Modules over the Takiff Block type Lie algebra

Abstract

In this paper, we introduce a family of infinite-dimensional Lie algebras $B(q)$ of Takiff Block type, containing Heisenberg-Virasoro algebra, $W$-algebra $W(2,2)$, and BMS-Kac-Moody algebra as subalgebras. For $q\in\mathbb C^*$, we classify the $ B(q)$-modules that are free of rank one over $U(\mathfrak h)$, where $\mathfrak h=\mathbb C L_{0,0}\oplus\mathbb C W_{0,0}$, and determine their irreducibility and isomorphism classes. We also establish irreducibility and isomorphism criteria for their tensor products with irreducible restricted modules. Finally, restricting the above $B(-1)$-modules to several natural subalgebras of the central quotient $B(-1)/(\mathbb C C_1\oplus\mathbb C C_2)$ yields families of non-weight modules.