Representations of non-finitely graded Heisenberg-Virasoro type Lie algebras

Abstract

We construct and study non-finitely graded Lie algebras $\mathcal{HV}(a,b;ε)$ related to Heisenberg-Virasoro type Lie algebras, where $a,b$ are complex numbers, and $ε= \pm 1$. Using combinatorial techniques, we completely classify the free $\mathcal{U}(\mathfrak h)$-modules of rank one over $\mathcal{HV}(a,b;ε)$. It turns out that these modules are more varied and complex than those over non-finitely graded Virasoro algebras, and in particular admit infinitely many free parameters if $b=1$ and $ε=-1$. Meanwhile, we also determine the simplicity and isomorphism classes of these modules.

Theorems & Definitions (16)

• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Proposition 3.1
• Remark 3.2
• Theorem 3.3
• Lemma 3.4
• Lemma 3.5
• Theorem 3.6