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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.