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 $q\in {\mathbb{C}}^{\ast }$, 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.