Relationships between generalized Heisenberg algebras and the classical Heisenberg algebra

Volume 134 / 2014

Marc Fabbri, Frank Okoh Colloquium Mathematicum 134 (2014), 255-265 MSC: Primary 17B65; Secondary 17B69. DOI: 10.4064/cm134-2-9


A Lie algebra is called a generalized Heisenberg algebra of degree $n$ if its centre coincides with its derived algebra and is $n$-dimensional. In this paper we define for each positive integer $n$ a generalized Heisenberg algebra $\mathcal {H}_{n}$. We show that $\mathcal {H}_{n}$ and $\mathcal {H}_{1}^{n}$, the Lie algebra which is the direct product of $n$ copies of $\mathcal {H}_{1}$, contain isomorphic copies of each other. We show that $\mathcal {H}_{n}$ is an indecomposable Lie algebra. We prove that $\mathcal {H}_{n}$ and $\mathcal {H}_{1}^{n}$ are not quotients of each other when $n \geq 2$, but $\mathcal {H}_{1}$ is a quotient of $\mathcal {H}_{n}$ for each positive integer $n$. These results are used to obtain several families of $\mathcal {H}_{n}$-modules from the Fock space representation of $\mathcal {H}_{1}$. Analogues of Verma modules for $\mathcal {H}_{n}$, $n \geq 2$, are also constructed using the set of rational primes.


  • Marc FabbriDepartment of Mathematics
    Pennsylvania State University
    University Park, PA 16802-6401, U.S.A.
  • Frank OkohDepartment of Mathematics
    Wayne State University
    Detroit, MI 48202-3622, U.S.A.

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