Graded q-differential algebras and algebraic models in noncommutative geometry
Kuupäev
2011-05-16
Autorid
Ajakirja pealkiri
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Kirjastaja
Abstrakt
Väitekirja uurimisvaldkonnaks on diferentsiaalgeomeetria. Diferentsiaalgeomeetria keskseteks mõisteteks on koahelate kompleks ja selle kohomoloogiad. Näiteks, siledal muutkonnal on määratud diferentsiaalvormide kompleks, mille kohomoloogiad on muutkonna topoloogia analüütiliseks karakteristikuks. Kohomoloogia mõiste baseerub diferentsiaali omadusel, et kaks korda rakendatud diferentsiaal annab nulli. Sellise geomeetrilise struktuuri algebraliseks mudeliks on diferentsiaalmoodul, miile diferentsiaali ruut võrdub nulliga. Käesolevas väitekirjas uuritakse diferentsiaalmooduli üldistust, kus diferentsiaal rahuldab tingimust, et diferentsiaal astmes võrdub nulliga, kus on täisarv, mis on suurem võrdne kahest. Antud üldistus tekkis mittekommutatiivse geomeetria raames ja selle teooria pani aluse sõltumatult M. Kapranov ja M. Dubois-Violette koos R. Kerneriga 1990. aastatel. Antud teooria tugineb mittekommutatiivses geomeetrias kasutatavale -deformeeritud struktuuridele: gradueeritud -kommutaator, gradueeritud -derivatsioon, gradueeritud -Leibnizi valem, kus on -inda astme algjuur. Selle teooria arengu tulemuseks olid välja töötatud järgmised struktuurid: -diferentsiaalkompleks, -simplitsiaalkompleks, koahelate -kompleks ja selle üldistatud kohomoloogiad, gradueeritud -diferentsiaalalgebra. Seostus ja tema kõverus on kihtkondade teooria põhimõisted ning mängivad olulist rolli mitte üksnes kaasaegses diferentsiaalgeomeetrias, vaid ka teoreetilises füüsikas, nimelt kalibratsiooniväljateoorias. Miitekommutatiivse geomeetria raames oli konstrueeritud seostuse üldistus, mida nimetatakse seostuseks moodulil. Käesolevas väitekirjas on pakutud seostuse moodulil üldistus, mida nimetatakse -seostuseks, meie lähenemine põhineb gradueeritud -diferentsiaalalgebral. Väitekirjas on defineeritud selle seostuse kõverus, on tõestatud, et kõverus rahuldab Bianchi samasuse analoogi, on näidatud, et igal projektiivsel moodulil eksisteerib -seostus. On uuritud -seostuse ja tema kõveruse lokaalne struktuur. -seostuse kõveruse komponendid on avaldatud -seostuse maatriksi kaudu. On toodud üks võimalus Cherni karakteri konstrueerimiseks -seostuse teooria raames.
The thesis is a study in differential geometry. One of the central notions in differential geometry are cochain complex and its cohomology. For instance, the cohomologies of the complex of differential forms defined on a smooth manifold are analytical characteristic of manifold's topology. The idea of cohomology is based on the following important property of the differential: it gives zero when applied twice. The algebraic model of this geometrical structure is differential module with differential satisfying the property that differential squared is equal to zero. In the present thesis we study the generalization of differential module where differential satisfies the property that differential to the th power is equal to zero, where is an integer greater than or equal to two. This generalization was proposed and studied within the framework of noncommutative geometry by M. Kapranov and M. Dubois-Violette with R. Kerner in 1990th. The theory is based on the following -deformed structures used in noncommutative geometry: graded -commutator, graded -derivation, graded -Leibniz rule, where is a primitive Nth root of unity. In elaboration of this theory following structures are developed: -differential complex, -simplicial complex, cochain -complex and its generalized cohomologies, graded -differential algebra. A connection and its curvature are basic elements of the theory of fibre bundles, and they play an important role not only in modern differential geometry but also in theoretical physics namely in gauge field theory. Within the framework of noncommutative geometry was constructed the generalization of the theory of connection, called connection on a module. In the present thesis we propose the generalization of a concept of connection on module, called -connection, our approach is based on a graded -differential algebra. We define the notion of curvature of -connection and proove that it satisfies the analog of Bianchi identity. We prove that every projective module admits an -connection. We study the local structure of -connection and its curvature. We express the components of the curvature of -connection in the terms of the matrix of -connection. We propose a possible approach to how one can construct a Chern character in the theory of -connection.
The thesis is a study in differential geometry. One of the central notions in differential geometry are cochain complex and its cohomology. For instance, the cohomologies of the complex of differential forms defined on a smooth manifold are analytical characteristic of manifold's topology. The idea of cohomology is based on the following important property of the differential: it gives zero when applied twice. The algebraic model of this geometrical structure is differential module with differential satisfying the property that differential squared is equal to zero. In the present thesis we study the generalization of differential module where differential satisfies the property that differential to the th power is equal to zero, where is an integer greater than or equal to two. This generalization was proposed and studied within the framework of noncommutative geometry by M. Kapranov and M. Dubois-Violette with R. Kerner in 1990th. The theory is based on the following -deformed structures used in noncommutative geometry: graded -commutator, graded -derivation, graded -Leibniz rule, where is a primitive Nth root of unity. In elaboration of this theory following structures are developed: -differential complex, -simplicial complex, cochain -complex and its generalized cohomologies, graded -differential algebra. A connection and its curvature are basic elements of the theory of fibre bundles, and they play an important role not only in modern differential geometry but also in theoretical physics namely in gauge field theory. Within the framework of noncommutative geometry was constructed the generalization of the theory of connection, called connection on a module. In the present thesis we propose the generalization of a concept of connection on module, called -connection, our approach is based on a graded -differential algebra. We define the notion of curvature of -connection and proove that it satisfies the analog of Bianchi identity. We prove that every projective module admits an -connection. We study the local structure of -connection and its curvature. We express the components of the curvature of -connection in the terms of the matrix of -connection. We propose a possible approach to how one can construct a Chern character in the theory of -connection.
Kirjeldus
Märksõnad
diferentsiaalgeomeetria, mittekommutatiivne algebra, kihtkonnad, kohomoloogia, dissertatsioonid, differential geometry, noncommuntative algebra, fibre bundles, cohomology