In this dissertation we generalise the basic theory of Hopf algebras to the context of autonomous pseudomonoids in monoidal bicategories. Autonomous pseudomonoids were introduced in [13] as generalisations of both autonomous monoidal categories and Hopf algebras. Much of the theory of autonomous pseudomonoids developed in [13] was inspired by the example of autonomous (pro)monoidal enriched categories. The present thesis aims to further develop the theory with results inspired by Hopf algebra theory instead. We study three important results in Hopf algebra theory: the so-called 'fundamental theorem of Hopf modules', the 'Drinfel'd quantum double' and its relation with the centre of monoidal categories, and 'Radford's formula'. The basic result of this work is a general fundamental theorem of Hopf modules that establishes conditions equivalent to the existence of a left dualization. With this result as a base, we are able to construct the centre (defined in [83]) and the lax centre of an autonomous pseudomonoid as an Eilenberg-Moore construction for certain monad. As an application we show that the Drinfel'd double of a finite-dimensional Hopf algebra is equivalent to the centre of the associated pseudomonoid. The next piece of theory we develop is a general Radford's formula for autonomous map pseudomonoids formula in the case of a (coquasi) Hopf algebra. We also introduce 'unimodular' autonomous pseudomonoids. In the last part of the dissertation we apply the general theory to enriched categories with a (chosen) class of (co)limits, with emphasis in the case of finite (co)limits. We construct tensor products of such categories by means of pseudo-commutative enriched monads (a slight generalisation of the pseudo-commutative 2-monads of [37], and showing that lax-idempotent 2-monads are pseudo-commutative. Finally we apply the general theory developed for pseudomonoids to deduce the main results of [27].

A little over 20 years ago Drinfeld presented the quantum (or Drinfeld) double construction. This construction takes any Hopf algebra and embeds it in a larger quasi-triangular Hopf algebra, which contains an algebraic solution to the constant Yang–Baxter equation. One such class of algebras consists of the Drinfeld doubles of finite groups, which are currently of interest due to their connections with non-Abelian anyons. The smallest non-commutative Drinfeld double of a finite group algebra is the Drinfeld double of D3 , the dihedral group of order six, which was recently used to construct solutions to the Yang–Baxter equation cor- responding to 2-state and 3-state integrable spin chains with periodic boundary conditions. In this thesis we construct R-matrices from the Drinfeld double of dihedral group algebras, D(Dn) and consider their associated integrable systems. The 3-state spin chain from D(D3) is generalised to include open boundaries and it is also shown that there exists a more general R-matrix for this algebra. For general D(Dn) an R-matrix is constructed as a descendant of the zero-field six-vertex model.

A little over 20 years ago Drinfeld presented the quantum (or Drinfeld) double construction. This construction takes any Hopf algebra and embeds it in a larger quasi-triangular Hopf algebra, which contains an algebraic solution to the constant Yang–Baxter equation. One such class of algebras consists of the Drinfeld doubles of finite groups, which are currently of interest due to their connections with non-Abelian anyons. The smallest non-commutative Drinfeld double of a finite group algebra is the Drinfeld double of D3 , the dihedral group of order six, which was recently used to construct solutions to the Yang–Baxter equation cor- responding to 2-state and 3-state integrable spin chains with periodic boundary conditions. In this thesis we construct R-matrices from the Drinfeld double of dihedral group algebras, D(Dn) and consider their associated integrable systems. The 3-state spin chain from D(D3) is generalised to include open boundaries and it is also shown that there exists a more general R-matrix for this algebra. For general D(Dn) an R-matrix is constructed as a descendant of the zero-field six-vertex model.

<p>This dissertation consists of two parts. In the first part, we discuss the boundary and domain wall theories of the generalized quantum double lattice realization of the two-dimensional topological orders based on Hopf algebras. The boundary Hamiltonian and domain wall Hamiltonian are constructed by using Hopf algebra pairings and generalized quantum double. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras, respectively. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.</p> <p><br></p> <p>In the second part, we introduce the weak Hopf algebra extension of symmetry, which arises naturally in anyonic quantum systems, and we establish weak Hopf symmetry breaking theory based on the fusion closed set of anyons. We present a thorough investigation of the quantum double model based on weak Hopf algebras, including the topological excitations and ribbon operators, and show that the vacuum sector of the model has weak Hopf symmetry. The gapped boundary and domain wall theories are also established. We show that the gapped boundary is algebraically determined by a comodule algebra, or equivalently, a module algebra, and the gapped domain wall is determined by the bicomodule algebra, or equivalently, a bimodule algebra. We also introduce the weak Hopf tensor network states, by which we solve the weak Hopf quantum double models on closed and open surfaces. Lastly, we discuss the duality of the quantum double phases.</p>