Noncommutative Galois Extension Approach to Ternary Grassmann Algebra and Graded q-Differential Algebra
Date
2016-07-07
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Abstract
Antud väitekirjas kasutatakse koordinaatdiferentsiaalarvutuse meetodit mittekommutatiivse diferentsiaalarvutuse konstrueerimiseks ja uurimiseks Grassmanni algebra, ternaarse Grassmanni algebra ja mittekommutatiivse ruumi korral. Mittekommutatiivse diferentsiaalarvutuse tähtsaks komponendiks on homomorfism algebrast maatriksite algebrasse. Käesolevas väitekirjas on leitud võrrandid ülalpool mainitud homomorfismi jaoks, mis tulenevad algebra kommutatsiooniseostest. Neid võrrandeid on uuritud ja lahendatud teise astme kommutatsiooniseoste korral. Saadud tulemused on rakendatud mittekommutatiivse diferentsiaalarvutuse uurimiseks Grassmanni algebra, ternaarse Grassmanni algebra ja mittekommutatiivse ruumi korral. Käesolevas väitekirjas uuritakse mittekommutatiivset Galois’ laiendit. Väitekirjas näidatakse seost mittekommutatiivse Galois laiendi ja gradueeritud q-diferentsiaalalgebrate vahel, kus q on N-järku algjuur ühest. Antud väitekirjas on tõestatud, et iga poolkommutatiivne Galois laiend on gradueeritud q-diferentsiaalalgebra. On uuritud selle algebra esimest järku ja kõrgemat järku diferentsiaalarvutus.
In the present thesis a method of coordinate differential calculus is used to construct and study a noncommutative differential calculus in the case of Grassmann algebra, ternary Grassmann algebra and noncommutative space. An important component of a noncommutative differential calculus is a homomorphism from an algebra to the algebra of matrices over this algebra. In the present thesis we find the equations for the entries of the matrix of a homomorphism which are derived by means of differentiating the commutation relations of algebra. These equations are solved in the case of quadratic commutation relations and solutions are applied to Grassmann algebra, ternary Grassmann algebra and quantum space to construct a noncommutative differential calculus. In the present thesis we propose a noncommutative Galois extension approach to graded q-differential algebra. It is proved that semi-commutative Galois extension is a graded q-differential algebra. We apply the structure of graded q-differential algebra to construct and study a noncommutative differential calculus over a noncommutative Galois extension.
In the present thesis a method of coordinate differential calculus is used to construct and study a noncommutative differential calculus in the case of Grassmann algebra, ternary Grassmann algebra and noncommutative space. An important component of a noncommutative differential calculus is a homomorphism from an algebra to the algebra of matrices over this algebra. In the present thesis we find the equations for the entries of the matrix of a homomorphism which are derived by means of differentiating the commutation relations of algebra. These equations are solved in the case of quadratic commutation relations and solutions are applied to Grassmann algebra, ternary Grassmann algebra and quantum space to construct a noncommutative differential calculus. In the present thesis we propose a noncommutative Galois extension approach to graded q-differential algebra. It is proved that semi-commutative Galois extension is a graded q-differential algebra. We apply the structure of graded q-differential algebra to construct and study a noncommutative differential calculus over a noncommutative Galois extension.
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mittekommutatiivne geomeetria, diferentsiaalgeomeetria, Grassmanni algebrad, noncommutative geometry, differential geometry, Grassmanni algebrad