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Register a new userLocalization of highest weight modules of a class of Extended Affine Lie Algebras
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In 2006, Gao and Zeng cite{GZ} gave the free field realizations of highest weight modules over a class of extended affine Lie algebras. In the present paper, applying the technique of localization to those free field realizations, we construct a class of new weight modules over the extended affine Lie algebras. We give necessary and sufficient conditions for these modules to be irreducible. In this way, we construct free field realizations for a class of simple weight modules with infinite weight multiplicities over the extended affine Lie algebras.
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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.
We classify all simple bounded highest weight modules of a basic classical Lie superalgebra $mathfrak g$. In particular, our classification leads to the classification of the simple weight modules with finite weight multiplicities over all classical Lie superalgebras. We also obtain some character formulas of strongly typical bounded highest weight modules of $mathfrak g$.
In [19], Zheng studied the bounded derived categories of constructible $bar{mathbb{Q}}_l$-sheaves on some algebraic stacks consisting of the representations of a enlarged quiver and categorified the integrable highest weight modules of the corresponding quantum group by using these categories. In this paper, we shall generalize Zhengs work to highest weight modules of a subalgebra of the double Ringel-Hall algebra associated to a quiver in a functional version.
The rank $n$ symplectic oscillator Lie algebra $mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite dimensional weight spaces over $mathfrak{g}_n$. When $dot z eq 0$, it is shown that there is an equivalence between the full subcategory $mathcal{O}_{mathfrak{g}_n}[dot z]$ of the BGG category $mathcal{O}_{mathfrak{g}_n}$ for $mathfrak{g}_n$ and the BGG category $mathcal{O}_{mathfrak{sp}_{2n}}$ for $mathfrak{sp}_{2n}$. Then using the technique of localization and the structure of generalized highest weight modules, we also give the classification of simple weight modules over $mathfrak{g}_n$ with finite-dimensional weight spaces.
The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $mathfrak{sl}_3$ and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications [arxiv:1005.0185, arxiv:1910.13781]. In particular, we prove that the simple Bershadsky-Polyakov algebras with admissible nonintegral $mathsf{k}$ are always rational in category $mathscr{O}$, whilst they always admit nonsemisimple relaxed highest-weight modules unless $mathsf{k}+frac{3}{2} in mathbb{Z}_{ge0}$.
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