TL;DR
This paper introduces a twisted Heisenberg double based on twisted Hopf algebras, establishing a Stone--von Neumann theorem and connecting it to various quantum algebras, expanding the algebraic framework.
Contribution
It presents a novel construction of twisted Heisenberg doubles and proves a fundamental representation theorem, linking it to known quantum algebra structures.
Findings
Established a Stone--von Neumann type theorem for the twisted Heisenberg double
Demonstrated the construction encompasses quantum Weyl and Heisenberg algebras
Showed the effect of shifting product and coproduct in the twisted setting
Abstract
We introduce a twisted version of the Heisenberg double, constructed from a twisted Hopf algebra and a twisted Hopf pairing. We state a Stone--von Neumann type theorem for a natural Fock space representation of this twisted Heisenberg double and deduce the effect on the algebra of shifting the product and coproduct of the original twisted Hopf algebra. We conclude by showing that the quantum Weyl algebra, quantum Heisenberg algebras, and lattice Heisenberg algebras are all examples of the general construction.
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