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After the classification of the finite-dimensional Nichols algebras ofdiagonal type arXiv:math/0411477, arXiv:math/0605795, the determination of itsdefining relations arXiv:1008.4144, arXiv:1104.0268, and the verification ofthe generation in degree one conjecture arXiv:1104.0268, there is still onestep missing in the classification of complex finite-dimensional Hopf algebraswith abelian group, without restrictions on the order of the latter: thecomputation of all deformations or liftings. A technique towards solving thisquestion was developed in arXiv:1212.5279, built on cocycle deformations. Inthis paper, we elaborate further and present an explicit algorithm to computeliftings. In our main result we classify all liftings of finite-dimensionalNichols algebras of Cartan type $A$, over a cosemisimple Hopf algebra $H$. Thisextends arXiv:math/0110136, where it was assumed that the parameter is a rootof unity of order $>3$ and that $H$ is a commmutative group algebra. When theparameter is a root of unity of order 2 or 3, new phenomena appear: the quantumSerre relations can be deformed, this allows in turn the power root vectors tobe deformed to elements in lower terms of the coradical filtration, but notnecessarily in the group algebra. These phenomena are already present in thecalculation of the liftings in type $A_2$ at a parameter of order 2 or 3 overan abelian group arXiv:math/0204075, arXiv:1003.5882, done by a differentmethod using a computer program. As a by-product of our calculations, wepresent new infinite families of finite-dimensional pointed Hopf algebras.

Liftings of Nichols algebras of diagonal type I. Cartan type A