| Summary: | In finite dimensions, Hopf algebras have a very nice duality theory, as the vector space dual of a finite-dimensional Hopf algebra is also a Hopf algebra in a canonical way. This breaks down in the infinite-dimensional setting, as here the dual need not be a Hopf algebra. Instead, one chooses a subalgebra of the vector space dual called the finite dual. This subalgebra is always canonically a Hopf algebra. In this thesis, we aim to better understand the finite dual by trying to understand how the finite dual of a crossed product relates to the finite duals of its components. We start by investigating what the assignment sending a Hopf algebra to its finite dual does to functions. Unlike in the finite-dimensional case, this is no longer a contravariant exact monoidal functor and might not even be a functor at all. However, many of the results true thanks to this in finite dimensions still always hold, while we can find necessary and sufficient conditions for others to hold as well as specific situations in which they are always true. Crossed products generalise the notion of a smash product, which can be viewed as the Hopf algebra equivalent of the semidirect product. Many Hopf algebras of interest can be written as crossed products. We study the finite dual of such a product and find numerous results when assuming conditions such as one of the components being finite-dimensional or the crossed product being a smash product. These can be combined for strong statements about the finite dual under certain assumptions. Finally, we consider Noetherian Hopf algebras which are finite modules over central Hopf subalgebras. Many of these algebras decompose as crossed products, so that we can use our previous results to study them. However, we also find results that are true without assuming such a decomposition. This allows us to calculate the finite duals of numerous examples, including a quantised enveloping algebra at a root of unity and all the prime affine regular Hopf algebras of Gelfand-Kirillov dimension one with prime PI degree.
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