EXTENSIONS OF THE CATEGORY OF COMODULES OF THE TAFT ALGEBRA

. We construct a family of non-equivalent pairwise extensions of the category of comodules of the Taft algebra, which are equivalent to representation categories of non-triangular quasi-Hopf algebras.


Introduction
Given a finite group G and a fusion category C, the G-extensions of C were classified in [3], however to give concrete examples of these classification in general is complicated.In the literature there are few examples of these extensions when C is non-semisimple.A different version, called Crossed Products was introduced in [4], where the parameters to construct those extensions are calculable when C is the category of comodules over a Hopf algebra H.The main difference with the work of [3] is that we need to calculate the Brauer-Picard group of the category comod(H), and for the work in [4] we only need the group of biGalois objects of the Hopf algebra.
Following this idea, in [5], we construct eight tensor categories which are extensions of the category of comodules over a supergroup algebra and in [6], we analyze when these categories are braided.
In this work we construct an infinite family (non equivalent pairwise) of C 2extensions of the category of comodules over the Taft algebra T (q), where C 2 is the cyclic group of two elements.As Abelian categories, they are two copies of the category of comodules of T (q) with tensor product described in Equations ( 6) and (7), and non-trivial associativity constrains.
Since T (q) is not a co-quasitriangular Hopf algebra, Comod(T (q)) is not braided then [6,Theorem 2.6] any extension of Comod(T (q)) is not braided.Therefore the categories described here are not braided.Nevertheless, each one is equivalent to the category of representations over some non-triangular quasi-Hopf algebra, using Frobenius-Perron dimension.In particular, this is another example of how results obtained in a categorical context produce results (of existence) in a context of Hopf algebras.Several examples of this have been introduced in the literature, for example in [2], the classification of braided unipotent tensor categories gives place to the classification of coconnected coquasitriangular Hopf algebras; and in [1] a result over modular categories allows to prove the Kaplansky's conjecture for quasitriangular semisimple Hopf algebras.

Preliminaries
2.1.Hopf algebras and BiGalois objects.In this work we work over the complex field C. Let H be a Hopf algebra and g ∈ G(H) be a group-like element.We denote C g the one-dimensional vector space generated by w g with left H-comodule given by λ : If g is a group-like element we can define a new H-biGalois object A g on the same underlying algebra A with unchanged right comodule structure and a new left Hcomodule structure given by λ g : A g → H ⊗ A g , λ g (a) = g −1 a (−1) g ⊗ a (0) for all a ∈ A. Recall [7] that two H-biGalois objects A, B are equivalent, if there exists an element g ∈ G(H) such that A g B as biGalois objects.
BiGal(H) is a group with the cotensor product H , where H is the unit, and for L ∈ BiGal(H), λ : L → H H L and ρ : L → L H H are the isomorphisms induced by the left and right coactions, with inverses induced by the counit.The subgroup of BiGal(H) consisting of H-biGalois objects equivalent to H is denoted by InnbiGal(H).This group is a normal subgroup of BiGal(H).We denote 2.2.Autoequivalences on categories.Let C be a finite tensor category.Given an invertible object σ ∈ C, we define the monoidal equivalence Proposition 2.1.Let F : C → C be an autoequivalence.
(1) F is pseudonatural equivalent to id C if and only if F is monoidal equivalent to Ad σ for some invertible object σ ∈ C.
(2) Ad σ is monoidal equivalent to Ad τ if and only if σ −1 ⊗τ admits an structure By the definition we have natural isomorphisms defines a pseudonatural equivalence of id C , i.e., an invertible object in Z(C), that satisfies the condition of the proposition.
2.3.C 2 -crossed product tensor categories.In [4], Galindo introduced a way to construct extensions of a given category.When the graded group is C 2 , the cyclic group of order 2, in [5], the authors give a complete classification if the tensor category in degree zero is the category of comodules of supergroup algebras.Theorem 2.2.[5, Section 5.1, Lemma 5.9] There is a correspondence between C 2crossed product tensor categories over Comod(H) and collections (L, g, f, γ) where The tensor categories associated to two collections (L, g, f, γ) and (L , g , f , γ ) are monoidally equivalent if, and only if, there exist a collection (A, h, ϕ, τ ) where and also the following equations are fulfilled

Taft algebra
Let N ≥ 2 be an integer and let q ∈ C be a primitive N -th root of unity.The Taft algebra T (q) is the C-algebra presented by generators X and Y with relations 0 and Y X = qXY .The algebra T (q) carries a Hopf algebra structure, determined by the group of group-like elements of T (q) is G(T (q)) = X Z/(N ), (3) T (q) T (q) * , (4) T (q) T (q ) if and only if q = q .For all α ∈ C * and β ∈ C, Schauenburg [7] proved that the T (q)-biGalois objects are the algebras A α,β := k x, y /(x N = α, y N = β, yx = qxy), with right ρ and left λ comodule structures The biGalois objects A α,β are representative sets of equivalence classes of biGalois objects [7, Theorem 2.2].There exists a group isomorphism A αα ,βα +β and there is a canonical isomorphism Schauenburg also calculates the group of Hopf algebra automorphism [7, Lemma 2.1], where r → f r with f r (X) = X and f r (Y ) = rY is a group isomorphism; and for X r ∈ G(T (q)), ϕ(Ad X r ) = f q −r .Also there exists a group homomorphism Aut Hopf (T (q)) → BiGal(T (q)) given by f → f H where as a vector space is H with left coaction given by v → f (v (−1) ) ⊗ v (0) .Regarding about bicomodule algebra isomorphisms of T (q), by [7, Theorem 2.2.3], they are precisely ι p for p N = 1 where Now, it is possible to calculate the inner and outer biGalois objects.

C 2 -Crossed product tensor categories
Now, we apply Theorem 2.2 to H = T (q), where the biGalois objects are parametrized by L = A α,β with α ∈ C * and β ∈ C. Fix A α,β , since G(T (q)) = {X s |s = 0, . . ., N − 1}, for a given s < N , A α,β T (q) C X s = C{x s ⊗ 1} is onedimensional; moreover the left coaction of T over • A α,β T (q) C X s is given by A α 2 ,βα+β by Equation ( 5), then there exists such f if, and only if, We obtain L ∈ {T (q), A −1,β |β ∈ C} and g ∈ {X s |s < N }.Now, we explicitly need to determine all comodule algebra morphism f .Since (L T (q) L) g T (q), f is parametrized by the bicomodule algebra automorphisms of T (q) described in Equation ( 4).
Lemma 4.1.Each collection (L, g, f, γ) where We explicitly described the tensor structure.As an Abelian category Since the category is C 2 -graded, we denoted the objects as V 1 or V u where V ∈ Comod(T (q)), C 2 = {1, u|u 2 = 1}.The tensor product is given The left comodule structure over The associativity is trivial except (V u ⊗ W u ) ⊗ Z u , which is defined using ι p and γ, see [5, Section 6.1].
As an Abelian category The tensor product is given Next, we determine in which cases these collections generates monoidally equivalent categories, applying the second part of Theorem 2.2.Notice that L in Lemma 4.1 has two options, then we consider in the next propositions these three possible cases.Combining them we obtain Theorem 4.5.
For each 1 = p ∈ C with p N = 1, s < N and γ ∈ {±1}, we obtain a family of non-equivalent categories the second set appears when p = p = 1, then the possible values for s are 0, 1.
as monoidal categories if, and only if, X s−s = X 2t for any t < N and p = p = 1.
Proof.Let A = A α,β be a biGalois object and h ∈ G(T (q)), then and consider the following diagram.
Notice that we use the same notation δ 0 to different morphisms, since they have the same definition but different domains and codomains.Then, Equation (2) (exterior of (11)) is equivalent to as in the previous proof, Equation ( 12) is valid if, and only if, s = s = 1.
For each 1 = p ∈ C with p N = 1, s < N , γ ∈ {±1} and β ∈ C × , we obtain a family of non-equivalent categories notice that the second set only depends on β, since for p = p = 1, as before, s is 0, 1.In (10) we calculate non-equivalent categories when associated biGalois objects are trivial, here in (13) when they are non-trivial.
(3) Non-trivial and trivial biGalois objects in the tuples.
As before, we obtain the following result.
Since F P dim(T (q)) is an integer, the Frobenius-Perron dimension of any of the categories listed before is 2F P dim(T (q)), then they are the category of representations of a quasi-Hopf algebra.