TY - GEN

T1 - On Rota's problem for linear operators in associative algebras

AU - Guo, Li

AU - Sit, William Y.

AU - Zhang, Ronghua

PY - 2011

Y1 - 2011

N2 - A long standing problem of Gian-Carlo 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. 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. With the aid of 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 have 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 Rota's problem.

AB - A long standing problem of Gian-Carlo 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. 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. With the aid of 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 have 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 Rota's problem.

KW - Rota's problem

KW - classification

KW - differential type operators

KW - operators

KW - rota-baxter type operators

UR - http://www.scopus.com/inward/record.url?scp=79959673378&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79959673378&partnerID=8YFLogxK

U2 - 10.1145/1993886.1993912

DO - 10.1145/1993886.1993912

M3 - Conference contribution

AN - SCOPUS:79959673378

SN - 9781450306751

T3 - Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

SP - 147

EP - 154

BT - ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation

T2 - 36th International Symposium on Symbolic and Algebraic Computation, ISSAC 2011

Y2 - 8 June 2011 through 11 June 2011

ER -