On spectrum of some sl2 and Uq(sl2) related Hamiltonians

  • Fahad Alamrani

Student thesis: Doctoral Thesis


We will deal with the representation theory of classical lie algebra and representation theory of quantum enveloping Uq(sl2). The lie algebra sl2 and quantum algebra Uq(sl2) are considered in a certain infinitely dimensional representation corresponding to “lowest weight 1”. The representation module is equivalent to the Fock space representation of the quantum mechanical oscillator. The canonical elements e and f of sl2, related to the “creation” and “annihilation” operators, appear to be anti-unitary in our construction, so that the operator H = e+fi is Hermitian, and therefore it can be interpreted as a Hamiltonian for a certain quantum mechanical system.

We will prove that this operator has continuous spectrum. Eigenstates of H are constructed explicitly. Our results are based on the representation of the fundamental solution set of the difference equation in the terms of slowly convergent semi-infinite matrix product and the analysis of its asymptotic. A provision of the basic algebraic principles that help on easing the insight of this research and catching up the fundamentals of this thesis is incorporated. We expose some basics on lie algebras and their representation theory, quantum mechanical aspects including the Fock space representation for quantum mechanical oscillator, and their relation to the representation theory of sl2. Also, we dialogue on quantum deformation of simple lie algebras and their common features.

At the end, we expose the category of modular representations of Uq(sl2). Adoption of the developed technique of analysis of slowly convergent semi-infinite matrix products to the framework of modular-type difference equations will the next step of our study. We sum up the overview of the thesis and we present major findings and similar studies. In conclusion, we summarise the major results of the research against the research questions investigated. Analysis of the major findings is presented.
Date of Award2022
Original languageEnglish
SupervisorShuangzhe Liu (Supervisor) & Dharmendra Sharma AM PhD (Supervisor)

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