Teaching

Introduction to Rings and Operator Algebras

 WS 2018/19

Georg Regensburger

 

For an algebraic treatment of linear systems of functional equations (like differential or difference equations), we need to be able to compute with matrices and modules over the corresponding operator algebras. In this course, we will learn some tools from ring and module theory to model and study such noncommutative operator algebras.

Topics we will discuss include:

  • Basic module theory over noncommutative rings
  • Skew polynomial rings (Ore extensions)
  • Factorization of skew polynomials
  • Matrices and modules over skew polynomials
  • Free algebras and integro-differential operators

Students should be familiar with basic algebraic notions such as rings, fields, polynomials, vector spaces, and determinants.

Grading will be based on an oral exam or small individual projects, which are to be presented after the end of the course.

Weekly lectures: Thursdays, 12:00 – 13:30, HT 177F.

References: 

P.M. Cohn, Introduction to Ring Theory, SUMS, Springer-Verlag, London, 2000. DOI

J. Gómez-Torrecillas, Basic module theory over non-commutative rings with computational aspects of operator algebras,  LNCS 8372, Springer, Heidelberg, pp. 23-82, with an appendix by V. Levandovskyy. arXiv | DOI

J. Bueso, J. Gómez-Torrecillas, A. Verschoren, Algorithmic methods in non-commutative algebra, Springer Netherlands, 2003. DOI

A. Quadrat, An introduction to constructive algebraic analysis and its applications, 2010. hal

 

Previous lectures:

WS 2015/16: (with Clemens Raab) Computational integro-differential algebra