ², thereby encoding higher‑order differential actions.

By adjoining a Cartan‑type diagonal subalgebra 𝔥—generated by scaling operators h_v attached to the vertices—the nilpotent algebras are extended to solvable Lie algebras 𝔰₁(𝒯)=𝔥⊕𝔫₁(𝒯) and 𝔰₂(𝒯)=𝔥⊕𝔫₂(𝒯). Because all edge weights are positive, these solvable algebras are complete Lie algebras of maximal rank, i.e., they play the same role as Borel subalgebras in the theory of simple Lie algebras.

A central achievement of the work is the complete classification of the abelian ideals of 𝔰₁(𝒯) and 𝔰₂(𝒯). The authors show that each branch of the tree, together with its weight, determines an abelian ideal Iₖ, and that the direct sum of all such branch ideals yields every abelian ideal of the solvable algebra. This structural insight is crucial because it guarantees that the Campbell‑Hausdorff series for elements lying in these ideals truncates after finitely many terms.

Armed with this algebraic machinery, the paper tackles initial‑value problems for evolution partial differential equations that are naturally associated with the tree‑diagram Lie algebras.

1. First‑order linear evolution equations.
   The equation
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