| 1. |
- Al-Shujary, Ahmed, 1980-
(författare)
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Kähler-Poisson Algebras
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis, we introduce Kähler-Poisson algebras and study their basic properties. The motivation comes from differential geometry, where one can show that the Riemannian geometry of an almost Kähler manifold can be formulated in terms of the Poisson algebra of smooth functions on the manifold. It turns out that one can identify an algebraic condition in the Poisson algebra (together with a metric) implying that most geometric objects can be given a purely algebraic formulation. This leads to the definition of a Kähler-Poisson algebra, which consists of a Poisson algebra and a metric fulfilling an algebraic condition. We show that every Kähler- Poisson algebra admits a unique Levi-Civita connection on its module of inner derivations and, furthermore, that the corresponding curvature operator has all the classical symmetries. Moreover, we present a construction procedure which allows one to associate a Kähler-Poisson algebra to a large class of Poisson algebras. From a more algebraic perspective, we introduce basic notions, such as morphisms and subalgebras, as well as direct sums and tensor products. Finally, we initiate a study of the moduli space of Kähler-Poisson algebras; i.e for a given Poisson algebra, one considers classes of metrics giving rise to non-isomorphic Kähler-Poisson algebras. As it turns out, even the simple case of a Poisson algebra generated by two variables gives rise to a nontrivial classification problem.
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| 2. |
- Tiger Norkvist, Axel, 1994-
(författare)
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The Noncommutative Geometry of Real Calculi
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Noncommutative geometry extends the traditional connections between algebra and geometry beyond the realm of commutative algebras, allowing for a broader exploration of geometric concepts in noncommutative settings. The geometric perspective facilitates the study and understanding of various mathematical structures, including operator algebras, quantum groups, and noncommutative spaces. Since its inception, noncommutative geometry has experienced remarkable growth, attracting mathematicians from diverse backgrounds who seek to delve into the geometric aspects of noncommutative structures. Through this lens, groundbreaking discoveries have deepened our understanding of fundamental mathematical principles and opened up new avenues of research. This ongoing exploration not only enriches our mathematical knowledge but also finds practical applications in theoretical physics, quantum field theory, and interdisciplinary fields.The primary focus of this thesis is to offer valuable insights into the derivation-based approach of real calculi, which employs modules over an algebra as an algebraic analogy for vector bundles over differential manifolds. An overarching goal is to give noncommutative counterparts of classical geometric concepts, with a specific emphasis being placed on a noncommutative adaptation of the Levi-Civita connection in (pseudo-)Riemannian geometry. An investigation into the existence of a Levi-Civita connection is conducted in the context of general projective modules, and in cases where it exists a theory of embeddings is developed and used to give a minimal embedding of the noncommutative torus into the noncommutative 3-sphere. The thesis also establishes the concept of morphisms of real calculi, which plays a crucial role in examining the relationship between projective modules and specific free modules in this framework. Moreover, the thesis provides an in-depth examination of matrix algebras, utilizing them as illustrative examples to showcase the process of determining isomorphism classes of real calculi in various scenarios and presenting classes of examples where a Levi-Civita connection does not exist.
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| 3. |
- Ilwale, Kwalombota, 1980-
(författare)
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Noncommutative Riemannian Geometry of Twisted Derivations
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- A twisted derivation is a generalized derivative satisfying a twisted version of the ordinary Leibniz rule for products. In particular, a (σ, τ )-derivation on an algebra A, is a derivation where Leibniz rule is twisted by two endomorphisms σ and τ on A. Such derivations play an important role in the theory of quantum groups, as well as in the context of discretized and deformed derivatives. In this thesis, we develop a (commutative and noncommutative) differential geometry based on (σ, τ )- derivations. To this end, we introduce the notion of (σ, τ )-algebra, consisting of an associative algebra together with a set of (σ, τ )-derivations, to construct connections satisfying a twisted Leibniz rule in analogy with (σ, τ )-derivations. We show that such connections always exist on projective modules and that it is possible to construct connections compatible with a hermitian form. To construct torsion and curvature of (σ, τ )-connections, we introduce the notion of (σ, τ )-Lie algebra and demonstrate that it is possible to construct a Levi-Civita (σ, τ )-connection. Having constructed the framework for studying (σ, τ)-connections, we demonstrate that the framework applied to commutative algebras can help to also give a good understanding of (σ, τ )-derivations on commutative algebras. In particular, we introduce a notion of symmetric (σ, τ )-derivations together with some regularity conditions. For example, we show that strongly regular (σ, τ )-derivations are always inner and there exist a symmetric (σ, τ )-connection on symmetric (σ, τ )- algebras. Finally, we introduce a (σ, τ)-Hochschild cohomology theory which in first degree captures the outer (σ, τ )-derivations of an associative algebra. Along the way, examples including both commutative and noncommutative algebras are presented to illustrate the novel concepts.
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| 4. |
- Al-Shujary, Ahmed, 1980-
(författare)
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Kähler-Poisson Algebras
- 2018
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The focus of this thesis is to introduce the concept of Kähler-Poisson algebras as analogues of algebras of smooth functions on Kähler manifolds. We first give here a review of the geometry of Kähler manifolds and Lie-Rinehart algebras. After that we give the definition and basic properties of Kähler-Poisson algebras. It is then shown that the Kähler type condition has consequences that allow for an identification of geometric objects in the algebra which share several properties with their classical counterparts. Furthermore, we introduce a concept of morphism between Kähler-Poisson algebras and show its consequences. Detailed examples are provided in order to illustrate the novel concepts.
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| 5. |
- Arnlind, Joakim, 1979-
(författare)
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Graph Techniques for Matrix Equations and Eigenvalue Dynamics
- 2008
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- One way to construct noncommutative analogues of a Riemannian manifold Σ is to make use of the Toeplitz quantization procedure. In Paper III and IV, we construct C-algebras for a continuously deformable class of spheres and tori, and by introducing the directed graph of a representation, we can completely characterize the representation theory of these algebras in terms of the corresponding graphs. It turns out that the irreducible representations are indexed by the periodic orbits and N-strings of an iterated map s:(reals) 2→(reals)2 associated to the algebra. As our construction allows for transitions between spheres and tori (passing through a singular surface), one easily sees how the structure of the matrices changes as the topology changes. In Paper II, noncommutative analogues of minimal surface and membrane equations are constructed and new solutions are presented -- some of which correspond to minimal tori embedded in S7. Paper I is concerned with the problem of finding differential equations for the eigenvalues of a symmetric N × N matrix satisfying Xdd=0. Namely, by finding N(N-1)/2 suitable conserved quantities, the time-evolution of X (with arbitrary initial conditions), is reduced to non-linear equations involving only the eigenvalues of Χ.
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| 6. |
- Tiger Norkvist, Axel, 1994-
(författare)
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Morphisms of real calculi from a geometric and algebraic perspective
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Noncommutative geometry has over the past four of decades grown into a rich field of study. Novel ideas and concepts are rapidly being developed, and a notable application of the theory outside of pure mathematics is quantum theory. This thesis will focus on a derivation-based approach to noncommutative geometry using the framework of real calculi, which is a rather direct approach to the subject. Due to their direct nature, real calculi are useful when studying classical concepts in Riemannian geometry and how they may be generalized to a noncommutative setting.This thesis aims to shed light on algebraic aspects of real calculi by introducing a concept of morphisms of real calculi, which enables the study of real calculi on a structural level. In particular, real calculi over matrix algebras are discussed both from an algebraic and a geometric perspective.Morphisms are also interpreted geometrically, giving a way to develop a noncommutative theory of embeddings. As an example, the noncommutative torus is minimally embedded into the noncommutative 3-sphere.
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