Citation
Kasatkin, Victor (2015) Some Constructions, Related to Noncommutative Tori; Fredholm Modules and the Beilinson–Bloch Regulator. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z91R6NGH. http://resolver.caltech.edu/CaltechTHESIS:05212015124038753
Abstract
A noncommutative 2torus is one of the main toy models of noncommutative geometry, and a noncommutative ntorus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6term exact sequence, which allows for the computation of the Ktheory of noncommutative tori. It follows that both even and odd Kgroups of ndimensional noncommutative tori are free abelian groups on 2^{n1} generators. In 1981, the PowersRieffel projector was described [19], which, together with the class of identity, generates the even Ktheory of noncommutative 2tori. In 1984, Elliott [10] computed trace and Chern character on these Kgroups. According to Rieffel [20], the odd Ktheory of a noncommutative ntorus coincides with the group of connected components of the elements of the algebra. In particular, generators of Ktheory can be chosen to be invertible elements of the algebra. In Chapter 1, we derive an explicit formula for the First nontrivial generator of the odd Ktheory of noncommutative tori. This gives the full set of generators for the odd Ktheory of noncommutative 3tori and 4tori.
In Chapter 2, we apply the gradedcommutative framework of differential geometry to the polynomial subalgebra of the noncommutative torus algebra. We use the framework of differential geometry described in [27], [14], [25], [26]. In order to apply this framework to noncommutative torus, the notion of the gradedcommutative algebra has to be generalized: the "signs" should be allowed to take values in U(1), rather than just {1,1}. Such generalization is wellknown (see, e.g., [8] in the context of linear algebra). We reformulate relevant results of [27], [14], [25], [26] using this extended notion of sign. We show how this framework can be used to construct differential operators, differential forms, and jet spaces on noncommutative tori. Then, we compare the constructed differential forms to the ones, obtained from the spectral triple of the noncommutative torus. Sections 2.12.3 recall the basic notions from [27], [14], [25], [26], with the required change of the notion of "sign". In Section 2.4, we apply these notions to the polynomial subalgebra of the noncommutative torus algebra. This polynomial subalgebra is similar to a free gradedcommutative algebra. We show that, when restricted to the polynomial subalgebra, Connes construction of differential forms gives the same answer as the one obtained from the gradedcommutative differential geometry. One may try to extend these notions to the smooth noncommutative torus algebra, but this was not done in this work.
A reconstruction of the BeilinsonBloch regulator (for curves) via Fredholm modules was given by Eugene Ha in [12]. However, the proof in [12] contains a critical gap; in Chapter 3, we close this gap. More specifically, we do this by obtaining some technical results, and by proving Property 4 of Section 3.7 (see Theorem 3.9.4), which implies that such reformulation is, indeed, possible. The main motivation for this reformulation is the longerterm goal of finding possible analogs of the second Kgroup (in the context of algebraic geometry and Ktheory of rings) and of the regulators for noncommutative spaces. This work should be seen as a necessary preliminary step for that purpose.
For the convenience of the reader, we also give a short description of the results from [12], as well as some background material on central extensions and ConnesKaroubi character.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Noncommutative geometry; Connes noncommutative geometry; noncommutative tori; Ktheory; generators; BeilinsonBloch regulator; gradedcommutative 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  6 May 2015 
Record Number:  CaltechTHESIS:05212015124038753 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:05212015124038753 
DOI:  10.7907/Z91R6NGH 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  8875 
Collection:  CaltechTHESIS 
Deposited By:  Victor Kasatkin 
Deposited On:  27 May 2015 21:38 
Last Modified:  02 Oct 2017 19:22 
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