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Some constructions, related to nonco...

Kasatkin, Victor.

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Some constructions, related to noncommutative tori; Fredholm modules and the Beilinson-Bloch regulator.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Some constructions, related to noncommutative tori; Fredholm modules and the Beilinson-Bloch regulator./
作者:	Kasatkin, Victor.
面頁冊數:	112 p.
附註:	Source: Dissertation Abstracts International, Volume: 76-10(E), Section: B.
Contained By:	Dissertation Abstracts International76-10B(E).
標題:	Theoretical mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3705076
ISBN:	9781321778571

Some constructions, related to noncommutative tori; Fredholm modules and the Beilinson-Bloch regulator.
Kasatkin, Victor.

Some constructions, related to noncommutative tori; Fredholm modules and the Beilinson-Bloch regulator. - 112 p.

Source: Dissertation Abstracts International, Volume: 76-10(E), Section: B.

Thesis (Ph.D.)--California Institute of Technology, 2015.

A noncommutative 2-torus is one of the main toy models of noncommutative geometry, and a noncommutative n-torus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6-term exact sequence, which allows for the computation of the K-theory of noncommutative tori. It follows that both even and odd K-groups of n-dimensional noncommutative tori are free abelian groups on 2 n-1 generators. In 1981, the Powers-Rieffel projector was described [19], which, together with the class of identity, generates the even K-theory of noncommutative 2-tori. In 1984, Elliott [10] computed trace and Chern character on these K-groups. According to Rieffel [20], the odd K-theory of a noncommutative n-torus coincides with the group of connected components of the elements of the algebra. In particular, generators of K-theory 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 K-theory of noncommutative tori. This gives the full set of generators for the odd K-theory of noncommutative 3-tori and 4-tori.

ISBN: 9781321778571Subjects--Topical Terms:

3173530
Theoretical mathematics.

Some constructions, related to noncommutative tori; Fredholm modules and the Beilinson-Bloch regulator.
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100 1 $a Kasatkin, Victor. $3 3192977
245 1 0 $a Some constructions, related to noncommutative tori; Fredholm modules and the Beilinson-Bloch regulator.
300 $a 112 p.
500 $a Source: Dissertation Abstracts International, Volume: 76-10(E), Section: B.
500 $a Adviser: Matilde Marcolli.
502 $a Thesis (Ph.D.)--California Institute of Technology, 2015.
520 $a A noncommutative 2-torus is one of the main toy models of noncommutative geometry, and a noncommutative n-torus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6-term exact sequence, which allows for the computation of the K-theory of noncommutative tori. It follows that both even and odd K-groups of n-dimensional noncommutative tori are free abelian groups on 2 n-1 generators. In 1981, the Powers-Rieffel projector was described [19], which, together with the class of identity, generates the even K-theory of noncommutative 2-tori. In 1984, Elliott [10] computed trace and Chern character on these K-groups. According to Rieffel [20], the odd K-theory of a noncommutative n-torus coincides with the group of connected components of the elements of the algebra. In particular, generators of K-theory 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 K-theory of noncommutative tori. This gives the full set of generators for the odd K-theory of noncommutative 3-tori and 4-tori.
520 $a In Chapter 2, we apply the graded-commutative 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 graded-commutative algebra has to be generalized: the "signs" should be allowed to take values in U(1), rather than just {-1,1}. Such generalization is well-known (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 2.1--2.3 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 graded-commutative 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 graded-commutative differential geometry. One may try to extend these notions to the smooth noncommutative torus algebra, but that was not done in this work.
520 $a A reconstruction of the Beilinson-Bloch 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 longer-term goal of finding possible analogs of the second K-group (in the context of algebraic geometry and K-theory of rings) and of the regulators for noncommutative spaces. This work should be seen as a necessary preliminary step for that purpose.
520 $a 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 Connes-Karoubi character.
520 $a [10] G. A. Elliott. On the K-theory of the C*-algebra generated by a projective representation of a torsion-free discrete abelian group. In Operator algebras and group representations, Vol. I (Neptun, 1980), volume 17 of Monogr. Stud. Math., pages 157--184. Pitman, Boston, MA, 1984.
520 $a [12] Eugene Ha. Fredholm modules and the Beilinson-Bloch regulator. 2014.
520 $a [14] J. Krasil'shchik and B. Prinari. Lectures on Linear Differential Operators over Commutative Algebras. The Diffiety Inst. Preprint Series, 1998.
520 $a [17] M. Pimsner and D. Voiculescu. Exact sequences for K-groups and Ext-groups of certain cross-product C*-algebras. J. Operator Theory, 4(1):93--118, 1980.
520 $a [19] M. A. Rieffel. C*-algebras associated with irrational rotations. Pacific J. Math., 93(2):415--429, 1981.
520 $a [20] M. A. Rieffel. Projective modules over higher-dimensional noncommutative tori. Canad. J. Math., 40(2):257--338, 1988.
520 $a [25] A. M. Vinogradov. The algebra of logic of the theory of linear differential operators. Dokl. Akad. Nauk SSSR, 205: 1025--1028, 1972.
520 $a [26] A. M. Vinogradov. Some homology systems connected with the differential calculus in commutative algebras. Uspekhi Mat. Nauk, 34(6(210)):145--150, 1979.
520 $a [27] A. M. Vinogradov and L. Vital'yano. Iterated differential forms: tensors. Dokl. Akad. Nauk 407(1): 16--18, 2006.
590 $a School code: 0037.
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Mathematics. $3 515831
690 $a 0642
690 $a 0405
710 2 $a California Institute of Technology. $b Mathematics. $3 2101204
773 0 $t Dissertation Abstracts International $g 76-10B(E).
790 $a 0037
791 $a Ph.D.
792 $a 2015
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3705076