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Register a new userIrreducible modules over the divergence zero algebras and their $q$-analogues
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In this paper, we study a class of $Z_d$-graded modules, which are constructed using Larssons functor from $sl_d$-modules $V$, for the Lie algebras of divergence zero vector fields on tori and quantum tori. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional $V$ using a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras.
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Let $mathbf{k}$ be an algebraically closed field, let $Lambda$ be a finite dimensional $mathbf{k}$-algebra, and let $widehat{Lambda}$ be the repetitive algebra of $Lambda$. For the stable category of finitely generated left $widehat{Lambda}$-modules $widehat{Lambda}$-underline{mod}, we show that the irreducible morphisms fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. We next use this fact in order to describe the shape of the Auslander-Reiten triangles in $widehat{Lambda}$-underline{mod}. We use the fact (and prove) that every Auslander-Reiten triangle in $widehat{Lambda}$-underline{mod} is induced from an Auslander-Reiten sequence of finitely generated left $widehat{Lambda}$-modules.
In this paper, we continue the study on toroidal vertex algebras initiated in cite{LTW}, to study concrete toroidal vertex algebras associated to toroidal Lie algebra $L_{r}(hat{frak{g}})=hat{frak{g}}otimes L_r$, where $hat{frak{g}}$ is an untwisted affine Lie algebra and $L_r=$mathbb{C}[t_{1}^{pm 1},ldots,t_{r}^{pm 1}]$. We first construct an $(r+1)$-toroidal vertex algebra $V(T,0)$ and show that the category of restricted $L_{r}(hat{frak{g}})$-modules is canonically isomorphic to that of $V(T,0)$-modules.Let $c$ denote the standard central element of $hat{frak{g}}$ and set $S_c=U(L_r(mathbb{C}c))$. We furthermore study a distinguished subalgebra of $V(T,0)$, denoted by $V(S_c,0)$. We show that (graded) simple quotient toroidal vertex algebras of $V(S_c,0)$ are parametrized by a $mathbb{Z}^r$-graded ring homomorphism $psi:S_crightarrow L_r$ such that Im$psi$ is a $mathbb{Z}^r$-graded simple $S_c$-module. Denote by $L(psi,0}$ the simple $(r+1)$-toroidal vertex algebra of $V(S_c,0)$ associated to $psi$. We determine for which $psi$, $L(psi,0)$ is an integrable $L_{r}(hat{frak{g}})$-module and we then classify irreducible $L(psi,0)$-modules for such a $psi$. For our need, we also obtain various general results.
We prove a character formula for the irreducible modules from the category $mathcal{O}$ over the simple affine vertex algebra of type $A_n$ and $C_n$ $(n geq 2)$ of level $k=-1$. We also give a conjectured character formula for types $D_4$, $E_6$, $E_7$, $E_8$ and levels $k=-1, cdots, -b$, where $b=2,3,4,6$ respectively.
The $(q, mathbf{Q})$-current algebra associated with the general linear Lie algebra was introduced by the second author in the study of representation theory of cyclotomic $q$-Schur algebras. In this paper, we study the $(q, mathbf{Q})$-current algebra $U_q(mathfrak{sl}_n^{langle mathbf{Q} rangle}[x])$ associated with the special linear Lie algebra $mathfrak{sl}_n$. In particular, we classify finite dimensional simple $U_q(mathfrak{sl}_n^{langle mathbf{Q} rangle}[x])$-modules.
In the present paper, we prove that any finite non-trivial irreducible module over a rank two Lie conformal algebra $mathcal{H}$ is of rank one. We also describe the actions of $mathcal{H}$ on its finite irreducible modules explicitly. Moreover, we show that all finite non-trivial irreducible modules of finite Lie conformal algebras whose semisimple quotient is the Virasoro Lie conformal algebra are of rank one.
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