Sharp commutator bounds on complex simple Lie algebras: the Böttcher--Wenzel inequality and the comass of the Cartan $3$-form
Abstract
For a complex simple Lie algebra g with compact conjugation τ, Killing form B, and Hermitian form H(x, y) = −B(x, τy), it is shown that H([x,y],[x,y]) ≤ (h∨)−1H(x,x)H(y,y), where h∨ is the dual Coxeter number, with equality exactly for H-orthogonal pairs in τ-stable long-root subalgebras isomorphic to sl(2, ℂ). Equivalently, the comass of the Cartan $3$-form with respect to H equals its comass on the compact real form: the spectral norm of the Cartan $3$-form does not increase under complexification. For the Frobenius norm on the image of a representation of Dynkin index ℓ the optimal constant is 2/ℓ; this contains the Böttcher--Wenzel and Bloch--Iserles inequalities, extends the latter to complex skew-symmetric matrices, and improves them substantially for the exceptional Lie algebras of types F4, E6, E7, and E8. The proof uses the Nahm algebra of g with the conjugation induced by τ: critical points of the relevant function correspond to τ-twisted idempotents, and its Hessian is governed by τ-twisted multiplication operators, whose anticommutation with multiplication by i turns the one-sided second order condition at a maximum into a two-sided spectral bound. Subalgebras isomorphic to sl(2, ℂ) of Dynkin index j give critical points with critical value 1/(jh∨), whose Morse indices are computed, and the set of maximum points is a nondegenerate critical manifold. Geometrically, the maximal complex sectional curvature of a compact simple Lie group equals its maximal sectional curvature.
Related
- Proof of Nobori's generalized Böttcher-Wenzel inequality conjecture
- A sharp weighted commutator inequality for rank-one matrices
- A uniform commutator bound in finite von Neumann algebras
- Commutator identities, Lie product identities, and Multiplicative Lie algebras
- Quantum expanders and dimension-free commutator bounds
BibTeX
@misc{fox2026sharp,
title = {{Sharp commutator bounds on complex simple Lie algebras: the Böttcher--Wenzel inequality and the comass of the Cartan $3$-form}},
author = {Daniel J. F. Fox},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11800},
eprint = {2610.11800},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11800},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp commutator bounds on complex simple Lie algebras: the Böttcher--Wenzel inequality and the comass of the Cartan $3$-form
AU - Daniel J. F. Fox
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.11800
UR - https://arxiv.org/abs/2610.11800
AB - For a complex simple Lie algebra g with compact conjugation τ, Killing form B, and Hermitian form H(x, y) = −B(x, τy), it is shown that H([x,y],[x,y]) ≤ (h^∨)^−1H(x,x)H(y,y), where h^∨ is the dual Coxeter number, with equality exactly for H-orthogonal pairs in τ-stable long-root subalgebras isomorphic to sl(2, ℂ). Equivalently, the comass of the Cartan $3$-form with respect to H equals its comass on the compact real form: the spectral norm of the Cartan $3$-form does not increase under complexification. For the Frobenius norm on the image of a representation of Dynkin index ℓ the optimal constant is 2/ℓ; this contains the Böttcher--Wenzel and Bloch--Iserles inequalities, extends the latter to complex skew-symmetric matrices, and improves them substantially for the exceptional Lie algebras of types F_4, E_6, E_7, and E_8. The proof uses the Nahm algebra of g with the conjugation induced by τ: critical points of the relevant function correspond to τ-twisted idempotents, and its Hessian is governed by τ-twisted multiplication operators, whose anticommutation with multiplication by i turns the one-sided second order condition at a maximum into a two-sided spectral bound. Subalgebras isomorphic to sl(2, ℂ) of Dynkin index j give critical points with critical value 1/(jh^∨), whose Morse indices are computed, and the set of maximum points is a nondegenerate critical manifold. Geometrically, the maximal complex sectional curvature of a compact simple Lie group equals its maximal sectional curvature.
ER -