Independence Polynomials of 2-step Nilpotent Lie Algebras
Marco Aldi, Thor Gabrielsen, Daniele Grandini, Joy Harris, Kyle Kelley
TL;DR
This work extends the graph-theoretic notion of independence to arbitrary finite-dimensional 2-step nilpotent Lie algebras by defining an independence polynomial $I(\mathfrak g,t)$ via basic cohomology, unifying combinatorics with Lie theory. It proves a Hansen-type upper bound generalizes to all such algebras and provides a new, algebraic lower bound for the independence number, along with an application to bound the dimension of abelian subalgebras. A metric-dependent basic Laplacian framework is introduced, linking $I(\mathfrak g,t)$ to a basic partition function $Z_{\mathfrak g,g}(s,t)$ and, in the Dani-Mainkar case, to a four-variable generalized subgraph counting polynomial; the approach yields exact calculations in Heisenberg algebras and reveals connections to Johnson graph spectra. The results bridge graph theory and Lie algebra cohomology, offering a quantum-mechanical interpretation and laying groundwork for hypergraph extensions and further cross-pollination between combinatorics and 2-step nilpotent Lie theory.
Abstract
Motivated by the Dani-Mainkar construction, we extend the notion of independence polynomial of graphs to arbitrary 2-step nilpotent Lie algebras. After establishing efficiently computable upper and lower bounds for the independence number, we discuss a metric-dependent generalization motivated by a quantum mechanical interpretation of our construction. As an application, we derive elementary bounds for the dimension of abelian subalgebras of 2-step nilpotent Lie algebras.