## Relationships between generalized Heisenberg algebras and the classical Heisenberg algebra

### Volume 134 / 2014

#### Abstract

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.