Noncommutative Differential Calculus And Its Applications

Symmetry and duality play an important part in theoretical physics and mathemat-ical physics. Quantum group is the promotion of the concept of basic symmetry of classicLie Group and Lie algebra. Based on the result of Alain Connes[1] on noncommutativegeometry, Manin[2], Wess and Zumino[3], etc. consider quantum group as linear trans-formations on quantum super surface, and specifically give di?erential forms, exteriordi?erential on the noncommutative space, then gradually establish a framework for non-commutative di?erential geometry, and give various q-gauge theories from di?erent pointof view.This dissertation is committed to construct a semi-discrete homotopic operator, andthus prove the exactness of semi-discrete di?erential complex. By using semi-discretedi?erential calculus, we get a semi-discrete integrable equation.In the first chapter, we brie?y introduce the exact complex, the history of noncom-mutative di?erential geometry and its applications on physics. In the second chapter, wefirst recall some basic concepts of di?erential geometry, then give the specific di?erentialforms, di?erential calculus and exactness of the complex on Euclidean space (i.e. Poincar′elemma on continuous space). In the third chapter, we give the di?erential forms, di?er-ential calculus and exactness of the complex on discrete space (i.e. Poincar′e lemma ondiscrete space). Combining the first two parts, in the fourth chapter, we give the di?eren-tial forms, di?erential calculus and exactness of the complex on semi-discrete space (i.e.Poincar′e lemma on semi-discrete space). Finally, in the fifth chapter, we illustrate an ap-plication of noncommutative di?erential calculus by obtaining the integrable Toda-Latticeequation.