Zeta functions of solvable Lie algebras over finite fields -- with calculations in detail

TL;DR

This paper explicitly computes zeta functions for subalgebras and ideals of low-dimensional solvable Lie algebras over finite fields, providing detailed formulas and exploring their theoretical implications.

Contribution

It offers detailed symbolic calculations of zeta functions for solvable Lie algebras of dimension up to 4 over finite fields, advancing understanding of their algebraic properties.

Findings

Explicit formulas for zeta functions of low-dimensional solvable Lie algebras

Analysis of properties and implications of these zeta functions

Connections to broader Lie algebra theory over finite fields

Abstract

Let $L$ be a solvable Lie algebra of dimension less than or equal to 4 over finite fields. We compute and record, in explicit symbolic form, the zeta functions enumerating subalgebras or ideals of $L$, and study their properties. We also discuss the implications of our data, in particular in relation to the general theory of Lie algebras over finite fields and zeta functions of Lie algebras over commutative rings.