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Wigner oscillators, twisted Hopf algebras, and second quantization. (English) Zbl 1152.81366

Summary: By correctly identifying the role of central extension in the centrally extended Heisenberg algebra h, we show that it is indeed possible to construct a Hopf algebraic structure on the corresponding enveloping algebra \(U(h)\) and eventually deform it through Drinfeld twist. This Hopf algebraic structure and its deformed version \(U^F(h)\) are shown to be induced from a more fundamental Hopf algebra obtained from the Schrödinger field/oscillator algebra and its deformed version, provided that the fields/oscillators are regarded as odd-elements of the super-algebra \(\text{osp}(1|2n)\). We also discuss the possible implications in the context of quantum statistics.

MSC:

81R05 Finite-dimensional groups and algebras motivated by physics and their representations
16T05 Hopf algebras and their applications
17B35 Universal enveloping (super)algebras
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81R60 Noncommutative geometry in quantum theory
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