Revista de la
Unión Matemática Argentina
Finite dimensional Hopf algebras over the Kac–Paljutkin algebra $H_8$
Yuxing Shi
Volume 60, no. 1 (2019), pp. 265–298

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Let $H_8$ 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 $H_8$ are given, and finite-dimensional Nichols algebras over $H_8$ are determined completely. It turns out that they are all of diagonal type. In fact, they are of Cartan types $A_1$, $A_2$, $A_2\times A_2$, $A_1\times \cdots \times A_1$, and $A_1\times \cdots \times A_1\times A_2$, 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 $H_8$ according to the lifting method.