2024/04/11 by Chih-Whi Chen, Yongjie Wang, Chen, Chih-Whi +1
Mathematics · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2404.07894
We investigate representations of the ℓ-th Takiff superalgebras \widetilde\mathfrak g_ℓ := \widetilde\mathfrak g⊗ \mathbb C[θ]/(θℓ+1), for ℓ>0, associated with a basic classical and a periplectic Lie superalgebras \widetilde\mathfrak g. We introduce the odd reflections and formulate a general notion of typical representations of the Takiff superalgebras \widetilde\mathfrak g_ℓ. As a consequence, we provide a complete description of the characters of the finite-dimensional modules over type I Takiff superalgebras. For the Lie superalgebras \widetilde\mathfrak g= \mathfrakgl(m|n) and \mathfrakosp(2|2n), we prove that the Kac induction functor of \widetilde\mathfrak g_ℓ leads to an equivalence from an arbitrary typical Jordan block of the category \mathcal O for \widetilde\mathfrak g_ℓ to a Jordan block of the category \mathcal O for the even subalgebra of \widetilde\mathfrak g_ℓ. We also obtain a classification of non-singular simple Whittaker modules over the Takiff superalgebras.