#### Date of Award

2011

#### Degree Type

Thesis

#### Degree Name

M.Sc.

#### Department

Mathematics and Statistics

#### First Advisor

Hu, Zhiguo (Mathematics and Statistics)

#### Keywords

Mathematics.

#### Rights

CC BY-NC-ND 4.0

#### Abstract

In this thesis, we study and apprehend Hopf algebras, multiplier Hopf algebras, and their dualities. A Hopf algebra A is a unital algebra with an comutiplication (A tensor A) &rarr A, which is the reverse of multiplication, and other structures. In the finite-dimensional case, we can construct the dual A' of A, which is also a Hopf algebra, and prove that A is isomorphic to its bidual A''. If we drop the assumption that A is unital and allow the comultiplication to have values in the multiplier algebra of (A tensor A), we end with a multiplier Hopf algebra. If the coopposite algebra of A is also a multiplier Hopf algebra, we say that A is regular, for which we can involve non-zero left (right) invariant linear functionals, called left (right) integrals. It is proved that if left (right) integrals exist, they are faithful and unique up to a scalar. For a regular multiplier Hopf algebra A with integrals, we can construct its dual A^ with the help of integrals. In this case, A^ is again a regular multiplier Hopf algebra, on which integrals can also be constructed, and it turns out that the bidual A^^ is canonically isomorphic to A.

#### Recommended Citation

Sun, Menghong, "Multiplier Hopf Algebras and Duality" (2011). *Electronic Theses and Dissertations*. 352.

https://scholar.uwindsor.ca/etd/352