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arXiv · 0810.3365

Central extensions of the Heisenberg algebra

Luigi Accardi·Andreas Boukas
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Abstract

We study the non-trivial central extensions $CEHeis$ of the Heisenberg algebra $Heis$ recently constructed in {AccBouCE}. We prove that a real form of $CEHeis$ is one the fifteen classified real four--dimensional solvable Lie algebras. We also show that $CEHeis$ can be realized (i) as a sub--Lie--algebra of the Schroedinger algebra and (ii) in terms of two independent copies of the canonical commutation relations (CCR). This gives a natural family of unitary representations of $CEHeis$ and allows an explicit determination of the associated group by exponentiation. In contrast with $Heis$, the group law for $CEHeis$ is given by nonlinear (quadratic) functions of the coordinates.

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Luigi Accardi, Andreas Boukas. 2008-10-19. Central extensions of the Heisenberg algebra. https://arxiv.org/abs/0810.3365

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