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arXiv · 2610.15320

Quadratic polynomials preserving free infinite divisibility

Wiktor Ejsmont·Franz Lehner·Mihai Popa·Kamil Szpojankowski
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Abstract

We characterize non-commutative polynomials of degree 2 which preserve free infinite divisibility as self-adjoint Lie polynomials of degree 2. This extends previous results which characterized homogeneous quadratic forms only. The proof is based on an involution of colored noncrossing partitions cancelling many terms; positive definiteness of the remaining terms is established by an explicit operator model on Fock space. Examples show that higher order Lie polynomials do not preserve free infinite divisibility.

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BibTeXRIS

Wiktor Ejsmont, Franz Lehner, Mihai Popa, Kamil Szpojankowski. 2026-10-11. Quadratic polynomials preserving free infinite divisibility. https://arxiv.org/abs/2610.15320

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