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Böttcher-Wenzel Inequality

The Böttcher-Wenzel inequality bounds the Frobenius norm of a matrix commutator. For complex n×n matrices,

||XY-YX||_F^2<=2||X||_F^2||Y||_F^2.

The constant 2 is optimal for n>=2 (Fox 2026).

Fox (2026) proved a sharp extension to a complex simple Lie algebra g. Let tau be conjugation with respect to a compact real form, let B be the Killing form, and set H(x,y)=-B(x,tauy) and ||x||^2=H(x,x). Then

||[x,y]||^2<=1/(h^ v )||x||^2||y||^2,

where h^ v is the dual Coxeter number. For nonzero x and y, equality holds precisely when they are H-orthogonal and contained in a tau-stable long-root subalgebra isomorphic to sl(2,C). Fox (2026) reports that Claude proposed proof strategies and lemmas and assisted calculations, with the author verifying the results.

See also

Commutator, Dual Coxeter Number, Frobenius Norm, Killing Form, Lie Algebra

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References

Fox, D. J. F. "Sharp Commutator Bounds on Complex Simple Lie Algebras: The Böttcher-Wenzel Inequality and the Comass of the Cartan 3-Form." 8 Oct 2026. https://arxiv.org/abs/2610.11800.

Cite this as:

Weisstein, Eric W. "Böttcher-Wenzel Inequality." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Bottcher-WenzelInequality.html

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