Classification of Hopf superalgebras associated with quantum special\n linear superalgebra at roots of unity using Weyl groupoid

2020/06/11 by A. Mazurenko, Mazurenko, Alexander, Vladimir Stukopin +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2006.06610

openalex publication_date 2020/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We summarize the definition of the Weyl groupoid using supercategory approach\nin order to investigate quantum superalgebras at roots of unity. We show how\nthe structure of a Hopf superalgebra on a quantum superalgebra is determined by\nthe quantum Weyl groupoid. The Weyl groupoid of mathfraksl(m|n) is\nconstructed to this end as some supercategory. We prove that in this case\nquantum superalgebras associated with Dynkin diagrams are isomorphic as\nsuperalgebras. It is shown how these quantum superalgebras considered as Hopf\nsuperalgebras are connected via twists and isomorphisms. We explicitly\nconstruct these twists using the Lusztig isomorphisms considered as elements of\nthe Weyl quantum groupoid. We build a PBW basis for each quantum superalgebra,\nand investigate how quantum superalgebras are connected with their classical\nlimits, i. e. Lie superbialgebras. We find explicit multiplicative formulas for\nuniversal R-matrices, describe relations between them for each realization\nand classify Hopf superalgebras and triangular structures for the quantum\nsuperalgebra U_q( mathfraksl(m|n)).\n

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