A well-known Petersons theorem says that the number of abelian ideals in a Borel subalgebra of a rank-$r$ finite dimensional simple Lie algebra is exactly $2^r$. In this paper, we determine the dimensional distribution of abelian ideals in a Borel su
balgebra of finite dimensional simple Lie algebras, which is a refinement of the Petersons theorem capturing more Lie algebra invariants.
For symplectic Lie algebras $mathfrak{sp}(2n,mathbb{C})$, denote by $mathfrak{b}$ and $mathfrak{n}$ its Borel subalgebra and maximal nilpotent subalgebra, respectively. We construct a relationship between the abelian ideals of $mathfrak{b}$ and the c
ohomology of $mathfrak{n}$ with trivial coefficients. By this relationship, we can enumerate the number of abelian ideals of $mathfrak{b}$ with certain dimension via the Poincare polynomials of Weyl groups of type $A_{n-1}$ and $C_n$.
Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural an
alogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this paper, we use Hodge Laplacian to study the cohomology of these Lie algebras. The total rank conjecture and $b_2$-conjecture for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler-Poincare principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Botts classical result in the case of special linear Lie algebras.
