Independence Polynomials of 2-step Nilpotent Lie Algebras

Daniele Grandini; Joy Harris; Kyle Kelley; Marco Aldi; Thor Gabrielsen

arxiv: 2504.19836 · v2 · submitted 2025-04-28 · 🧮 math.RA · math.CO· quant-ph

Independence Polynomials of 2-step Nilpotent Lie Algebras

Marco Aldi , Thor Gabrielsen , Daniele Grandini , Joy Harris , Kyle Kelley This is my paper

Pith reviewed 2026-05-22 18:13 UTC · model grok-4.3

classification 🧮 math.RA math.COquant-ph

keywords independence polynomial2-step nilpotent Lie algebraabelian subalgebraDani-Mainkar constructiongraph theorymetric generalizationquantum mechanical interpretation

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The pith

Independence polynomials extend from graphs to 2-step nilpotent Lie algebras and bound the dimensions of their abelian subalgebras.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper lifts the independence polynomial from graphs to any 2-step nilpotent Lie algebra through the Dani-Mainkar construction. This produces a polynomial whose coefficients and evaluations yield efficiently computable upper and lower bounds on an independence number that counts maximal sets of pairwise non-commuting elements. A further version incorporates a metric on the algebra, drawing from a quantum mechanical reading of the same construction. The resulting bounds then translate directly into elementary estimates for the dimension of the largest abelian subalgebra. A sympathetic reader cares because the approach supplies a combinatorial counting device for a basic structural question in Lie algebra theory.

Core claim

By extending the notion of independence polynomial of graphs to arbitrary 2-step nilpotent Lie algebras, one obtains a well-defined polynomial whose value at 1 furnishes bounds on the independence number; these bounds in turn give elementary upper estimates on the dimension of any abelian subalgebra.

What carries the argument

The independence polynomial obtained by lifting the Dani-Mainkar construction from the associated graph to the Lie algebra, where independent sets correspond to sets of elements whose brackets satisfy the nilpotency and commutativity conditions.

If this is right

Upper and lower bounds on the independence number become efficiently computable for any 2-step nilpotent Lie algebra.
The dimension of every abelian subalgebra is at most the independence number obtained from the polynomial evaluated at 1.
A metric-dependent version of the polynomial arises naturally and admits a quantum-mechanical reading.
These dimension bounds require only elementary counting and do not depend on solving the full classification problem for the algebra.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same lifting technique could be tested on other families of nilpotent Lie algebras to see whether similar counting polynomials exist.
The quantum-mechanical motivation might connect the polynomial to operator-algebra questions outside pure Lie theory.
The polynomial might function as a coarse invariant that separates some non-isomorphic 2-step nilpotent Lie algebras.

Load-bearing premise

The Dani-Mainkar construction lifts directly to every 2-step nilpotent Lie algebra and defines a well-behaved independence polynomial without extra structural restrictions on the algebra.

What would settle it

Compute the lifted independence polynomial for the three-dimensional Heisenberg algebra and compare its derived upper bound against the known dimension of the largest abelian subalgebra, which is two.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends independence polynomials to 2-step nilpotent Lie algebras via Dani-Mainkar but the resulting bounds may depend on basis choice rather than being intrinsic.

read the letter

The core move is to associate a graph to a 2-step nilpotent Lie algebra using the bracket map from the Dani-Mainkar construction and then carry over the independence polynomial. They derive elementary upper and lower bounds on the independence number and apply them to bound the dimension of abelian subalgebras, with a side discussion of a metric version tied to a quantum interpretation. That extension itself is new in this setting and the bounds look straightforward to compute, which could be handy for explicit calculations on low-dimensional examples or for quick checks during classification work. The quantum-motivated generalization is a reasonable extra direction even if it stays speculative. The main soft spot is the one flagged in the stress test. If the graph (and therefore the independence number) changes with the choice of basis for the generating space or the center, then the bounds are not invariants of the algebra and the claimed application to abelian subalgebras becomes presentation-dependent. The abstract does not spell out how they ensure the construction is well-defined up to isomorphism, so that point needs explicit verification in the proofs. No obvious circularity or heavy parameter fitting shows up. This is aimed at people already working on combinatorial aspects of nilpotent Lie algebras or on independence polynomials in algebraic settings. A reader who needs concrete bounds for small cases or who wants to explore the quantum link might get something out of it. I would send it to peer review; the central idea is coherent enough that referees can check the invariance question and tighten the bounds if needed.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the independence polynomial from graphs to arbitrary 2-step nilpotent Lie algebras by associating a graph to the algebra via the Dani-Mainkar construction applied to the bracket map from the generating vector space V to the center. It derives efficiently computable upper and lower bounds on the independence number, introduces a metric-dependent generalization motivated by a quantum-mechanical interpretation, and applies the construction to obtain elementary bounds on the maximum dimension of abelian subalgebras.

Significance. If the construction is shown to be basis-independent and the bounds are rigorously derived, the work would provide a concrete link between graph-theoretic invariants and the structure theory of nilpotent Lie algebras, with potential utility for bounding abelian subalgebras. The computability emphasis and the quantum-motivated generalization are noted strengths, though the latter appears exploratory.

major comments (2)

[§2] Definition of the independence polynomial (likely §2): the graph obtained from the Dani-Mainkar construction on the bracket map must be shown to be independent of the choice of basis for V and the center. If different bases produce non-isomorphic graphs, the independence number ceases to be an intrinsic invariant of the Lie algebra, rendering the derived bounds on abelian subalgebra dimension presentation-dependent rather than well-defined.
[§3] Bounds on the independence number (likely §3): the upper and lower bounds are stated to be efficiently computable, but the manuscript must supply explicit derivations or algorithms that do not rely on auxiliary choices; without these, it is unclear whether the bounds hold for the algebra itself or only for specific presentations.

minor comments (2)

[Generalization section] The quantum-mechanical motivation for the metric-dependent generalization would benefit from a concrete low-dimensional example illustrating how the metric enters the polynomial.
[Introduction] A brief recall of the original Dani-Mainkar construction (for graphs or for Lie algebras) in the introduction would improve accessibility for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying these important points regarding the intrinsic nature of the construction. We address each major comment below.

read point-by-point responses

Referee: [§2] Definition of the independence polynomial (likely §2): the graph obtained from the Dani-Mainkar construction on the bracket map must be shown to be independent of the choice of basis for V and the center. If different bases produce non-isomorphic graphs, the independence number ceases to be an intrinsic invariant of the Lie algebra, rendering the derived bounds on abelian subalgebra dimension presentation-dependent rather than well-defined.

Authors: We agree that establishing basis-independence is essential for the construction to yield an intrinsic invariant. The Dani-Mainkar graph is obtained by viewing the bracket as a linear map from V to Hom(V, Z) and recording the support in a basis of V; a change of basis in V or Z induces a linear transformation on this map. We will add a new lemma in §2 proving that any two such graphs arising from different bases are isomorphic (via the change-of-basis matrix), hence share the same independence polynomial. This will also confirm that the subsequent bounds on abelian subalgebra dimension are well-defined for the Lie algebra itself. revision: yes
Referee: [§3] Bounds on the independence number (likely §3): the upper and lower bounds are stated to be efficiently computable, but the manuscript must supply explicit derivations or algorithms that do not rely on auxiliary choices; without these, it is unclear whether the bounds hold for the algebra itself or only for specific presentations.

Authors: We accept that the current presentation of the bounds would benefit from greater explicitness. The upper and lower bounds in §3 are obtained by applying the standard Caro-Wei and Turán-type inequalities to the Dani-Mainkar graph and then translating the resulting numerical estimates back into statements about the Lie algebra. In the revision we will insert a short subsection containing (i) the precise algorithm that takes the structure constants of the bracket map as input and outputs the numerical bounds without further auxiliary choices, and (ii) a short proof that the output is independent of the initial basis once the isomorphism invariance established in §2 is used. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained with no circular steps

full rationale

The paper extends the independence polynomial via the Dani-Mainkar construction applied to the bracket map of a 2-step nilpotent Lie algebra, then derives upper and lower bounds on the independence number using standard combinatorial arguments on the resulting graph. These bounds are applied to obtain estimates on the dimension of abelian subalgebras. No fitted parameters are involved, no self-citations serve as load-bearing premises for the central claims, and the construction does not reduce any derived quantity to the input definition by tautology. The metric-dependent generalization is presented as an optional extension rather than a foundational step.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, ad-hoc axioms, or invented entities; relies on standard definitions from graph theory and Lie algebra literature.

axioms (1)

domain assumption Dani-Mainkar construction applies to arbitrary 2-step nilpotent Lie algebras
Invoked to motivate the extension in the abstract.

pith-pipeline@v0.9.0 · 5596 in / 1047 out tokens · 32042 ms · 2026-05-22T18:13:28.454187+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We extend the notion of independence polynomial of graphs to arbitrary 2-step nilpotent Lie algebras... using the dimension of the graded pieces of the basic cohomology as coefficients
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 27... α(g) ≤ 1/2 + √(1/4 + b² + b − 2d)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Pure Appl

Marco Aldi and Samuel Bevins, 2-step nilpotent L∞-algebras and hypergraphs , J. Pure Appl. Algebra 228 (2024), no. 6, Paper No. 107593, 12

work page 2024
[2]

Pure Appl

Marco Aldi, Andrew Butler, Jordan Gardiner, Daniele Grandini, Mon ica Lichtenwalner, and Kevin Pan, On the cohomology of Lie algebras associated with graphs , J. Pure Appl. Algebra 229 (2025), no. 1, Paper No. 107838, 15

work page 2025
[3]

Marco Aldi, Thor Gabrielsen, Daniele Grandini, Joy Harris, and Kyle K elley, An eﬃciently com- putable lower bound for the independence number of hypergra phs, available at arXiv:2502.11814

work page arXiv
[4]

Jorge Luis Arocha, Properties of the independence polynomial of a graph , Cienc. Mat. (Havana) 5 (1984), no. 3, 103–110 (Spanish, with English summary)

work page 1984
[5]

6, North-Holland Publishing Co., Amsterdam-London; Am erican Elsevier Publishing Co., Inc., New York, 1976

Claude Berge, Graphs and hypergraphs , Second revised edition, North-Holland Mathematical Li- brary, Vol. 6, North-Holland Publishing Co., Amsterdam-London; Am erican Elsevier Publishing Co., Inc., New York, 1976. Translated from the French by Edward M inieka

work page 1976
[6]

A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Ergebnisse der Math- ematik und ihrer Grenzgebiete (3) [Results in Mathematics and Relate d Areas (3)], vol. 18, Springer-Verlag, Berlin, 1989

work page 1989
[7]

Graph Theory 15 (1991), no

Yair Caro and Zsolt Tuza, Improved lower bounds on k-independence, J. Graph Theory 15 (1991), no. 1, 99–107

work page 1991
[8]

Plick, and Ali Shokoufandeh, A note on the Caro-Tuza bound on the independence number of uniform hypergraphs , Australas

B´ ela Csaba, Thomas A. Plick, and Ali Shokoufandeh, A note on the Caro-Tuza bound on the independence number of uniform hypergraphs , Australas. J. Combin. 52 (2012), 235–242

work page 2012
[9]

S. G. Dani and Meera G. Mainkar, Anosov automorphisms on compact nilmanifolds associated with graphs , Trans. Amer. Math. Soc. 357 (2005), no. 6, 2235–2251

work page 2005
[10]

The- sis (Ph.D.)–University of California, San Diego

Alexander Eustis, Hypergraph Independence Numbers, ProQuest LLC, Ann Arbor, MI, 2013. The- sis (Ph.D.)–University of California, San Diego

work page 2013
[11]

Roumaine Math

Pierre Hansen, Upper bounds for the stability number of a graph , Rev. Roumaine Math. Pures Appl. 24 (1979), no. 8, 1195–1199

work page 1979
[12]

1, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA,

Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipa nde, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow, Mirror symmetry , Clay Mathematics Monographs, vol. 1, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA,

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INDEPENDENCE POLYNOMIALS OF 2-STEP NILPOTENT LIE ALGEBRAS 13

With a preface by Vafa. INDEPENDENCE POLYNOMIALS OF 2-STEP NILPOTENT LIE ALGEBRAS 13

work page
[14]

Karp, Reducibility among combinatorial problems , Complexity of computer computa- tions (Proc

Richard M. Karp, Reducibility among combinatorial problems , Complexity of computer computa- tions (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorkto wn Heights, N.Y., 1972), The IBM Research Symposia Series, Plenum, New York-London, 1972, p p. 85–103. MR378476

work page 1972
[15]

Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theo rem, Ann. of Math. (2) 74 (1961), 329–387

work page 1961
[16]

Levit and Eugen Mandrescu, The independence polynomial of a graph—a survey , Pro- ceedings of the 1st International Conference on Algebraic Infor matics, Aristotle Univ

Vadim E. Levit and Eugen Mandrescu, The independence polynomial of a graph—a survey , Pro- ceedings of the 1st International Conference on Algebraic Infor matics, Aristotle Univ. Thessa- loniki, Thessaloniki, 2005, pp. 233–254

work page 2005
[17]

Mainkar, Graphs and two-step nilpotent Lie algebras , Groups Geom

Meera G. Mainkar, Graphs and two-step nilpotent Lie algebras , Groups Geom. Dyn. 9 (2015), no. 1, 55–65

work page 2015
[18]

Katsumi Nomizu, On the cohomology of compact homogeneous spaces of nilpoten t Lie groups , Ann. of Math. (2) 59 (1954), 531–538

work page 1954
[19]

Pure Appl

Hannes Pouseele and Paulo Tirao, Compact symplectic nilmanifolds associated with graphs , J. Pure Appl. Algebra 213 (2009), no. 9, 1788–1794

work page 2009
[20]

L. J. Santharoubane, Cohomology of Heisenberg Lie algebras , Proc. Amer. Math. Soc. 87 (1983), no. 1, 23–28

work page 1983
[21]

Algebra 185 (1996), no

Stefan Sigg, Laplacian and homology of free two-step nilpotent Lie algeb ras, J. Algebra 185 (1996), no. 1, 144–161

work page 1996
[22]

Thesis (Dr

Martin Trinks, Graph Polynomials and Their Representations , Technische Universit¨ at Bergakademie Freiberg, 2012. Thesis (Dr. rer. nat.)

work page 2012
[23]

Paul Tur´ an,Eine Extremalaufgabe aus der Graphentheorie , Mat. Fiz. Lapok 48 (1941), 436–452 (Hungarian, with German summary). Marco Aldi Department of Mathematics and Applied Mathematics Virginia Commonwealth University Richmond, V A 23284, USA maldi2@vcu.edu Thor Gabrielsen Department of Mathematics Colby College Waterville, ME 04901 trgabr26@colby.edu...

work page 1941

[1] [1]

Pure Appl

Marco Aldi and Samuel Bevins, 2-step nilpotent L∞-algebras and hypergraphs , J. Pure Appl. Algebra 228 (2024), no. 6, Paper No. 107593, 12

work page 2024

[2] [2]

Pure Appl

Marco Aldi, Andrew Butler, Jordan Gardiner, Daniele Grandini, Mon ica Lichtenwalner, and Kevin Pan, On the cohomology of Lie algebras associated with graphs , J. Pure Appl. Algebra 229 (2025), no. 1, Paper No. 107838, 15

work page 2025

[3] [3]

Marco Aldi, Thor Gabrielsen, Daniele Grandini, Joy Harris, and Kyle K elley, An eﬃciently com- putable lower bound for the independence number of hypergra phs, available at arXiv:2502.11814

work page arXiv

[4] [4]

Jorge Luis Arocha, Properties of the independence polynomial of a graph , Cienc. Mat. (Havana) 5 (1984), no. 3, 103–110 (Spanish, with English summary)

work page 1984

[5] [5]

6, North-Holland Publishing Co., Amsterdam-London; Am erican Elsevier Publishing Co., Inc., New York, 1976

Claude Berge, Graphs and hypergraphs , Second revised edition, North-Holland Mathematical Li- brary, Vol. 6, North-Holland Publishing Co., Amsterdam-London; Am erican Elsevier Publishing Co., Inc., New York, 1976. Translated from the French by Edward M inieka

work page 1976

[6] [6]

A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Ergebnisse der Math- ematik und ihrer Grenzgebiete (3) [Results in Mathematics and Relate d Areas (3)], vol. 18, Springer-Verlag, Berlin, 1989

work page 1989

[7] [7]

Graph Theory 15 (1991), no

Yair Caro and Zsolt Tuza, Improved lower bounds on k-independence, J. Graph Theory 15 (1991), no. 1, 99–107

work page 1991

[8] [8]

Plick, and Ali Shokoufandeh, A note on the Caro-Tuza bound on the independence number of uniform hypergraphs , Australas

B´ ela Csaba, Thomas A. Plick, and Ali Shokoufandeh, A note on the Caro-Tuza bound on the independence number of uniform hypergraphs , Australas. J. Combin. 52 (2012), 235–242

work page 2012

[9] [9]

S. G. Dani and Meera G. Mainkar, Anosov automorphisms on compact nilmanifolds associated with graphs , Trans. Amer. Math. Soc. 357 (2005), no. 6, 2235–2251

work page 2005

[10] [10]

The- sis (Ph.D.)–University of California, San Diego

Alexander Eustis, Hypergraph Independence Numbers, ProQuest LLC, Ann Arbor, MI, 2013. The- sis (Ph.D.)–University of California, San Diego

work page 2013

[11] [11]

Roumaine Math

Pierre Hansen, Upper bounds for the stability number of a graph , Rev. Roumaine Math. Pures Appl. 24 (1979), no. 8, 1195–1199

work page 1979

[12] [12]

1, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA,

Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipa nde, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow, Mirror symmetry , Clay Mathematics Monographs, vol. 1, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA,

work page

[13] [13]

INDEPENDENCE POLYNOMIALS OF 2-STEP NILPOTENT LIE ALGEBRAS 13

With a preface by Vafa. INDEPENDENCE POLYNOMIALS OF 2-STEP NILPOTENT LIE ALGEBRAS 13

work page

[14] [14]

Karp, Reducibility among combinatorial problems , Complexity of computer computa- tions (Proc

Richard M. Karp, Reducibility among combinatorial problems , Complexity of computer computa- tions (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorkto wn Heights, N.Y., 1972), The IBM Research Symposia Series, Plenum, New York-London, 1972, p p. 85–103. MR378476

work page 1972

[15] [15]

Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theo rem, Ann. of Math. (2) 74 (1961), 329–387

work page 1961

[16] [16]

Levit and Eugen Mandrescu, The independence polynomial of a graph—a survey , Pro- ceedings of the 1st International Conference on Algebraic Infor matics, Aristotle Univ

Vadim E. Levit and Eugen Mandrescu, The independence polynomial of a graph—a survey , Pro- ceedings of the 1st International Conference on Algebraic Infor matics, Aristotle Univ. Thessa- loniki, Thessaloniki, 2005, pp. 233–254

work page 2005

[17] [17]

Mainkar, Graphs and two-step nilpotent Lie algebras , Groups Geom

Meera G. Mainkar, Graphs and two-step nilpotent Lie algebras , Groups Geom. Dyn. 9 (2015), no. 1, 55–65

work page 2015

[18] [18]

Katsumi Nomizu, On the cohomology of compact homogeneous spaces of nilpoten t Lie groups , Ann. of Math. (2) 59 (1954), 531–538

work page 1954

[19] [19]

Pure Appl

Hannes Pouseele and Paulo Tirao, Compact symplectic nilmanifolds associated with graphs , J. Pure Appl. Algebra 213 (2009), no. 9, 1788–1794

work page 2009

[20] [20]

L. J. Santharoubane, Cohomology of Heisenberg Lie algebras , Proc. Amer. Math. Soc. 87 (1983), no. 1, 23–28

work page 1983

[21] [21]

Algebra 185 (1996), no

Stefan Sigg, Laplacian and homology of free two-step nilpotent Lie algeb ras, J. Algebra 185 (1996), no. 1, 144–161

work page 1996

[22] [22]

Thesis (Dr

Martin Trinks, Graph Polynomials and Their Representations , Technische Universit¨ at Bergakademie Freiberg, 2012. Thesis (Dr. rer. nat.)

work page 2012

[23] [23]

Paul Tur´ an,Eine Extremalaufgabe aus der Graphentheorie , Mat. Fiz. Lapok 48 (1941), 436–452 (Hungarian, with German summary). Marco Aldi Department of Mathematics and Applied Mathematics Virginia Commonwealth University Richmond, V A 23284, USA maldi2@vcu.edu Thor Gabrielsen Department of Mathematics Colby College Waterville, ME 04901 trgabr26@colby.edu...

work page 1941