283 research outputs found
Conjugacy classes of p-cycles of type D in alternating groups
We classify the conjugacy classes of p-cycles of type D in alternating
groups. This finishes the open cases in arXiv:0812.4628. We also determine all
the subracks of those conjugacy classes which are not of type D.Comment: Second paragraph of subsection 2.2 rewritten. 4-th sentence of
subsection 2.4 rewritten. More explanations added in Remark 2.4. Lemma 2.5
and Corollary 2.7 added. Appendix removed and put it as Remark 3.1. Remark
3.2 (former 3.1) reorganized. References: [Da], [EGSS], [H], [IS] added,
[GPPS] removed. Communications in Algebra (2014
On Nichols algebras associated to simple racks
This is a report on the present state of the problem of determining the
dimension of the Nichols algebra associated to a rack and a cocycle. This is
relevant for the classification of finite-dimensional complex pointed Hopf
algebras whose group of group-likes is non-abelian. We deal mainly with simple
racks. We recall the notion of rack of type D, collect the known lists of
simple racks of type D and include preliminary results for the open cases. This
notion is important because the Nichols algebra associated to a rack of type D
and any cocycle has infinite dimension. For those racks not of type D, the
computation of the cohomology groups is needed. We discuss some techniques for
this problem and compute explicitly the cohomology groups corresponding to some
conjugacy classes in symmetric or alternating groups of low order.Comment: 26 pages, minor change
On Nichols algebras over PGL(2,q) and PSL(2,q)
We compute necessary conditions on Yetter-Drinfeld modules over the groups
\mathbf{PGL}(2,q)=\mathbf{PGL}(2,\FF_q) and
\mathbf{PSL}(2,q)=\mathbf{PSL}(2,\FF_q) to generate finite dimensional
Nichols algebras. This is a first step towards a classification of pointed Hopf
algebras with group of group-likes isomorphic to one of these groups.
As a by-product of the techniques developed in this work, we prove that there
is no non-trivial finite-dimensional pointed Hopf algebra over the Mathieu
groups and .Comment: Minor change
On Nichols algebras over SL(2,Fq) and GL(2,Fq)
We compute necessary conditions on Yetter-Drinfeld modules over the groups
SL(2,Fq) and GL(2,Fq) to generate finite dimensional Nichols algebras. This is
a first step towards a classification of pointed Hopf algebras with a group of
group-likes isomorphic to one of these groups.Comment: Major exposition revision, including referees remarks. To appear in
J. Math. Phys. 13 page
Braided racks, Hurwitz actions and Nichols algebras with many cubic relations
We classify Nichols algebras of irreducible Yetter-Drinfeld modules over
groups such that the underlying rack is braided and the homogeneous component
of degree three of the Nichols algebra satisfies a given inequality. This
assumption turns out to be equivalent to a factorization assumption on the
Hilbert series. Besides the known Nichols algebras we obtain a new example. Our
method is based on a combinatorial invariant of the Hurwitz orbits with respect
to the action of the braid group on three strands.Comment: v2: 35 pages, 6 tables, 14 figure
The logbook of Pointed Hopf algebras over the sporadic groups
In this notes we give details of the proofs performed with GAP of the
theorems of our paper "Pointed Hopf Algebras over the Sporadic Simple Groups".Comment: 22 pages, final versio
Finite-dimensional pointed Hopf algebras with alternating groups are trivial
It is shown that Nichols algebras over alternating groups A_m, m>4, are
infinite dimensional. This proves that any complex finite dimensional pointed
Hopf algebra with group of group-likes isomorphic to A_m is isomorphic to the
group algebra. In a similar fashion, it is shown that the Nichols algebras over
the symmetric groups S_m are all infinite-dimensional, except maybe those
related to the transpositions considered in [FK], and the class of type (2,3)
in S_5. We also show that any simple rack X arising from a symmetric group,
with the exception of a small list, collapse, in the sense that the Nichols
algebra of (X,q) is infinite dimensional, for q an arbitrary cocycle.
arXiv:0904.3978 is included here.Comment: Changes in version 7: We eliminate the Subsection 3.3 and references
to type C throughout the paper. We remove Lemma 3.24, Proposition 3.28 and
Example 3.29 (old numbering), since they are not needed in the present paper.
Several minor mistakes are corrected. The proof of Step 2 in Theorem 4.1 is
adjuste
Characterization of variable EST SSR markers for Norway spruce (Picea abies L.)
<p>Abstract</p> <p>Background</p> <p>Norway spruce is widely distributed across Europe and the predominant tree of the Alpine region. Fast growth and the fact that timber can be harvested cost-effectively in relatively young populations define its status as one of the economically most important tree species of Northern Europe. In this study, EST derived simple sequence repeat (SSR) markers were developed for the assessment of putative functional diversity in Austrian Norway spruce stands.</p> <p>Results</p> <p>SSR sequences were identified by analyzing 14,022 publicly available EST sequences. Tri-nucleotide repeat motifs were most abundant in the data set followed by penta- and hexa-nucleotide repeats. Specific primer pairs were designed for sixty loci. Among these, 27 displayed polymorphism in a testing population of 16 <it>P. abies </it>individuals sampled across Austria and in an additional screening population of 96 <it>P. abies </it>individuals from two geographically distinct Austrian populations. Allele numbers per locus ranged from two to 17 with observed heterozygosity ranging from 0.075 to 0.99.</p> <p>Conclusions</p> <p>We have characterized variable EST SSR markers for Norway spruce detected in expressed genes. Due to their moderate to high degree of variability in the two tested screening populations, these newly developed SSR markers are well suited for the analysis of stress related functional variation present in Norway spruce populations.</p
Pointed Hopf algebras over some sporadic simple groups
Any finite-dimensional complex pointed Hopf algebra with group of group-likes
isomorphic to a sporadic group, with the possible exception of the Fischer
groups Fi22, the Baby Monster B and the Monster M, is a group algebra.Comment: 4 pages, v4: Minor changes, more groups include
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