A q-analogue of convolution on the line

In this paper we study a q-analogue of the convolution product on the line in detail. A convolution product on the braided line was defined algebraically by Kempf and Majid. We adapt their definition in order to give an analytic definition for the q-convolution and we study convergence extensively. Since the braided line is commutative as an algebra, all results can be viewed both as results in classical q-analysis and in braided algebra. We define various classes of functions on which the convolution is well-defined and we show that they are algebras under the defined product. One particularly nice family of algebras, a decreasing chain depending on a parameter running through (0,1], turns out to have 1/2 as the critical parameter value above which the algebras are commutative. Morerover, the commutative algebras in this family are precisely the algebras in which each function is determined by its q-moments. We also treat the relationship between q-convolution and q-Fourier transform. Finally, in the Appendix, we show an equivalence between the existence of an analytic continuation of a function defined on a q-lattice, and the behaviour of its q-derivatives.Comment: 31 pages; many small corrections; accepted by Methods and Applications of Analysi

Cocycle twisting of E(n)-module algebras and applications to the Brauer group

We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group $BQ(k,E(n))$ of E(n) that are isomorphic. For a triangular structure $R$ on E(n) we prove that the subgroup $BM(k,E(n),R)$ of $BQ(k,E(n))$ arising from $R$ is isomorphic to a direct product of $BW(k)$, the Brauer-Wall group of the ground field $k$, and $Sym_n(k)$, the group of $n \times n$ symmetric matrices under addition. For a general quasi-triangular structure $R'$ on E(n) we construct a split short exact sequence having $BM(k,E(n), R')$ as a middle term and as a left term a central extension of the group of symmetric matrices of order $r<n$ ($r$ depending on $R'$). We finally describe how the image of the Hopf automorphism group inside $BQ(k,E(n))$ acts on $Sym_n(k)$.Comment: Accidentally an old version of the paper was posted. Main corrections are in Section 2 and in Section 4.

On spherical twisted conjugacy classes

Let G be a simple algebraic group over an algebraically closed field of good odd characteristic, and let theta be an automorphism of G arising from an involution of its Dynkin diagram. We show that the spherical theta-twisted conjugacy classes are precisely those intersecting only Bruhat cells corresponding to twisted involutions in the Weyl group. We show how the analogue of this statement fails in the triality case. We generalize to good odd characteristic J-H. Lu's dimension formula for spherical twisted conjugacy classes.Comment: proof of Lemma 6.4 polished. The journal version is available at http://www.springerlink.com/content/k573l88256753640

Size, shape and surface chemistry of nano-gold dictate its cellular interactions, uptake and toxicity

Colloidal gold is undoubtedly one of the most extensively studied nanomaterials, with 1000s of different protocols currently available to synthesise gold nanoparticles (AuNPs). While developments in the synthesis of AuNPs have progressed rapidly in recent years, our understanding of their biological impact, with particular respect to the effect of shape, size, surface characteristics and aggregation states, has struggled to keep pace. It is generally agreed that when AuNPs are exposed to biological systems, these parameters directly influence their pharmacokinetic and pharmacodynamic properties by influencing AuNPs distribution, circulation time, metabolism and excretion in biological systems. However, the rules governing these properties, and the science behind them, are poorly understood. Therefore, a systematic understanding of the implications of these variables at the nano-bio interface has recently become a topic of major interest. This Review Article attempts to ignite a discussion around the influence of different physico-chemical parameters on biological activity of AuNPs, while focussing on critical aspects of cellular interactions, uptake and cytotoxicity. The review also discusses emerging trends in AuNP uptake and toxicity that are leading to technological advances through AuNP-based therapy, diagnostics and imaging

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type II. Unipotent classes in the symplectic groups

We show that Nichols algebras of most simple Yetter-Drin-feld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterium to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes in Chevalley and Steinberg groups

Quotients for sheets of conjugacy classes

We provide a description of the orbit space of a sheet S for the conjugation action of a complex simple simply connected algebraic group G. This is obtained by means of a bijection between S/G and the quotient of a shifted torus modulo the action of a subgroup of the Weyl group and it is the group analogue of a result due to Borho and Kraft. We also describe the normalisation of the categorical quotient \overline{S}//G for arbitrary simple G and give a necessary and sufficient condition for S//G to be normal in analogy to results of Borho, Kraft and Richardson. The example of G_2 is worked out in detail

Quotients for sheets of conjugacy classes

We provide a description of the orbit space of a sheet S for the conjugation action of a complex simple simply connected algebraic group G. This is obtained by means of a bijection between S 15G and the quotient of a shifted torus modulo the action of a subgroup of the Weyl group and it is the group analogue of a result due to Borho and Kraft. We also describe the normalisation of the categorical quotient // for arbitrary simple G and give a necessary and sufficient condition for //G to be normal in analogy to results of Borho, Kraft and Richardson. The example of G2 is worked out in detail

The fundamental group and torsion group of Beauville surfaces

We give a survey on the fundamental group of surfaces isogenous to a higher product. If the surfaces are regular, e.g. if they are Beauville surfaces, the first homology group is a finite group. We present a MAGMA script which calculates the first homology groups of regular surfaces isogenous to a product.Comment: 14 pages; MAGMA script included; v2: minor corrections, final version to appear in the Proceedings of the Conference "Beauville Surfaces and Groups", Newcastle University (UK), 7-9th June 201