Finite dimensional Hopf algebras over the Kac–Paljutkin algebra H 8

Yuxing Shi ^[1]
1. [1] Jiangxi Normal University
  
  Jiangxi Normal University
  
  China
Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 60, Nº. 1, 2019, págs. 265-298
Idioma: inglés
DOI: 10.33044/revuma.v60n1a17
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Resumen
- Let H8 be the Kac–Paljutkin algebra [Trudy Moskov. Mat. Obšč. 15 (1966), 224–261], which is the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra. All simple Yetter–Drinfel'd modules over H8 are given, and finite-dimensional Nichols algebras over H8 are determined completely. It turns out that they are all of diagonal type. In fact, they are of Cartan types A1, A2, A2×A2, A1×⋯×A1, and A1×⋯×A1×A2, respectively. By the way, we calculate Gelfand–Kirillov dimensions for some Nichols algebras. As an application, we complete the classification of the finite-dimensional Hopf algebras over H8 according to the lifting method.