Realization of U&lt;sub&gt;q&lt;/sub&gt;(sp&lt;sub&gt;2n&lt;/sub&gt;) within the Differential Algebra on Quantum Symplectic Space

Zhang, Jiao; Hu, Naihong

doi:10.3842/sigma.2017.084

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“…1010) (his motivation mainly from [33] & [34]), but not yet sufficient. It should be noticed that the design of our QDO in subsections 2.6, 3.4 (also see [13] & [37]) leads to our quantum differential (form) d satisfying the twisted Leibniz rule (see [7], pp. 5), which is different from both [33] and [34].…”

Section: 3mentioning

confidence: 99%

Quantum (dual) Grassmann superalgebra as $\mathcal U_q(\mathfrak{gl}(m|n))$-module algebra and beyond

Ge¹,

Hu²,

Zhang³

et al. 2019

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We introduce and define the quantum affine (m|n)-superspace (or say quantum Manin superspace) A m|n q and its dual object, the quantum Grassmann superalgebra Ω q (m|n). Correspondingly, a quantum Weyl algebra W q (2(m|n)) of (m|n)-type is introduced as the quantum differential operators (QDO for short) algebra Diff q (Ω q ) defined over Ω q (m|n), which is a smash product of the quantum differential Hopf algebra D q (m|n) (isomorphic to the bosonization of the quantum Manin superspace) and the quantum Grassmann superalgebra Ω q (m|n). An interested point of this approach here is that even though W q (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 Ω q is made into the U q (g)-module (super)algebra structure, Ω q = Ω q (m|n) for q generic, or Ω q (m|n, 1) for q root of unity, and g = gl(m|n) or sl(m|n), the general or special linear Lie superalgebra. This QDO approach provides us with explicit realization models for some simple U q (g)-modules, together with the concrete information on their dimensions. Similar results hold for the quantum dual Grassmann superalgebra Ω ! q as U q (g)-module algebra. This paper is a sequel to [13], some examples of pointed Hopf algebras can arise from the QDOs, whose idea is an expansion of the spirit noted by Manin in [20], & [21]. For instance, as byproducts, if considering the bosonizations of A m|n q, Ω q (m|n), D q (m|n) and their finite dimensional restricted objects when char(q) = ℓ > 2, we can obtain some examples of pointed Hopf algebras, e.g., the known multi-rank (generalized) Taft Hopf algebras. ContentsG. FENG, N.H. HU, M.R. ZHANG, AND X.T. ZHANG 2.5. Quantum (restricted) divided power algebras 7 2.6. q-Derivatives on A q (m) 8 3. Quantum Grassmann superalgebra and quantum Weyl superalgebra 8 3.1. Quantum exterior superalgebras 8 3.2. Quantum affine (m|n)-superspace A m|n q 9 3.3. Quantum Grassmann superalgebra 3.4. Quantum differential operators on Ω q 3.5. Quantum differential Hopf algebra D q (m|n) 3.6. Bosonization of quantum affine (m|n)-superspace 3.7. Multi-rank Taft (Hopf) algebra of (m|n)-type 3.8. Multi-rank Taft (Hopf) algebra of ℓ-type 3.9. Bosonization of quantum Grassmann superalgebra 3.10. Quantum Weyl algebra W q (2(m|n)) of (m|n)-type 4. The U q -module algebra structure over Ω q and its simple modules 4.1. Bosonization U q (gl(m|n)) of quantum superalgebra U q (gl(m|n)) 4.2. Ω q as U q -module algebra structure via quantum Weyl algebra 4.3. The submodule structures on homogeneous spaces of Ω q 5. Manin dual of quantum affine (m|n)-superspace A m|n q , quantum Grassmann dual superalgebra Ω ! q (m|n) and its module algebra 5.1. Manin dual of quantum affine (m|n)-superspace A m|n q 5.2. Quantum dual Grassmann superalgebra as U q -module algebra 5.3. The submodule structures on homogeneous spaces of Ω ! q

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