Quadratic polynomials preserving free infinite divisibility
Wiktor Ejsmont, Franz Lehner, Mihai Popa, Kamil Szpojankowski
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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