Andruskiewitsch and Mombelli (2007) have established a general theory of module categories
over the representation category Rep(H) of a finite-dimensional Hopf algebra H: For an
indecomposable exact module category M over Rep(H), there exists a right H-simple
H-comodule algebra A with trivial coinvariants such that M is equivalent to Rep(A).
The Taft algebra Tq at a root of unity q is one of the simplest examples of pointed Hopf
algebras. Indecomposable exact module categories over Rep(Tq) have been classified by
Etingof and Ostrik (2004). The small quantum group H = uq(sl2) at a root of unity q
of odd order would be the next simplest example of pointed Hopf algebras (after the Taft
algebra) and has applications in various areas. For the coradically graded Hopf algebra
U of H, Mombelli (2010) has already classified right U -simple U -comodule algebras with
trivial coinvariants, and consequently classified exact module categories over Rep(U ). Since
Rep(U ) and Rep(H) are categorically Morita equivalent, one can (in principle) obtain a list
of indecomposable exact module categories over Rep(H) from Mombelli’s list.
In this talk, we will give an explicit list of right H-simple H-comodule algebras with trivial
coinvariants. The strategy is as follows. First, we note that H is a 2-cocycle deformation
of U and we can explicitly write down such a 2-cocycle σ. For a (right U -simple) U -
comodule algebra A, we can deform the algebra structure of A by using σ, which we
denote by σA. Then one sees that the resulting algebra σA is a (right H-simple) H-
comodule algebra. Moreover, Rep(σA) is the indecomposable exact module category over
Rep(H) corresponding to the indecomposable exact module category Rep(A) over Rep(U )
under the categorical Morita equivalence between Rep(H) and Rep(U ). Therefore, for
each A in Mombelli’s list, we will give an explicit description of σA. We note that the
determination of the σ-cocycle deformation σA of A is not a trivial problem. For example,
there is a 3-parameter family A(α, β, λ) (α, β, λ ∈ C) of U -comodule algebras generated
by a single element w subject to wN = λ and such that the U -coaction is given as w 7 →
(αx + βy) ⊗ 1 + g−1 ⊗ w, where N is the order of q and U = ⟨x, y, g±1⟩. After the 2-cocycle
deformation, the algebra σA(α, β, λ) is still generated by a single element w and a similar
H-coaction. However, the minimal polynomial of w becomes more complicated:
N −1Y
i=0
w − (ξ+q2i + ξ−q−2i),
where ξ± ∈ C are chosen so that they satisfy ξN
+ + ξN
− = λ and ξ+ξ−(1 − q2) = αβ.
This talk is based on ongoing joint work with Kenichi Shimizu (Shibaura Institute of
Technology) and Daisuke Nakamura (Okayama University of Science).
