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Journal of Lie Theory 16 (2006), No. 4, 691--618
Copyright Heldermann Verlag 2006

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

Xiaoping Xu
Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, P. R. China

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.

Keywords: Tree diagram, Lie algebra of differential operators, abelian ideal, evolution partial differential equation, Campbell-Hausdorff.

MSC: 17B30, 35F15, 35G15; 35C15, 35Q58

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