Date of Award
Mathematics and Statistics
Hu, Zhiguo (Mathematics and Statistics)
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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.
Sun, Menghong, "Multiplier Hopf Algebras and Duality" (2011). Electronic Theses and Dissertations. 352.