Relationships between generalized Heisenberg algebras and the classical Heisenberg algebra

Marc Fabbri¹ ; Frank Okoh²

¹ Department of Mathematics Pennsylvania State University University Park, PA 16802-6401, U.S.A.
² Department of Mathematics Wayne State University Detroit, MI 48202-3622, U.S.A.

Colloquium Mathematicum, Tome 134 (2014) no. 2, pp. 255-265.

Résumé

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.

Détail

DOI : 10.4064/cm134-2-9

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