Lifting via cocycle deformation

Abstract

We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such lifting.

Representations of copointed Hopf algebras arising from the tetrahedron rack

“…We denote by A k G = A as right k G -module via the right multiplication. [AAGMV,Lemma 4.1]. Hence we can think of R as a left k G -submodule of A and therefore…”

“…Any spherical Hopf algebra H has an associated tensor category Rep(H) which is a quotient of Rep(H), see [AAGMV,BaW1,BaW2] for the background of this subject. Moreover, Rep(H) is semisimple but rarely is a fusion category in the sense of [ENO], i. e. Rep(H) rarely has a finite number of irreducibles.…”

Finite-dimensional Pointed or Copointed Hopf algebras over affine racks

“…Thus the first part of the lemma is proved. Then π 2 (z)t −1 z generates a normal subalgebra which forms the coinvariants by [AAGMV,Remark 5.5]. It is a polynomial algebra by [AAGMV,Lemma 5.13].…”

“…Hence π n (z ′ ) is central in B n (q, X). The lemma follows using [AAGMV,Remark 5.5,Lemma 5.13] as in Lemma 4.3.…”

“…Assume that the ideal J (V ) defining the Nichols algebra B(V ) is finitely generated and let G be a minimal set of homogeneous generators of J (V ). In [AAGMV] a strategy was developed to compute all the liftings of B(V ) over H as cocycle deformations of B(V )#H. We briefly recall this strategy, see [AAGMV,Section 5] for details.…”