2019/09/23 by Feng Ge, Feng, Ge, Naihong Hu +5
Mathematics · Physics and Astronomy · #17A70 #17B10 #17B37 #20G05 #20G42 #81R50 (Primary) #81R60 #81T70 #81T75 (Secondary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1909.10276
openalex publication_date 2019/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce and define the quantum affine (m|n)-superspace (or say quantum Manin superspace) Aqm|n and its dual object, the quantum Grassmann superalgebra Ωq(m|n). Correspondingly, a quantum Weyl algebra \mathcal Wq(2(m|n)) of (m|n)-type is introduced as the quantum differential operators (QDO for short) algebra \textrmDiffq(Ωq) defined over Ωq(m|n), which is a smash product of the quantum differential Hopf algebra \mathfrak Dq(m|n) (isomorphic to the bosonization of the quantum Manin superspace) and the quantum Grassmann superalgebra Ωq(m|n). An interested point of this approach here is that even though \mathcal Wq(2(m|n)) itself is in general no longer a Hopf algebra, so are some interesting sub-quotients existed inside. This point of view gives us one of main expected results, that is, the quantum (restricted) Grassmann superalgebra Ωq is made into the \mathcal Uq(\mathfrak g)-module (super)algebra structure,Ωq=Ωq(m|n) for q generic, or Ωq(m|n, \bold 1) for q root of unity, and \mathfrak g=\mathfrakgl(m|n) or \mathfrak sl(m|n), the general or special linear Lie superalgebra. This QDO approach provides us with explicit realization models for some simple \mathcal Uq(\mathfrak g)-modules, together with the concrete information on their dimensions. Similar results hold for the quantum dual Grassmann superalgebra Ωq^! as \mathcal Uq(\mathfrak g)-module algebra.In the paper some examples of pointed Hopf algebras can arise from the QDOs, whose idea is an expansion of the spirit noted by Manin in \citeMa, & \citeMa1.