Fachbereich Mathematik,

Universitaet Kaiserslautern,

Erwin-Scroedinger Str. 48,

Postfach 3049,

67653 Kaiserslautern,

GERMANY

E-mail: levandov@mathematik.uni-kl.de

**Applications of Non-commutative Computer Algebra**

**Abstract:**

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.