92 research outputs found
On finite-dimensional Hopf algebras
This is a survey on the state-of-the-art of the classification of
finite-dimensional complex Hopf algebras. This general question is addressed
through the consideration of different classes of such Hopf algebras. Pointed
Hopf algebras constitute the class best understood; the classification of those
with abelian group is expected to be completed soon and there is substantial
progress in the non-abelian case.Comment: 25 pages. To be presented at the algebra session of ICM 2014.
Submitted versio
On pointed Hopf algebras associated with alternating and dihedral groups
We classify finite-dimensional complex pointed Hopf algebra with group of
group-like elements isomorphic to A_5. We show that any pointed Hopf algebra
with infinitesimal braiding associated with the conjugacy class of \in
is infinite-dimensional if the order of is odd except for in . We also study pointed Hopf algebras over the dihedral groups.Comment: v2: minor corrections, we remove Table 1 and change Remark 3.4; v3:
minor corrections, we modify reference [FGV]; final version to appear in Rev.
Uni\'on Mat. Argent., 17 page
Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type I. Non-semisimple classes in PSL(n,q)
We show that Nichols algebras of most simple Yetter-Drinfeld modules over the
projective special linear group over a finite field, corresponding to
non-semisimple orbits, have infinite dimension. We spell out a new criterium to
show that a rack collapses.Comment: minor changes, to appear in Journal of Algebr
On infinite-dimensional Hopf algebras
This is a survey on pointed Hopf algebras with finite Gelfand-Kirillov
dimension and related aspects of the theory of infinite-dimensional Hopf
algebras.Comment: Comments are welcome! Version 2: references and a few details are
adde
Twisting Hopf algebras from cocycle deformations
Let be a Hopf algebra. Any finite-dimensional lifting of arising as a cocycle deformation of
defines a twist in the Hopf algebra , via
dualization. We follow this recipe to write down explicit examples and show
that it extends known techniques for defining twists. We also contribute with a
detailed survey about twists in braided categories.Comment: 20 page
Quantum subgroups of a simple quantum group at roots of 1
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.Comment: 29 pages, accepted in Compositio Mathematic
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