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The index of nilpotent Lie algebras

Dietrich Burde, Karel Dekimpe

TL;DR

This work provides explicit formulas for the index $\chi(\mathfrak{g})$ of several nilpotent Lie algebras, tying the invariant to concrete combinatorial and algebraic data. By translating the problem into ranks of bracket matrices and skew-symmetric forms, the authors derive exact values for $\chi(F_{g,2})$ and $\chi(F_{g,3})$, as well as for 2-step graph Lie algebras via the matching number and Lovász’s skew-symmetric adjacency matrix results. They also determine the index and abelian-subalgebra dimensions for metabelian free-nilpotent quotients $M_{g,c}$ and analyze filiform algebras, including a construction yielding a range of indices $\chi(\mathfrak{g}_{n,k})=n-k+1$ for odd $k$, plus known results for graded cases. The findings connect algebraic invariants to combinatorial structures and graded-decomposition data, aiding representation-theoretic and invariant-theoretic applications of nilpotent Lie algebras.

Abstract

The index of a Lie algebra is an important invariant which arises in several areas, e.g. in the study of coadjoint orbits for a Lie group, in invariant theory and in representation theory. We study the index for several classes of nilpotent Lie algebras. In particular, we give explicit formulas for free-nilpotent Lie algebras of nilpotency class two and three, or of solvability class two, for graph Lie algebras, and for filiform nilpotent Lie algebras.

The index of nilpotent Lie algebras

TL;DR

This work provides explicit formulas for the index of several nilpotent Lie algebras, tying the invariant to concrete combinatorial and algebraic data. By translating the problem into ranks of bracket matrices and skew-symmetric forms, the authors derive exact values for and , as well as for 2-step graph Lie algebras via the matching number and Lovász’s skew-symmetric adjacency matrix results. They also determine the index and abelian-subalgebra dimensions for metabelian free-nilpotent quotients and analyze filiform algebras, including a construction yielding a range of indices for odd , plus known results for graded cases. The findings connect algebraic invariants to combinatorial structures and graded-decomposition data, aiding representation-theoretic and invariant-theoretic applications of nilpotent Lie algebras.

Abstract

The index of a Lie algebra is an important invariant which arises in several areas, e.g. in the study of coadjoint orbits for a Lie group, in invariant theory and in representation theory. We study the index for several classes of nilpotent Lie algebras. In particular, we give explicit formulas for free-nilpotent Lie algebras of nilpotency class two and three, or of solvability class two, for graph Lie algebras, and for filiform nilpotent Lie algebras.
Paper Structure (6 sections, 18 theorems, 77 equations)

This paper contains 6 sections, 18 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The index of free-nilpotent Lie algebras
  4. The index of graph Lie algebras
  5. The index of metabelian free-nilpotent Lie algebras
  6. The index of filiform nilpotent Lie algebras

Key Result

Proposition 2.3

Let $\mathfrak{g}$ be a Lie algebra over an algebraically closed field $K$ of characteristic zero. Then $\alpha(\mathfrak{g})$ and $\chi(\mathfrak{g})$ satisfy the following properties.

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 35 more