Connected Hopf algebras of finite Gelfand-Kirillov dimension

Following the seminal work of Zhuang, connected Hopf algebras of finite GK-dimension over algebraically closed fields of characteristic zero have been the subject of several recent papers. This thesis is concerned with continuing this line of research and promoting connected Hopf algebras as a natural, intricate and interesting class of algebras. We begin by discussing the theory of connected Hopf algebras which are either commutative or cocommutative, and then proceed to review the modern theory of arbitrary connected Hopf algebras of finite GK-dimension initiated by Zhuang. We next focus on the (left) coideal subalgebras of connected Hopf algebras of finite GK-dimension. They are shown to be deformations of commutative polynomial algebras. A number of homological properties follow immediately from this fact. Further properties are described, examples are considered and invariants are constructed. A connected Hopf algebra is said to be "primitively thick" if the difference between its GK-dimension and the vector-space dimension of its primitive space is precisely one . Building on the results of Wang, Zhang and Zhuang,, we describe a method of constructing such a Hopf algebra, and as a result obtain a host of new examples of such objects. Moreover, we prove that such a Hopf algebra can never be isomorphic to the enveloping algebra of a semisimple Lie algebra, nor can a semisimple Lie algebra appear as its primitive space. It has been asked in the literature whether connected Hopf algebras of finite GK-dimension are always isomorphic as algebras to enveloping algebras of Lie algebras. We provide a negative answer to this question by constructing a counterexample of GK-dimension 5. Substantial progress was made in determining the order of the antipode of a finite dimensional pointed Hopf algebra by Taft and Wilson in the 1970s. Our final main result is to show that the proof of their result can be generalised to give an analogous result for arbitrary pointed Hopf algebras.

Perceptions of climate change, environmental variability and the role of agricultural adaptation strategies by small-scale farmers in Africa: the case of Mwanga district in Northern Tanzania

Abstract The potential impacts of climate change and environmental variability are already evident in most parts of the world, which is witnessing increasing temperature rates and prolonged flood or drought conditions that affect agriculture activities and nature-dependent livelihoods. This study was conducted in Mwanga District in the Kilimanjaro region of Tanzania to assess the nature and impacts of climate change and environmental variability on agriculture-dependent livelihoods and the adaptation strategies adopted by small-scale rural farmers. To attain its objective, the study employed a mixed methods approach in which both qualitative and quantitative techniques were used. The study shows that farmers are highly aware of their local environment and are conscious of the ways environmental changes affect their livelihoods. Farmers perceived that changes in climatic variables such as rainfall and temperature had occurred in their area over the period of three decades, and associated these changes with climate change and environmental variability. Farmers’ perceptions were confirmed by the evidence from rainfall and temperature data obtained from local and national weather stations, which showed that temperature and rainfall in the study area had become more variable over the past three decades. Farmers’ knowledge and perceptions of climate change vary depending on the location, age and gender of the respondents. The findings show that the farmers have limited understanding of the causes of climatic conditions and environmental variability, as some respondents associated climate change and environmental variability with social, cultural and religious factors. This study suggests that, despite the changing climatic conditions and environmental variability, farmers have developed and implemented a number of agriculture adaptation strategies that enable them to reduce their vulnerability to the changing conditions. The findings show that agriculture adaptation strategies employ both planned and autonomous adaptation strategies. However, the study shows that increasing drought conditions, rainfall variability, declining soil fertility and use of cheap farming technology are among the challenges that limit effective implementation of agriculture adaptation strategies. This study recommends further research on the varieties of drought-resilient crops, the development of small-scale irrigation schemes to reduce dependence on rain-fed agriculture, and the improvement of crop production in a given plot of land. In respect of the development of adaptation strategies, the study recommends the involvement of the local farmers and consideration of their knowledge and experience in the farming activities as well as the conditions of their local environment. Thus, the findings of this study may be helpful at various levels of decision making with regard to the development of climate change and environmental variability policies and strategies towards reducing farmers’ vulnerability to current and expected future changes.

Decomposable approximations and coloured isomorphisms for C*-algebras

In this thesis we introduce nuclear dimension and compare it with a stronger form of the completely positive approximation property. We show that the approximations forming this stronger characterisation of the completely positive approximation property witness finite nuclear dimension if and only if the underlying C*-algebra is approximately finite dimensional. We also extend this result to nuclear dimension at most omega. We review interactions between separably acting injective von Neumann algebras and separable nuclear C*-algebras. In particular, we discuss aspects of Connes' work and how some of his strategies have been used by C^*-algebraist to estimate the nuclear dimension of certain classes of C*-algebras. We introduce a notion of coloured isomorphisms between separable unital C*-algebras. Under these coloured isomorphisms ideal lattices, trace spaces, commutativity, nuclearity, finite nuclear dimension and weakly pure infiniteness are preserved. We show that these coloured isomorphisms induce isomorphisms on the classes of finite dimensional and commutative C*-algebras. We prove that any pair of Kirchberg algebras are 2-coloured isomorphic and any pair of separable, simple, unital, finite, nuclear and Z-stable C*-algebras with unique trace which satisfy the UCT are also 2-coloured isomorphic.

2-categories and cyclic homology

The topic of this thesis is the application of distributive laws between comonads to the theory of cyclic homology. The work herein is based on the three papers 'Cyclic homology arising from adjunctions', 'Factorisations of distributive laws', and 'Hochschild homology, lax codescent,and duplicial structure', to which the current author has contributed. Explicitly, our main aims are: 1) To study how the cyclic homology of associative algebras and of Hopf algebras in the original sense of Connes and Moscovici arises from a distributive law, and to clarify the role of different notions of bimonad in this generalisation. 2) To extend the procedure of twisting the cyclic homology of a unital associative algebra to any duplicial object defined by a distributive law. 3) To study the universality of Bohm and Stefan’s approach to constructing duplicial objects, which we do in terms of a 2-categorical generalisation of Hochschild (co)homology. 4) To characterise those categories whose nerve admits a duplicial structure.