Differential calculus d3 = 0 on binary and ternary associative algebras

Abstract

Antud väitekiri on pühendatud mittekommutatiivse geomeetria raames tekkinud diferent-siaalarvutusele diferentsiaaliga d, mis rahuldab tingimust d^3=0. Antud diferentsiaalarvutus tugineb gradueeritud Q-diferentsiaalalgebra mõistele, kus Q on kuupjuur ühest. Gradueeritud Q-diferentsiaalalgebrat võib vaadelda gradueeritud diferentsiaalalgebra üldistusena, kusjuures gradueeritud diferentsiaalalgebrad mängivad tähtsat rolli nii kaasaegses diferentsiaalgeomeet- rias kui ka väljateooriates.

Esimeses peatükis antakse esimest järku diferentsiaalarvutuse assotsiatiivsetel algebratel de-tailne kirjeldus, seejärel vaadeldakse koordinaatdiferentsiaalarvutust, mis tekib juhul, kui al-gebra A on lõplikult tekitatud algebra ja A-bimoodul M kui parempoolne A-moodul on vaba. Näidatakse, et koordinaatdiferentsiaalarvutuse korral diferentsiaal d tekitab parempoolsed osatuletised algebral A, mis rahuldavad teatud homomorfismiga deformeeritud Leibnizi vale-mit. Antakse universaalse esimest järku diferentsiaalarvutuse ja gradueeritud diferentsiaal-algebra struktuuri detailne kirjeldus.

Teises peatükis uuritakse gradueeritud diferentsiaalalgebra üldistust, mis tekkis mittekommu-tatiivse geomeetria raames 1990-ndatel aastatel. Seda üldistust nimetatakse gradueeritud q-di-ferentsiaalalgebraks ja teises peatükis antakse selle üldistuse definitsioon. Gradueeritud q-di-ferentsiaalalgebra diferentsiaal d rahuldab tingimust d^N=0, kus N≠2. Teises peatükis on vaa-deldud esimest mittetriviaalset üldistust, kus q on kuupjuur ühest ja diferentsiaal rahuldab d^3=0. Gradueeritud Q-diferentsiaalalgebra rakendustes mittekommutatiivses geomeetrias ja teoreetilises füüsikas on tähtis probleem, kuidas konstrueerida gradueeritud q-diferentsiaal-algebra, kui on antud esimest järku diferentsiaalarvutus assotsiatiivsel algebral. Teises peatü-kis näidatakse, kuidas konstrueerida gradueeritud Q-diferentsiaalalgebrat, kui on antud esi-mest järku diferentsiaalarvutus algebral, mis on lõplikult tekitatud teatuid seoseid rahuldavate moodustajate poolt.

Kolmandas peatükis rakendatakse teises peatükis välja töötatud meetodit diferentsiaalvormi-de algebra analoogi konstrueerimiseks järgmistel ruumidel: anioonilisel (anyonic) ühedimen-sionaalsel ruumil, ruutalgebral, h-deformeeritud kvanttasandil, q-deformeeritud kvanttasandil. On antud diferentsiaalvormide algebra analoogi detailne kirjeldus, kusjuures on leitud kõik kommutatsiooniseosed koordinaatide ja nende diferentsiaalide vahel. Ühedimensionaalse ruu-mi korral on näidatud, et diferentsiaalvormide algebra struktuuri erinevad komponendid on kooskõlas, kui ühedimensionaalne ruum on aniooniline. On näidatud, et anioonilise ruumi korral diferentsiaalvormide algebra analoog sisaldab esimest järku diferentsiaalarvutust, mida kasutatakse murdsupersümmeetrilistes teooriates.

Viimases peatükis vaadeldakse diferentsiaalarvutust d^3=0 ternaarsetel algebratel. On antud ternaarse algebra struktuuri ja sellega seotud mõistete (ternaarne assotsiatiivsus, involutsioon, struktuurikonstandid) lühikirjeldus. On defineeritud A-mooduli mõiste üle ternaarse algebra. Moodulit üle ternaarse algebra nimetatakse trimooduliks. Tuuakse sisse ternaarse algebra derivatsiooni mõiste ja uuritakse selliste derivatsioonide ruumi struktuuri. Konstrueeritakse ternaarseid diferentsiaalalgebraid ja universaalne diferentsiaalarvutus ternaarsel algebral.

In present theses we study a generalization of a graded differential algebra which is called graded Q-differential algebra and this generalization was proposed and studied within the framework of noncommutative geometry in 1990s. We consider the first non-trivial generali-zation of graded Q-differential algebra, where Q is a primitive cubic root of unity and diffe-rential satisfies d^3=0. We answer to an important question which arises in applications of graded Q-differential algebras to noncommutative geometry and theoretical physics: how to construct a graded Q-differential algebra if we are given a first order differential calculus.

In Chapter 1 we give a detailed review of noncommutative geometry approach to a first order differential calculus on associative algebras and we also give the detailed description of the structure of a graded differential algebra with d^2=0. We give description of an universal first order differential calculus which is constructed with the help of tensor product of given associative algebra. We end this chapter with a detailed description of the structure of a graded differential algebra and universal differential envelope.

In Chapter 2 we give the detailed description of construction of graded Q-differential algebra if we are given an algebra generated by finite set of variables which obey some relations. Our approach is based on tensor algebra and the construction of graded differential algebra for a given semi-coordinate first order differential calculus which is described in the previous chapter.

In Chapter 3 we apply the described before method to construction of an algebra of differen-tial forms with exterior differential satisfying d^3=0 on the following noncommutative spa-ces: the anyonic line, the quadratic algebra, the q- and h-deformed quantum planes. We give an explicit description of the structure of algebra of differential forms with differential satisfying d^3=0 in the case of the mentioned above noncommutative spaces by finding all commutation relations between coordinates and their differentials. We show that in the case of one dimensional space the structure of an algebra of differential forms with differential satisfying d^3=0 is self-consistent, if this one dimensional space is the any-onic line. We show that the first order differential calculus induced by our algebra of differential forms is well known first order differential calculus of fractional supersymmetry in dimension one.

In Chapter 4 we wish to stress the fact that the natural structure on which the Z_3-grading takes its full meaning is a ternary algebra, which means a linear vector space over complex numbers on which a ternary composition law is defined. Although ternary laws can be mode-led in ordinary algebras with an associative binary law by defining corresponding ternary ideals and dividing the algebra by the equivalence relations induced by these ideals, one may also introduce ternary composition laws for the entities which can not be derived from a bi-nary law. We begin this chapter with a brief description of the structure of a ternary algebra giving the notions such as associative ternary algebra, structure constants of a ternary algebra, ternary algebra with involution and so on. Then we propose a notion of analog of module over a ternary algebra algebra which we call tri-module. We also define derivations of ternary al-gebra and construct several examples of ternary differential algebras. We propose a construc-tion of universal envelope for associative ternary algebra and the universal differential calcu-lus on a ternary algebra. Although we believe that these novel algebraic constructions might be pertinent for the description of quark fields and new models of elementary interactions in particle physics, we shall stress here mathematical rather than physical aspects, keeping hope that further developments and physical applications of ternary structures will follow soon.