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

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 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

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 $M_{20}$ and $M_{21}=\mathbf{PSL}(3,4)$.Comment: Minor change

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

Three-gluon vertex in arbitrary gauge and dimension

One-loop off-shell contributions to the three-gluon vertex are calculated, in arbitrary covariant gauge and in arbitrary space-time dimension, including quark-loop contributions (with massless quarks). It is shown how one can get the results for all on-shell limits of interest directly from the general off-shell expression. The corresponding general expressions for the one-loop ghost-gluon vertex are also obtained. They allow for a check of consistency with the Ward--Slavnov--Taylor identity.Comment: 41 pages, LaTex, plus 3 figures in separate file. Misprints (signs) in eqs.(4.26), (C.2), (C.4), (C.5) are corrected. To appear in Phys. Rev.

Muon anomalous magnetic dipole moment in supersymmetric theories

We study the muon anomalous magnetic dipole moment in supersymmetric theories. The impact of the recent Brookhaven E821 experimental measurement on both model-independent and model-dependent supersymmetric parameter spaces is discussed in detail. We find that values of tan\beta as low as 3 can be obtained while remaining within the E821 one-sigma bound. This requires a light smuon; however, we show that, somewhat surprisingly, no model-independent bound can be placed on the mass of the lightest chargino for any tan\beta greater than or equal to 3. We also show that the maximum contributions to the anomalous magnetic moment are insensitive to CP-violating phases. We provide analyses of the supersymmetric contribution to the muon anomalous magnetic moment in dilaton-dominated supergravity models and gauge-mediated supersymmetry-breaking models. Finally, we discuss how other phenomena, such as $B(b\to s\gamma)$, relic abundance of the lightest superpartner, and the Higgs mass may be correlated with the anomalous magnetic moment, but do not significantly impact the viability of a supersymmetric explanation, or the mass limits obtainable on smuons and charginos.Comment: 28 page

Very light CP-odd scalar in the Two-Higgs-Doublet Model

We show that a general two-Higgs-doublet model (THDM) with a very light CP-odd scalar (A) can be compatible with the rho parameter, Br(b --> s\gamma), R_b, A_b, (g-2) of muon, Br(Upsilon --> A gamma), and the direct search via the Yukawa process at LEP. For its mass around 0.2 GeV, the muon (g-2) and Br(Upsilon --> A \gamma) data require tan(beta) to be about 1. Consequently, A can behave like a fermiophobic CP-odd scalar and predominantly decay into a photon pair ("gamma gamma"), which registers in detectors of high energy collider experiments as a single photon signature when the momentum of A is large. We compute the partial decay width of Z --> A A A and the production rate of f \bar{f} --> Z A A --> Z +"gamma gamma", f^' {\bar f} --> W^{\pm} A A --> W^\pm + "gamma gamma" and f \bar f --> H^+ H^- --> W^+ W^- A A --> W^+ W^- + "gamma gamma" at high energy colliders such as LEP, Tevatron, LHC, and future Linear Colliders. Other production mechanisms of a light A, such as gg --> h --> AA --> "gamma gamma", are also discussed.Comment: Some improvementes, references updated, 3 new figures, one new appendix, abstract and conclusions unchaged. Version to appear in Physical Review