Fachbereich Mathematik,
Universitaet Kaiserslautern,
Erwin-Scroedinger Str. 48,
Postfach 3049,
67653 Kaiserslautern,

Applications of Non-commutative Computer Algebra

We present several important applications of the non-commutative computer algebra together with the implementation in the Computer Algebra System Singular:Plural. The algebras with Poincare-Birkhoff-Witt bases appear in a variety of applications, ranging from the Theoretical Physics and Quantum Algebras to the Differential Equations and Control Theory. We will show, how many important constructions can be formulated in a convenient framework of GR-algebras and proceed with the description of the most fundamental applications of Groebner bases in these algebras. We present our implementation in the system Singular:Plural and compute several interesting examples live.