TL;DR
This paper proves that N-graded Lie algebras of type FP with entropy ≤ 1 are finite-dimensional, and Koszul Lie algebras with entropy ≤ 1 are abelian, using a generalized Witt formula and necklace polynomial analysis.
Contribution
It introduces a generalized Witt formula for N-graded Lie algebras of type FP and establishes new entropy bounds leading to finiteness and abelian properties.
Findings
N-graded Lie algebras of type FP with entropy ≤ 1 are finite-dimensional
Koszul Lie algebras with entropy ≤ 1 are abelian
Generalized Witt formula and necklace polynomial analysis underpin these results
Abstract
It will be shown that every N-graded Lie algebra generated in degree 1 of type FP with entropy less or equal to 1 must be finite-dimensional (cf. Thm. A). As a consequence every Koszul Lie algebra with entropy less or equal to 1 must be abelian (cf. Cor. C). These results are obtained from a generalized Witt formula (cf. Thm. D) for N-graded Lie algebras of type FP and the analysis of necklace polynomials at roots of unity.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Nonlinear Waves and Solitons · Advanced Topics in Algebra