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Homogeneity and Derivations on Analytic Algebras

  • In the present master’s thesis we investigate the connection between derivations and homogeneities of complete analytic algebras. We prove a theorem, which describes a specific set of generators for the module of derivations of an analytic algebra, which map the maximal ideal of R into itself. It turns out, that this set has a structure similar to a Cartan subalgebra and contains information regarding multi-homogeneity. In order to prove this theorem, we extend the notion of grading by Scheja and Wiebe to projective systems and state the connection between multi-gradings and pairwise commuting diagonalizable derivations. We prove a theorem similar to Cartan’s Conjugacy Theorem in the setup of infinite-dimensional Lie algebras, which arise as projective limits of finite-dimensional Lie algebras. Using this result, we can show that the structure of the aforementioned set of generators is an intrinsic property of the analytic algebra. At the end we state an algorithm, which is theoretically able to compute the maximal multi-homogeneity of a complete analytic algebra.
Author:Raul - Paul Epure
URN (permanent link):urn:nbn:de:hbz:386-kluedo-52298
Advisor:Mathias Schulze
Document Type:Master's Thesis
Language of publication:English
Publication Date:2018/05/04
Year of Publication:2018
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2018/05/04
Number of page:85, XXIV
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):32-XX SEVERAL COMPLEX VARIABLES AND ANALYTIC SPACES (For infinite-dimensional holomorphy, see 46G20, 58B12) / 32Bxx Local analytic geometry [See also 13-XX and 14-XX] / 32B05 Analytic algebras and generalizations, preparation theorems
68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section {04 in that areag 68-00 General reference works (handbooks, dictionaries, bibliographies, etc.) / 68Wxx Algorithms (For numerical algorithms, see 65-XX; for combinatorics and graph theory, see 05C85, 68Rxx) / 68W30 Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10]
Licence (German):Creative Commons 4.0 - Namensnennung, nicht kommerziell (CC BY-NC 4.0)