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.

Tracing outsideness: young women's institutional journeys and the geographies of closed space

Understanding confinement and its complex workings between individuals and society has been the stated aim of carceral geography and wider studies on detention. This project contributes ethnographic insights from multiple sites of incarceration, working with an under-researched group within confined populations. Focussing on young female detainees in Scotland, this project seeks to understand their experiences of different types of ‘closed’ space. Secure care, prison and closed psychiatric facilities all impact on the complex geographies of these young women’s lives. The fluid but always situated relations of control and care provide the backdrop for their journeys in/out and beyond institutional spaces. Understanding institutional journeys with reference to age and gender allows an insight into the highly mobile, often precarious, and unfamiliar lives of these young women who live on the margins. This thesis employs a mixed-method qualitative approach and explores what Goffman calls the ‘tissue and fabric’ of detention as a complex multi-institutional practice. In order to be able to understand the young women’s gendered, emotional and often repetitive experiences of confinement, analysis of the constitution of ‘closed space’ represents a first step for inquiry. The underlying nature of inner regimes, rules and discipline in closed spaces, provide the background on which confinement is lived, perceived and processed. The second part of the analysis is the exploration of individual experiences ‘on the inside’, ranging from young women’s views on entering a closed institution, the ways in which they adapt or resist the regime, and how they cope with embodied aspects of detention. The third and final step considers the wider context of incarceration by recovering the young women’s journeys through different types of institutional spaces and beyond. The exploration of these journeys challenges and re-develops understandings of mobility and inertia by engaging the relative power of carceral archipelagos and the figure of femina sacra. This project sits comfortably within the field of carceral geography while also pushing at its boundaries. On a conceptual level, a re-engagement with Goffman’s micro-analysis challenges current carceral-geographic theory development. Perhaps more importantly, this project pushes for an engagement with different institutions under the umbrella of carceral geography, thus creating new dialogues on issues like ‘care’ and ‘control’. Finally, an engagement with young women addresses an under-represented population within carceral geography in ways that raise distinctly problematic concerns for academic research and penal policy. Overall, this project aims to show the value of fine grained micro-level research in institutional geographies for extending thinking and understanding about society’s responses to a group of people who live on the margins of social and legal norms.