Irreducible Modules Over Witt Algebras Wn$\mathcal {W}_{n}$ and Over 𝖘𝖑n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$

By H. Tan, Kaiming Zhao. 2013

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In this paper, by using the “twisting technique” we obtain a class of new modules Ab over the Witt algebras Wn$\mathcal {W}_{n}$ from modules A over the Weyl algebras Kn$\mathcal {K}_{n}$ (of Laurent polynomials) for any b∈ℂ$b\in \mathbb {C}$. We give necessary and sufficient conditions for Ab to be irreducible, and determine necessary and sufficient conditions for two such irreducible Wn$\mathcal {W}_{n}$-modules to be isomorphic. Since 𝔰𝔩n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$ is a subalgebra of Wn$\mathcal {W}_{n}$, all the above irreducible Wn$\mathcal {W}_{n}$-modules Ab can be considered as 𝔰𝔩n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$-modules. For a class of such 𝔰𝔩n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$-modules, denoted by Ω1−a(λ1, λ2, ⋯ ,λn) where a∈ℂ,λ1,λ2,⋯,λn∈ℂ∗$a\in \mathbb {C}, \lambda _{1},\lambda _{2},\cdots ,\lambda _{n} \in \mathbb {C}^{*}$, we determine necessary and sufficient conditions for these 𝔰𝔩n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$-modules to be irreducible. If the 𝔰𝔩n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$-module Ω1−a(λ1, λ2,⋯ ,λn) is reducible, we prove that it has a unique nontrivial submodule W1−a(λ1, λ2,...λn) and the quotient module is the finite dimensional 𝔰𝔩n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$-module with highest weight mΛn for some non-negative integer m∈ℤ+$m\in \mathbb {Z}_{+}$. We also determine necessary and sufficient conditions for two 𝔰𝔩n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$-modules of the form Ω1−a(λ1, λ2,⋯ ,λn) or of the form W1−a(λ1, λ2,...λn) to be isomorphic.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/69bf9b53debc21e5d383628211f5a114823b645f