Some Fock spaces with depth two action
Michael Anshelevich, Jacob Mashburn

TL;DR
This paper studies operators on a specialized Fock space acting on the first two tensor components, unifies previous constructions, and extends key results to a more general setting with matrix-valued generating functions.
Contribution
It unifies and generalizes earlier work on Fock space operators acting on the first two components, extending results to matrix-valued generating functions and broader contexts.
Findings
Derived quadratic relations for free cumulant generating functions
Obtained resolvent forms for Wick polynomial generating functions
Classified cases with tracial vacuum states
Abstract
The subject of this paper are operators represented on a Fock space which act only on the two leading components of the tensor. We unify the constructions from arXiv:math/0702158, arXiv:0709.4334, arXiv:0812.0895, and arXiv:1003.2998, and extend a number of results from these papers to our more general setting. The results include the quadratic relation satisfied by (the kernel of) the free cumulant generating function, the resolvent form of the generating function for the Wick polynomials, and classification results for the case when the vacuum state on the operator algebra is tracial. We handle the generating functions in infinitely many variables by considering their matrix-valued versions.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Noncommutative and Quantum Gravity Theories
