Linkable Dynkin diagrams

Didt, Daniel

doi:10.1016/s0021-8693(02)00148-5

Cited by 6 publications

(5 citation statements)

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“…This is a Hopf algebra, cf. [D,AS5]. We order the vertices in the Dynkin diagram so that the root vectors of the k th component are e S k +i,S k +j , 1 ≤ i < j ≤ n k + 1, where S k = n 1 + · · · + n k−1 .…”

Section: The Mixed Casementioning

confidence: 99%

Pointed Hopf Algebras and Quasi-isomorphisms

Didt

2005

Algebr Represent Theor

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We show that a large class of finite dimensional pointed Hopf algebras is quasi-isomorphic to their associated graded version coming from the coradical filtration, i.e. they are 2-cocycle deformations of the latter. This supports a slightly specialized form of a conjecture in [M]. IntroductionRecently there has been a lot of progress in determining the structure of pointed Hopf algebras. This has led to a discovery of whole new classes of such Hopf algebras and to some important classification results. See the survey article [AS4] for an introduction and references. A lot of these classes contain infinitely many non-isomorphic Hopf algebras of the same dimension, thus disproving an old conjecture of Kaplansky. Masuoka showed in [M], and in a private note, that for certain of these new families the Hopf algebras are all 2-cocycle deformations of each other. This led him to weaken Kaplansky's conjecture, stating that up to quasi-isomorphisms there should only be a finite number of Hopf algebras of a given dimension. This was disproved in [EG] for a family of Hopf algebras of dimension 32. However, our results support Masuoka's conjecture in a really big class of * This work will be part of the author's Ph.D. thesis written under the supervision of Professor H.-J. Schneider. The author is a member of the Graduiertenkolleg "Mathematik im Bereich ihrer Wechselwirkung mit der Physik" at Munich University.

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Section: The Mixed Casementioning

confidence: 99%

Pointed Hopf Algebras and Quasi-isomorphisms

Didt

2005

Algebr Represent Theor

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“…We collect some useful facts from [2,8]: any vertex i is linkable to at most one vertex j; if i, j are linkable, then q ii = q j j , q i j q ji = 1; if i and k, respectively, j and l, are linkable, then a i j = a kl , a ji = a lk .…”

Section: Preliminariesmentioning

confidence: 99%

“…ad c (x i ) 1−a i j (x j ) = 0, for all i ∼ j, i j, (8) x i x j − q i j x j x i = λ i j (1 − g i g j ), for all i ≁ j. (9) where ad c (x i )(x j ) = x i x j −q a i j x j x i .…”

Section: Hopf Algebra U(d λ 2 )mentioning

confidence: 99%

Representation theory of a Class of Hopf algebras

Wang,

Chen

2009

Preprint

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“…However, in the more general case where the relations (5.9) have a non-zero right side, one needs general linking data. See [13] for a combinatorial description of all linkings of Dynkin diagrams.…”

Section: Definition 57 -Let Us Fix a Decompositionmentioning

confidence: 99%

“…(iv) The question of finding all the possible linking elements attached to a fixed collection g i , χ i , 1 i θ, (a ij ) 1 i,j θ , is also of combinatorial nature, see Section 5, and also [13]. Once these two problems are solved effectively, the isomorphism classes of the Hopf algebras u(D) can be determined using [5,Prop.…”

Section: Introductionmentioning

confidence: 99%

Finite quantum groups over abelian groups of prime exponent

Andruskiewitsch

Schneider

2002

Annales Scientifiques de l’École Normale Supérieure

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We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p > 17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius-Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. An important step in the classification result is to show that all these Hopf algebras are generated by group-like and skew-primitive elements.  2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Nous classifions les algèbres de Hopf complexes de dimension finie dont le groupe des éléments groupoïdaux est abélien d'exposant premier p, p > 17. Les algèbres de Hopf que nous trouvons appartiennent à une famille d'algèbres de Hopf pointées que nous construisons à partir de diagrammes de Dynkin. Comme cas particuliers de notre construction nous obtenons tous les noyaux de Frobenius-Lusztig des algèbres de Lie semi-simples et leurs sous-algèbres paraboliques. Une étape importante dans notre classification consiste à montrer que toutes ces algèbres de Hopf sont engendrées par des éléments groupoïdaux et des éléments primitifs tordus.

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Linkable Dynkin diagrams

Cited by 6 publications

References 6 publications

Pointed Hopf Algebras and Quasi-isomorphisms

Pointed Hopf Algebras and Quasi-isomorphisms

Representation theory of a Class of Hopf algebras

Finite quantum groups over abelian groups of prime exponent

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