Vector fields and Lie group representations
Kuupäev
2012-10-12
Autorid
Ajakirja pealkiri
Ajakirja ISSN
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Kirjastaja
Abstrakt
Käesoleva väitekirja uurimisobjektiks on vektorväli, mis on üks olulisematest mõistetest diferentsiaalgeomeetrias ja globaalanalüüsis ning mis leiab rakendust pideva keskkonna mehaanikas ja dünaamilistes süsteemides. Töös on uurimisel vektorvälja kolm omapära. Esiteks, vektorväli on diferentsiaaloperaator ja me tõlgendame vektorvälja liikumise stopp-kaadrina. Teiseks, vektorväli projetseerib muutkonna invariantide ruumi ja sellega kaasneb singulaarsuste klassifitseerimine. Kolmandaks, vektorväli võib ise olla projetseeritav ja see võimaldab teistest struktuuridest (näit. lõpmatute džettide ruumist) saada vajalikke invariante ja sümmeetriaid. Väitekirja keskseks mõisteks on tensorväljade Lie tuletised. Töös tuletatakse Lie diferentsiaalarvutuse põhivalemid mitteholonoomses (koordinaatidevabas) baasis. Lie tuletiste abil kirjeldatakse, kuidas mitteholonoomne baas ise muutub vektorvälja voos (derivatsioonivalemid). Tutvustatakse tensorväljade integreerimine, mida vaadeldakse kui Lie diferentseerimise pöördoperatsiooni. Nimelt, tavalise integreeritava funktsiooni määramata ja määratud integraali mõisted on laiendatud üldisema tensorvälja juhule, kus valemites esinevad Lie tuletised. Lihtsamatel juhtudel on antud tensorvälja integraali geomeetriline tõlgendus. Lie tuletised on difeomorfismide rühma esituse infinitesimaalne versioon tensorväljadel. Seoses sellega töös uuritakse Lie rühma puutujarühma struktuur ning näidatakse, kasutades infinitesimaalmeetodeid (S.Lie mõttes), kuidas lineaarrühm GL(n,R) toimib iseendal vasak- ja paremnihetega ning siseautomorfismidega (adjungeeritud esitus). Väitekirjas uuritakse lõpmatute džettide ruumi struktuur. Nimelt, on teada, et analoogiliselt protsessile, kuidas sile kujutus tekitab lõpmatu džeti, siledate kujutuste kompositsioon tekitab lõpmatute džettide kompositsiooni. Töös defineeritakse lõpmatute džettide kompositsioon rekurrentse valemi abil nii, et definitsioon ei sõltuks konkreetsete kujutuste valikust. Näidatakse, kuidas vastavad täisdiferentsiaaloperaatorid ja Cartan'i välisdiferentsiaalvormid on seotud džettide kompositsioonil.
One of the most significant tools in differential geometry and global analysis, in continuous environment mechanics and dynamical systems is the notion of vector field. In the current thesis the vector field is considered as follows. First, the vector field can be interpreted as a differential operator and as a stop-frame of a movement. Second, a manifold can be projected by vector field onto the space of invariants, and this involves classification of singularities. Third, the vector field itself can be projectable, which allows to obtain the necessary symmetries and invariants from other structures (for instance, from the jet space). The central notion of this thesis is the Lie derivatives of tensor fields. The Lie differentiation technique is developed in nonholonomic (noncoordinate) basis, where an important role is played by derivation formulas together with the nonholonomy object. There is introduced an integration of tensor fields as a reverse process to the Lie differentiation. This definition generalizes the notion of integral of an ordinary function. A few geometrical examples included in the text clarify the topic being discussed. The Lie derivatives are infinitesimal versions of representation of a diffeomorphism group on tensor fields. In this regard, the structure of a tangent group of a Lie group is studied, and is shown how general linear group GL(n,R) acts on itself by left and right translations, and by interior automorphisms (adjoint representation). In this thesis the structure of the space of infinite jets is studied. In particular, analogously to the process, when a smooth map induces an infinite jet, the composition of smooth maps induces a composition of jets. The problem is how to define the jet composition in such a way that the definition does not depend on a choice of maps. The problem leads to another question: how the corresponding total differentiation operators and Cartan forms are related under the jet composition. The answers are given in the form of convenient recurrence formulas.
One of the most significant tools in differential geometry and global analysis, in continuous environment mechanics and dynamical systems is the notion of vector field. In the current thesis the vector field is considered as follows. First, the vector field can be interpreted as a differential operator and as a stop-frame of a movement. Second, a manifold can be projected by vector field onto the space of invariants, and this involves classification of singularities. Third, the vector field itself can be projectable, which allows to obtain the necessary symmetries and invariants from other structures (for instance, from the jet space). The central notion of this thesis is the Lie derivatives of tensor fields. The Lie differentiation technique is developed in nonholonomic (noncoordinate) basis, where an important role is played by derivation formulas together with the nonholonomy object. There is introduced an integration of tensor fields as a reverse process to the Lie differentiation. This definition generalizes the notion of integral of an ordinary function. A few geometrical examples included in the text clarify the topic being discussed. The Lie derivatives are infinitesimal versions of representation of a diffeomorphism group on tensor fields. In this regard, the structure of a tangent group of a Lie group is studied, and is shown how general linear group GL(n,R) acts on itself by left and right translations, and by interior automorphisms (adjoint representation). In this thesis the structure of the space of infinite jets is studied. In particular, analogously to the process, when a smooth map induces an infinite jet, the composition of smooth maps induces a composition of jets. The problem is how to define the jet composition in such a way that the definition does not depend on a choice of maps. The problem leads to another question: how the corresponding total differentiation operators and Cartan forms are related under the jet composition. The answers are given in the form of convenient recurrence formulas.
Kirjeldus
Märksõnad
vektorväljad, Lie' rühmad, rühmade esitused, vector fields, Lie' groups, representations of groups