Submodule structures of C[s,t]\mathbb C[s,t] over W(0,b)W(0,b) and a new class of irreducible modules over the Virasoro algebra

For any $a,b\in\mathbb C$, $W(a,b)$ is the Lie algebra with basis $\{L_m,M_m\,|\,m\in\mathbb Z\}$ and relations $[L_m,L_n]=(n-m)L_{m+n},$ $[L_m,W_n]=(a+n+bm)W_{m+n}$, $[W_m,W_n]=0$ for $m,n\in\mathbb Z$. For any $\lambda\in\mathbb C^*,$ $\alpha\in\mathbb C$, $h:=h(t)\in\mathbb C[t]$, there exists a non-weight module over $W(0,b)$ (resp., $W(0,1)$), denoted by $\Phi(\lambda,\alpha,h)$ (resp. $\Theta(\lambda,h)$), which is defined on the space $\mathbb C[s,t]$ of polynomials on variables $s,t$ and is free of rank one over the enveloping algebra $U(\mathbb C L_0\oplus\mathbb C W_0)$ of $\mathbb C L_0\oplus\mathbb C W_0$. In the present paper, by introducing two sequences of useful operators on $\mathbb C[s,t]$, we determine all submodules of $\mathbb C[s,t]$. We also study submodules of $\mathbb C[s,t]$ regarded as modules over the Virasoro algebra $\mathscr V\!$ (with the trivial action of the center), and prove that these submodules are finitely generated if and only if ${\rm deg}\,h(t)\geq1$. In addition, it is proven that $\Phi(\lambda, \alpha,h)$ is an irreducible $\mathscr V\!$-module if and only if $b=-1$, ${\rm deg}\, h(t)=1$, $\alpha\neq0$. Finally, we obtain a large family of new irreducible modules over the Virasoro algebra $\mathscr V\!$, by taking various tensor products of a finite number of irreducible modules $\Phi(\lambda_i,\alpha_i, h_i)$ for $\lambda_i,\alpha_i\in\mathbb C^*,$ $h_i\in\mathbb C[t]$ with an irreducible $\mathscr V\!$-module $V$, where $V$ satisfies that there exists a nonnegative integer $R_V$ such that $L_m$ acts locally finitely on $V$ for $m\geq R_V$.Comment: 22 page

Automorphism groups of Witt algebras

The automorphism groups ${\rm Aut\,}A_n$ and ${\rm Aut\,}W_n$ of the polynomial algebra $A_n=C[x_1,x_2,\cdots, x_n]$ and the rank $n$ Witt algebra $W_n={\rm Der\,}A_n$ are studied in this paper. It is well-known that ${\rm Aut\,}A_n$ for $n\ge3$ and ${\rm Aut\,}W_n$ for $n\ge2$ are open. In the present paper, by characterizing the semigroup ${\rm End\,}W_n\setminus\{0\}$ of nonzero endomorphisms of $W_n$ via the semigroup of the so-called Jacobi tuples, we establish an isomorphism between ${\rm Aut}\,A_n$ and ${\rm Aut\,}W_n$ for any positive integer $n$. In particular, this enables us to work out the automorphism group ${\rm Aut\,}W_2$ of $W_2$

Verma modules over a class of Block type Lie algebras

Irreducibilities of Verma modules over a class of Block type Lie algebras are completely determined. The approach developed in the present paper can be used to deal with non-weight modules.Comment: Pages 1

Non-weight modules over the affine-Virasoro algebra of type A1A_1

In this paper, we study a class of non-weight modules over the affine-Virasoro algebra of type $A_1$, which are free modules of rank one when restricted to the Cartan subalgebra (modulo center). We give the classification of such modules. Moreover, the simplicity and the isomorphism classes of these modules are determined.Comment: 14 page

Loop super-Virasoro Lie conformal superalgebra

The loop super-Virasoro conformal superalgebra $\mathfrak{cls}$ associated with the loop super-Virasoro algebra is constructed in the present paper. The conformal superderivation algebra of $\mathfrak{cls}$ is completely determined, which is shown to consist of inner superderivations. And nontrivial free and free $\mathbb{Z}$-graded $\mathfrak{cls}$-modules of rank two are classified. We also give a classification of irreducible free $\mathfrak{cls}$-modules of rank two and all irreducible submodules of each free $\mathbb{Z}$-graded $\mathfrak{cls}$-module of rank two.Comment: Pages 1

Some finite properties for vertex operator superalgebras

Vertex operator superalgebras are studied and various results on rational Vertex operator superalgebras are obtained. In particular, the vertex operator super subalgebras generated by the weight 1/2 and weight 1 subspaces are determined. It is also established that if the even part $V_{\bar 0}$ of a vertex operator superalgebra $V$ is rational, so is $V.$Comment: 18 page

A class of non-weight modules over the Virasoro algebra

For any triple $(\mu,\lambda,\alpha)$ of complex numbers and an $\mathfrak a$-module ${V}$, a class of non-weight modules $\mathcal{M}\big(V,\mu,\Omega(\lambda,\alpha)\big)$ over the Virasoro algebra $\mathcal L$ is constructed in this paper. We prove if $V$ is a nontrivial simple $\mathfrak a$-module satisfying: for any $v\in V$ there exists $r\in\Z_+$ such that $L_{r+i}v=0$ for all $i\geq1$, then $\mathcal{M}\big(V,\mu,\Omega(\lambda,\alpha)\big)$ is simple if and only if $\mu\neq1, \lambda\neq0,\alpha\neq0,$. We also give the necessary and sufficient conditions for two such simple $\mathcal L$-modules being isomorphic. Finally, we prove that these simple $\mathcal L$-modules $\mathcal{M}\big(V,\mu,\Omega(\lambda,\alpha)\big)$ are new by showing they are not isomorphic to any other known simple non-weight module provided that $V$ is not a highest weight $\mathfrak a$-module with highest weight nonzero

Modules over the algebra Vir(a,b)\mathcal{V}ir(a,b)

For any two complex numbers $a$ and $b$, $\mathcal{V} ir(a,b)$ is a central extension of $\mathcal{W}(a,b)$ which is universal in the case $(a,b)\neq (0,1)$, where $\mathcal{W}(a,b)$ is the Lie algebra with basis $\{L_n,W_n\mid n\in\Z\}$ and relations $[L_m,L_n]=(n-m)L_{m+n}$, $[L_m,W_n]=(a+n+bm)W_{m+n}$, $[W_m,W_n]=0$. In this paper, we construct and classify a class of non-weight modules over the algebra $\mathcal{V} ir(a,b)$ which are free $U(\mathbb{C} L_0\oplus\mathbb{C} W_0)$-modules of rank $1$. It is proved that such modules can only exist for $a=0$.Comment: 12page

Irreducible weight modules with a finite-dimensional weight space over the twisted N=1 Schr\"{o}dinger-Neveu-Schwarz algebra

It is shown that there are no simple mixed modules over the twisted N=1 Schr\"{o}dinger-Neveu-Schwarz algebra, which implies that every irreducible weight module over it with a nontrivial finite-dimensional weight space, is a Harish-Chandra module

A new class of Z-graded Lie conformal algebras of infinite rank

In this paper, a new class of $\Z$-graded Lie conformal algebras \CW(a,c) of infinite rank is constructed. The conformal derivations and one-dimensional central extensions of \CW(a,c) are completely determined. And all conformal modules of rank one over \CW(a,c) (a\neq0) are proved to be trivial and all such nontrivial (irreducible) modules over \CW(0,c) are classified.Comment: 11 page