Classifying Hopf algebras of a given dimension

Classifying all Hopf algebras of a given finite dimension over the complex numbers is a challenging problem which remains open even for many small dimensions, not least because few general approaches to the problem are known. Some useful techniques include counting the dimensions of spaces related to the coradical filtration, studying sub- and quotient Hopf algebras, especially those sub-Hopf algebras generated by a simple subcoalgebra, working with the antipode, and studying Hopf algebras in Yetter-Drinfeld categories to help to classify Radford biproducts. In this paper, we add to the classification tools in our previous work [arXiv:1108.6037v1] and apply our results to Hopf algebras of dimension rpq and 8p where p,q,r are distinct primes. At the end of this paper we summarize in a table the status of the classification for dimensions up to 100 to date.Comment: This version of the paper contains a correction on the published version. The statement and proof of Proposition 2.17 are changed and the proof of the results that follow from it are corrected accordingly. We thank H.-S. Ng for kindly communicating the gap to us and for the careful reading of our pape

Cocycle deformations for liftings of quantum linear spaces

Let $A$ be a Hopf algebra over a field $K$ of characteristic 0 and suppose there is a coalgebra projection $\pi$ from $A$ to a sub-Hopf algebra $H$ that splits the inclusion. If the projection is $H$-bilinear, then $A$ is isomorphic to a biproduct R #_{\xi}H where $(R,\xi)$ is called a pre-bialgebra with cocycle in the category $_{H}^{H}\mathcal{YD}$. The cocycle $\xi$ maps $R \otimes R$ to $H$. Examples of this situation include the liftings of pointed Hopf algebras with abelian group of points $\Gamma$ as classified by Andruskiewitsch and Schneider [AS1]. One asks when such an $A$ can be twisted by a cocycle $\gamma:A\otimes A\rightarrow K$ to obtain a Radford biproduct. By results of Masuoka [Ma1, Ma2], and Gr\"{u}nenfelder and Mastnak [GM], this can always be done for the pointed liftings mentioned above. In a previous paper [ABM1], we showed that a natural candidate for a twisting cocycle is {$\lambda \circ \xi$} where $\lambda\in H^{\ast}$ is a total integral for $H$ and $\xi$ is as above. We also computed the twisting cocycle explicitly for liftings of a quantum linear plane and found some examples where the twisting cocycle we computed was different from {$\lambda \circ \xi$}. In this note we show that in many cases this cocycle is exactly $\lambda\circ\xi$ and give some further examples where this is not the case. As well we extend the cocycle computation to quantum linear spaces; there is no restriction on the dimension

The role of Empodisma robustum litter in CO₂ exchange at Kopuatai bog

Respiration from the decomposition of standing dead litter in peatlands influences the ecosystem carbon balance through its contribution to total ecosystem respiration (ER). This research determined the proportion of ER estimated at Kopuatai bog that is sourced from the decomposition of Empodisma robustum litter in the canopy. Canopy harvests were carried out to measure the mass of standing litter in the canopy; laboratory litter incubations were used to measure respiration rates over a range of temperatures and moisture contents; and a simple model was developed to estimate annual litter respiration using inputs of canopy wetness duration and canopy temperature. E. robustum litter comprised an average of 51% of the total canopy biomass, with 0.92 kg m⁻² standing litter dry matter in 1.8 kg m⁻² of total canopy dry matter. The majority of this litter is located in the lower part of the canopy. Very low respiration rates were measured for the E. robustum litter, although respiration was significantly higher in litter which was more physically decomposed (R₁₀ = 0.44 (± 0.1) μmol kg⁻¹ s⁻¹) than that which was freshly dead (R₁₀ = 0.24 (± 0.05) μmol kg⁻¹ s⁻¹). Litter respiration showed a strong temperature response, and was moisture-limited below approximately 50% moisture content. The model of litter respiration estimated that standing dead E. robustum litter contributed 59 g C m⁻² yr⁻¹ (8.8%) to annual total ER. This represents an estimated litter turnover time of 7 – 8 years. While the contribution of litter respiration to ER is relatively small, the resulting large mass of recalcitrant litter in the canopy may contribute to E. robustum’s ability to engineer its environment

Portfolio of compositions

This portfolio contains two works. The first, arising from equations describing chaotic systems, is a four-movement composition entitled Bifurcation of Nature. This is written for six amplified cellos, six amplified horns and field recordings. The second entitled Angle to the Sun, in two movements, and written for an acoustic ensemble, Flute, Bb Clarinet, Bass Clarinet, Horn, Cello and soundscape, was developed from genetic concepts. This commentary discusses my compositional strategies in terms of a “Beyond Domain” approach and my investigations into the combination of instrumental forces and recorded media using ideas within the fields of spectralism, spectromorphology and mimetic discourse. Concepts of imagery and political allusion also form part of this discussion

Support for graphicacy: a review of textbooks available to accounting students

This Teaching Note reports on the support available in textbooks for graphicacy that will help students understand the complexities of graphical displays. Graphical displays play a significant role in financial reporting, and studies have found evidence of measurement distortion and selection bias. To understand the complexities of graphical displays, students need a sound understanding of graphicacy and support from the textbooks available to them to develop that understanding. The Teaching Note reports on a survey that examined the textbooks available to students attending two Scottish universities. The support of critical graphicacy skills was examined in conjunction with textbook characteristics. The survey, which was not restricted to textbooks designated as required reading, examined the textbooks for content on data measurement and graphical displays. The findings highlight a lack of support for graphicacy in the textbooks selected. The study concludes that accounting educators need to scrutinize more closely the selection of textbooks and calls for more extensive research into textbooks as a pedagogic tool