Normality and Short Exact Sequences of Hopf-Galois Structures

Every Hopf-Galois structure on a finite Galois extension K/k where G = Gal(K/k) corresponds uniquely to a regular subgroup N ≤ B = Perm(G), normalized by λ(G) ≤ B, in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra which acts on K/k is HN = (K[N])λ(G). For a given such N we consider the Hopf-Galois structure arising from a subgroup P ⊳ N that is also normalized by λ(G). This subgroup gives rise to a Hopf sub-algebra HP ⊆ HN with fixed field F = KHP . By the work of Chase and Sweedler, this yields a Hopf-Galois structure on the extension K/F where the action arises by base changing HP to F ⊗k HP which is an F-Hopf algebra. We examine this analogy with classical Galois theory, and also examine how the Hopf-Galois structure on K/F relates to that on K/k. We will also pay particular attention to how the Greither-Pareigis enumeration/construction of those HP acting on K/F relates to that of the HN which act on K/k. In the process we also examine short exact sequences of the Hopf algebras which act, whose exactness is directly tied to the descent theoretic description of these algebras

Opposite Skew Left Braces and Applications

Given a skew left brace B, we introduce the notion of an \opposite" skew left brace B0, which is closely related to the concept of the opposite of a group, and provide several applications. Skew left braces are closely linked with both solutions to the Yang-Baxter Equation and Hopf-Galois structures on Galois field extensions. We show that the set-theoretic solution to the YBE given by B0 is the inverse to the solution given by B. Every Hopf-Galois structure on a Galois field extension L=K gives rise to a skew left brace B; if the underlying Hopf algebra is not commutative, then one can construct an additional, \commuting" Hopf-Galois structure (see [10], which relates the Hopf-Galois module structures of each); the corresponding skew left brace to this second structure is precisely B0. We show how left ideals (and a newly introduced family of quasi-ideals) of B0 allow us to identify the intermediate fields of L=K which occur as fixed fields of sub-Hopf algebras under this correspondence. Finally, we use the opposite to connect the inverse solution to the YBE and the structure of the Hopf algebra H acting on L=K; this allows us to identify the group-like elements of H

Surprisingly Simple Spectra

The large N limit of the anomalous dimensions of operators in ${\cal N}=4$ super Yang-Mills theory described by restricted Schur polynomials, are studied. We focus on operators labeled by Young diagrams that have two columns (both long) so that the classical dimension of these operators is O(N). At large N these two column operators mix with each other but are decoupled from operators with $n\ne 2$ columns. The planar approximation does not capture the large N dynamics. For operators built with 2, 3 or 4 impurities the dilatation operator is explicitly evaluated. In all three cases, in a certain limit, the dilatation operator is a lattice version of a second derivative, with the lattice emerging from the Young diagram itself. The one loop dilatation operator is diagonalized numerically. All eigenvalues are an integer multiple of $8g_{YM}^2$ and there are interesting degeneracies in the spectrum. The spectrum we obtain for the one loop anomalous dimension operator is reproduced by a collection of harmonic oscillators. This equivalence to harmonic oscillators generalizes giant graviton results known for the BPS sector and further implies that the Hamiltonian defined by the one loop large $N$ dilatation operator is integrable. This is an example of an integrable dilatation operator, obtained by summing both planar and non-planar diagrams.Comment: 34 page

Microstructural damage of the posterior corpus callosum contributes to the clinical severity of neglect

One theory to account for neglect symptoms in patients with right focal damage invokes a release of inhibition of the right parietal cortex over the left parieto-frontal circuits, by disconnection mechanism. This theory is supported by transcranial magnetic stimulation studies showing the existence of asymmetric inhibitory interactions between the left and right posterior parietal cortex, with a right hemispheric advantage. These inhibitory mechanisms are mediated by direct transcallosal projections located in the posterior portions of the corpus callosum. The current study, using diffusion imaging and tract-based spatial statistics (TBSS), aims at assessing, in a data-driven fashion, the contribution of structural disconnection between hemispheres in determining the presence and severity of neglect. Eleven patients with right acute stroke and 11 healthy matched controls underwent MRI at 3T, including diffusion imaging, and T1-weighted volumes. TBSS was modified to account for the presence of the lesion and used to assess the presence and extension of changes in diffusion indices of microscopic white matter integrity in the left hemisphere of patients compared to controls, and to investigate, by correlation analysis, whether this damage might account for the presence and severity of patients' neglect, as assessed by the Behavioural Inattention Test (BIT). None of the patients had any macroscopic abnormality in the left hemisphere; however, 3 cases were discarded due to image artefacts in the MRI data. Conversely, TBSS analysis revealed widespread changes in diffusion indices in most of their left hemisphere tracts, with a predominant involvement of the corpus callosum and its projections on the parietal white matter. A region of association between patients' scores at BIT and brain FA values was found in the posterior part of the corpus callosum. This study strongly supports the hypothesis of a major role of structural disconnection between the right and left parietal cortex in determining 'neglect'

Whole genomes of deep-sea sponge-associated bacteria exhibit high novel natural product potential

Abstract Global antimicrobial resistance is a health crisis that can change the face of modern medicine. Exploring diverse natural habitats for bacterially-derived novel antimicrobial compounds has historically been a successful strategy. The deep-sea presents an exciting opportunity for the cultivation of taxonomically novel organisms and exploring potentially chemically novel spaces. In this study, the draft genomes of 12 bacteria previously isolated from the deep-sea sponges Phenomena carpenteri and Hertwigia sp. are investigated for the diversity of specialized secondary metabolites. In addition, early data support the production of antibacterial inhibitory substances produced from a number of these strains, including activity against clinically relevant pathogens Acinetobacter baumannii, Escherichia coli, Klebsiella pneumoniae, Pseudomonas aeruginosa and Staphylococcus aureus. Draft whole-genomes are presented of 12 deep-sea isolates, which include four potentially novel strains: Psychrobacter sp. PP-21, Streptomyces sp. DK15, Dietzia sp. PP-33, and Micrococcus sp. M4NT. Across the 12 draft genomes, 138 biosynthetic gene clusters were detected, of which over half displayed less than 50% similarity to known BGCs, suggesting that these genomes present an exciting opportunity to elucidate novel secondary metabolites. Exploring bacterial isolates belonging to the phylum Actinomycetota, Pseudomonadota and Bacillota from understudied deep-sea sponges provided opportunities to search for new chemical diversity of interest to those working in antibiotic discovery

Isomorphism problems for Hopf-Galois structures on separable field extensions

Let L=K be a finite separable extension of fields whose Galois closure E=K has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on L=K has the form E[N]G for some group N of order [L : K]. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and K-algebras that appear in the classification of Hopf-Galois structures on a cyclic extension of degree pn, for p an odd prime number

Impact of growth media and pressure on the diversity and antimicrobial activity of isolates from two species of hexactinellid sponge

Access to deep-sea sponges brings with it the potential to discover novel antimicrobial candidates, as well as novel cold- and pressure-adapted bacteria with further potential clinical or industrial applications. In this study, we implemented a combination of different growth media, increased pressure and high-throughput techniques to optimize recovery of isolates from two deep-sea hexactinellid sponges, Pheronema carpenteri and Hertwigia sp., in the first culture-based microbial analysis of these two sponges. Using 16S rRNA gene sequencing for isolate identification, we found a similar number of cultivable taxa from each sponge species, as well as improved recovery of morphotypes from P. carpenteri at 22–25 °C compared to other temperatures, which allows a greater potential for screening for novel antimicrobial compounds. Bacteria recovered under conditions of increased pressure were from the phyla Proteobacteria , Actinobacteria and Firmicutes , except at 4 %O2/5 bar, when the phylum Firmicutes was not observed. Cultured isolates from both sponge species displayed antimicrobial activity against Micrococcus luteus, Staphylococcus aureus and Escherichia coli .</jats:p