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Register a new user$lambda$-Differential operators and $lambda$-differential modules for the Virasoro algebra
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The concept of $lambda$-differential operators is a natural generalization of differential operators and difference operators. In this paper, we determine the $lambda$-differential Lie algebraic structure on the Witt algebra and the Virasoro algebra for invertible $lambda$. Then we consider several families of modules over the Virasoro algebra with explicit module actions and determine the $lambda$-differential module structures on them.
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In this paper, we explore a canonical connection between the algebra of $q$-difference operators $widetilde{V}_{q}$, affine Lie algebra and affine vertex algebras associated to certain subalgebra $mathcal{A}$ of the Lie algebra $mathfrak{gl}_{infty}$. We also introduce and study a category $mathcal{O}$ of $widetilde{V}_{q}$-modules. More precisely, we obtain a realization of $widetilde{V}_{q}$ as a covariant algebra of the affine Lie algebra $widehat{mathcal{A}^{*}}$, where $mathcal{A}^{*}$ is a 1-dimensional central extension of $mathcal{A}$. We prove that restricted $widetilde{V_{q}}$-modules of level $ell_{12}$ correspond to $mathbb{Z}$-equivariant $phi$-coordinated quasi-modules for the vertex algebra $V_{widetilde{mathcal{A}}}(ell_{12},0)$, where $widetilde{mathcal{A}}$ is a generalized affine Lie algebra of $mathcal{A}$. In the end, we show that objects in the category $mathcal{O}$ are restricted $widetilde{V_{q}}$-modules, and we classify simple modules in the category $mathcal{O}$.
We set up a framework for discussing `$q$-analogues of the usual covariant differential operators for hermitian symmetric spaces. This turns out to be directly related to the deformation quantization associated to quadratic algebras satisfying certain conditions introduced by Procesi and De Concini.
We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra $V_{mathcal{L}}(ell_{123},0)$.Then, for any integer $t>1$, we introduce a new Lie algebra $mathcal{L}_{t}$, and show that $sigma_{t}$-twisted $V_{mathcal{L}}(ell_{123},0)$($ell_{2}=0$)-modules are in one-to-one correspondence with restricted $mathcal{L}_{t}$-modules of level $ell_{13}$, where $sigma_{t}$ is an order $t$ automorphism of $V_{mathcal{L}}(ell_{123},0)$. At the end, we give a complete list of irreducible $sigma_{t}$-twisted $V_{mathcal{L}}(ell_{123},0)$($ell_{2}=0$)-modules.
Every irreducible finite-dimensional representation of the quantized enveloping algebra U_q(gl_n) can be extended to the corresponding quantum affine algebra via the evaluation homomorphism. We give in explicit form the necessary and sufficient conditions for irreducibility of tensor products of such evaluation modules.
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