Details:
Title  Differential type operators and Gröbner–Shirshov bases  Author(s)  Giovanni De Micheli, William Y. Sit, Ronghua Zhang  Type  Article in Journal  Abstract  A long standing problem of GianCarlo Rota for associative algebras is the classification of all linear operators that can be defined on them. In the 1970s, there were only a few known operators, for example, the derivative operator, the difference operator, the average operator, and the Rota–Baxter operator. A few more appeared after Rota posed his problem. However, little progress was made to solve this problem in general. In part, this is because the precise meaning of the problem is not so well understood. In this paper, we propose a formulation of the problem using the framework of operated algebras and viewing an associative algebra with a linear operator as one that satisfies a certain operated polynomial identity. This framework also allows us to apply theories of rewriting systems and Gröbner–Shirshov bases. To narrow our focus more on the operators that Rota was interested in, we further consider two particular classes of operators, namely, those that generalize differential or Rota–Baxter operators. These two classes of operators correspond to those that possess Gröbner–Shirshov bases under two different monomial orderings. Using this framework and computer algebra, we are able to come up with a list of these two classes of operators, and provide some evidence that these lists may be complete. Our search has revealed quite a few new operators of these types whose properties are expected to be similar to the differential operator and Rota–Baxter operator respectively. Recently, a more unified approach has emerged in related areas, such as difference algebra and differential algebra, and Rota–Baxter algebra and Nijenhuis algebra. The similarities in these theories can be more efficiently explored by advances on Rotaʼs problem.  Keywords  Rotaʼs problem, Rewriting systems, Gröbner–Shirshov bases, Operators, Classification, Differential type operators, Rota–Baxter type operators  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717112001319 
Language  English  Journal  Journal of Symbolic Computation  Volume  52  Number  0  Pages  97  123  Year  2013  Note  International Symposium on Symbolic and Algebraic Computation  Edition  0  Translation 
No  Refereed 
No 
