In this paper, we study a new kind of vertex operator algebra related to the twisted Heisenberg-Virasoro algebra, which we call the twisted Heisenberg-Virasoro vertex operator algebra, and its modules. Specifically, we present some results concerning the relationship between the restricted module categories of twisted Heisenberg-Virasoro algebras of rank one and rank two and several different kinds of module categories of their corresponding vertex algebras. We also study fully the structures of the twisted Heisenberg-Virasoro vertex operator algebra, give a characterization of it as a tensor product of two well-known vertex operator algebras, and solve the commutant problem.

We establish a natural connection of the q-Virasoro algebra D q introduced by Belov and Chaltikian with affine Kac-Moody Lie algebras. More specifically, for each abelian group S together with a one-to-one linear character χ, we define an infinite-dimensional Lie algebra D S which reduces to D q when S = Z. Guided by the theory of equivariant quasi modules for vertex algebras, we introduce another Lie algebra g S with S as an automorphism group and we prove that D S is isomorphic to the S-covariant algebra of the affine Lie algebra g S . We then relate restricted D S -modules of level ℓ ∈ C to equivariant quasi modules for the vertex algebra V g S (ℓ, 0) associated to g S with level ℓ. Furthermore, we show that if S is a finite abelian group of order 2l + 1, D S is isomorphic to the affine Kac-Moody algebra of type B(1) l .

In this paper, we study a certain deformation D of the Virasoro algebra that was introduced and called q-Virasoro algebra by Nigro, in the context of vertex algebras. Among the main results, we prove that for any complex number ℓ, the category of restricted D-modules of level ℓ is canonically isomorphic to the category of quasi modules for a certain vertex algebra of affine type. We also prove that the category of restricted D-modules of level ℓ is canonically isomorphic to the category of Z-equivariant φ-coordinated quasi modules for the same vertex algebra. In the process, we introduce and employ a certain infinite dimensional Lie algebra which is defined in terms of generators and relations and then identified explicitly with a subalgebra of gl ∞ .

<p style='text-indent:20px;'>We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra <inline-formula><tex-math id="M1">$ V_{\mathcal{L}}(\ell_{123},0) $</tex-math></inline-formula>. Then, for any integer <inline-formula><tex-math id="M2">$ t&gt;1 $</tex-math></inline-formula>, we introduce a new Lie algebra <inline-formula><tex-math id="M3">$ \mathcal{L}_{t} $</tex-math></inline-formula>, and show that <inline-formula><tex-math id="M4">$ \sigma_{t} $</tex-math></inline-formula>-twisted <inline-formula><tex-math id="M5">$ V_{\mathcal{L}}(\ell_{123},0) $</tex-math></inline-formula>(<inline-formula><tex-math id="M6">$ \ell_{2} = 0 $</tex-math></inline-formula>)-modules are in one-to-one correspondence with restricted <inline-formula><tex-math id="M7">$ \mathcal{L}_{t} $</tex-math></inline-formula>-modules of level <inline-formula><tex-math id="M8">$ \ell_{13} $</tex-math></inline-formula>, where <inline-formula><tex-math id="M9">$ \sigma_{t} $</tex-math></inline-formula> is an order <inline-formula><tex-math id="M10">$ t $</tex-math></inline-formula> automorphism of <inline-formula><tex-math id="M11">$ V_{\mathcal{L}}(\ell_{123},0) $</tex-math></inline-formula>. At the end, we give a complete list of irreducible <inline-formula><tex-math id="M12">$ \sigma_{t} $</tex-math></inline-formula>-twisted <inline-formula><tex-math id="M13">$ V_{\mathcal{L}}(\ell_{123},0) $</tex-math></inline-formula>(<inline-formula><tex-math id="M14">$ \ell_{2} = 0 $</tex-math></inline-formula>)-modules.</p>

No abstract