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PreprintCC BY-NC-ND 4.0arXiv

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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Cite

BibTeXRIS
BibTeX
@misc{ejsmont2026quadratic,
  title = {{Quadratic polynomials preserving free infinite divisibility}},
  author = {Wiktor Ejsmont and Franz Lehner and Mihai Popa and Kamil Szpojankowski},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15320},
  eprint = {2610.15320},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15320},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Quadratic polynomials preserving free infinite divisibility
AU  - Wiktor Ejsmont
AU  - Franz Lehner
AU  - Mihai Popa
AU  - Kamil Szpojankowski
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15320
UR  - https://arxiv.org/abs/2610.15320
AB  - 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.
ER  -