General unitary quantum groups and generalized Fermion algebras

Abstract

Since the middle of the twentieth century, physicists have concentrated on finding quantum counterparts of classical systems. When a classical system is quantized its invariance group may still be a classical group. In the nineteen-eighties it was shown that when some classical systems are quantized, their classical group becomes a quantum group so that the system is invariant under a quantum group. So quantum groups play an important role in carrying physical properties to the quantum world. In this thesis, after presenting structure of Hopf algebra as a quantum group, structure of matrix quantum groups are investigated using inhomogeneous quantum groups, fermionic inhomogeneous orthogonal quantum invariance groups FIO, and bosonic inhomogeneous symplectic quantum invariance groups BISp. Then using invariance quantum groups of orthofermion algebra a general structure for unitary quantum groups are constructed in chapter three. Commuting fermion algebra is defined using the Heisenberg spin algebra and its inhomogeneous quantum group is defined in chapter four. Chapter three and chapter four of the thesis is based on original research.