arXiv subjects
Publications and source records attributed to Gaston Andres Garcia.
We complete the classification of quantum subgroups of $SL_q(2)$ with $q$ a root of unity of arbitrary order, that is, Hopf algebra quotients of the quantum function algebras $\mathcal{O}_{q} (SL_2(\mathbb{C}))$.
This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra K_{n}, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K_{3}. This includes a family of 12-dimensional Nichols algebras $B_ξ$ depending on 3rd roots of unity. Here, $B_{1}$ is isomorphic to the well-known Fomin-Kirillov algebra, and $B_ξ \simeq B_{ξ^{2}} $ as graded algebras but $B_1$ is not isomorphic to $B_ξ $ as algebra for $ξ\neq 1$. As a byproduct we obtain new Hopf algebras of dimension 216.
We describe how to find quantum determinants and antipode formulas from braided vector spaces using the FRT-construction and finite-dimensional Nichols algebras. It generalizes the construction of quantum function algebras using quantum grassmanian algebras. Given a finite-dimensional Nichols algebra B, our method provides a Hopf algebra H such that B is a braided Hopf algebra in the category of H-comodules. It also serves as source to produce Hopf algebras generated by cosemisimple subcoalgebras, which are of interest for the generalized lifting method. We give several examples, among them quantum function algebras from Fomin-Kirillov algebras associated with the symmetric groups on three letters.
We classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero such that its coradical is isomorphic to the algebra of functions over a dihedral group D_m, with m=4a> 11. We obtain this classification by means of the lifting method, where we use cohomology theory to determine all possible deformations. Our result provides an infinite family of new examples of finite-dimensional copointed Hopf algebras over dihedral groups.
Let p be an odd prime and k an algebraically closed field of characteristic zero. We classify nonsemisimple Hopf algebras over k of dimension 8p with the Chevalley property, and give partial results on nonsemisimple Hopf algebras of dimension 24.
Using the standard filtration associated with a generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf subalgebra isomorphic to the smallest non-pointed non-cosemisimple Hopf algebra ${\mathcal{K}}$ and the corresponding infinitesimal module is an indecomposable object in ${}^{\mathcal{K}}_{\mathcal{K}}\mathcal{YD}$ (we assume that the diagrams are Nichols algebras). As a byproduct, we obtain new Nichols algebras of dimension 8 and new Hopf algebras of dimension 64.
Let $G$ be a connected, simply connected simple complex algebraic group and let $ε$ be a primitive $\ell$th root of unity with $\ell$ odd and coprime with $3$ if $G$ is of type $G_{2}$. We determine all Hopf algebra quotients of the twisted multiparameter quantum function algebra $\mathcal{O}_ε^φ(G)$ introduced by Costantini and Varagnolo. This extends the results of Andruskiewitsch and the first author, where the untwisted case is treated.
Classifying all Hopf algebras of a given finite dimension over the complex numbers is a challenging problem which remains open even for many small dimensions, not least because few general approaches to the problem are known. Some useful techniques include counting the dimensions of spaces related to the coradical filtration, studying sub- and quotient Hopf algebras, especially those sub-Hopf algebras generated by a simple subcoalgebra, working with the antipode, and studying Hopf algebras in Yetter-Drinfeld categories to help to classify Radford biproducts. In this paper, we add to the classification tools in our previous work [arXiv:1108.6037v1] and apply our results to Hopf algebras of dimension rpq and 8p where p,q,r are distinct primes. At the end of this paper we summarize in a table the status of the classification for dimensions up to 100 to date.
We describe quantum groups given by multiparametric deformations of enveloping algebras of Kac-Moody algebras as a family of pointed Hopf algebras introduced by Andruskiewitsch and Schneider associated to a generalized Cartan matrix. We show that under some assumptions, these Hopf algebras depend only on one parameter on each connected component of the Dynkin diagram, up to a cocycle deformation. In particular, we obtain in this way a known result of Hu, Pei and Rosso.
We introduce a method to construct explicitly multiplicative 2-cocycles for bosonizations of Nichols algebras B(V) over Hopf algebras H. These cocycles arise as liftings of H-invariant linear functionals on V tensor V and give a close formula to deform braided commutator-type relations. Using this construction, we show that all known finite dimensional pointed Hopf algebras over the dihedral groups D_m with m=4t > 11, over the symmetric group S_3 and some families over S_4 are cocycle deformations of bosonizations of Nichols algebras.
The classification of all Hopf algebras of a given finite dimension over an algebraically closed field of characteristic 0 is a difficult problem. If the dimension is a prime, then the Hopf algebra is a group algebra. If the dimension is the square of a prime then the Hopf algebra is a group algebra or a Taft Hopf algebra. The classification is also complete for dimension 2p or 2p^2, p a prime. Partial results for some other cases are available. For example, for dimension p^3 the classification of the semisimple Hopf algebras was done by Masuoka, and the pointed Hopf algebras were classified by Andruskiewitsch and Schneider, Caenepeel and Dascalescu, and Stefan and van Oystaeyen independently. Many classification results for the nonsemisimple, nonpointed, non-copointed case have been proved by the second author but the classification in general for dimension p^3 is still incomplete, up to now even for dimension 27. In this paper we outline some results and techniques which have been useful in approaching this problem and add a few new ones. We give some further results on Hopf algebras of dimension p^3 and finish the classification for dimension 27.
Let k be an algebraically closed field of characteristic 0. We conclude the classification of finite dimensional pointed Hopf algebras whose group of group-likes is S_4. We also describe all pointed Hopf algebras over S_5 whose infinitesimal braiding is associated to the rack of transpositions.
Let a, b be non-zero complex numbers and l an odd natural number bigger that 2. We determine all Hopf algebra quotients of the quantized coordinate algebra O_{a,b}(GL_{n}) when a^{-1}b is a primitive l-th root of unity and a, b satisfy certain mild conditions, and we caracterize all finite-dimensional quotients when a^{-1}b is not a root of unity. As a byproduct we give a new family of non-semisimple and non-pointed Hopf algebras with non-pointed duals which are quotients of O_{a, b}(GL_{n}).
Let G be a connected, simply connected, simple complex algebraic group and let e be a primitive l-th root of 1, with l odd and 3 does not divide l if G is of type G_{2}. We determine all Hopf algebra quotients of the quantized coordinate algebra of G at e.
We complete the classification of Hopf algebras of dimension 16 over an algebraically closed field of characteristic zero. We show that a non-semisimple Hopf algebra of dimension 16, has either the Chevalley property or its dual is pointed.
We discuss some general results on finite-dimensional Hopf algebras over an algebraically closed field k of characteristic zero and then apply them to Hopf algebras H of dimension p^{3} over k. There are 10 cases according to the group-like elements of H and H^{*}. We show that in 8 of the 10 cases, it is possible to determine the structure of the Hopf algebra. We give also a partial classification of the quasitriangular Hopf algebras of dimension p^{3} over k, after studying extensions of a group algebra of order p by a Taft algebra of dimension p^{2}. In particular, we prove that every ribbon Hopf algebra of dimension p^{3} over k is either a group algebra or a Frobenius-Lusztig kernel. Finally, using some previous results on bounds for the dimension of the first term H_{1} in the coradical filtration of H, we give the complete classification of the quasitriangular Hopf algebras of dimension 27.