### Abstract

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.

Original language | English (US) |
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Title of host publication | ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation |

Pages | 147-154 |

Number of pages | 8 |

DOIs | |

State | Published - Jul 1 2011 |

Event | 36th International Symposium on Symbolic and Algebraic Computation, ISSAC 2011 - San Jose, CA, United States Duration: Jun 8 2011 → Jun 11 2011 |

### Publication series

Name | Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC |
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### Other

Other | 36th International Symposium on Symbolic and Algebraic Computation, ISSAC 2011 |
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Country | United States |

City | San Jose, CA |

Period | 6/8/11 → 6/11/11 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation*(pp. 147-154). (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC). https://doi.org/10.1145/1993886.1993912

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*ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation.*Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC, pp. 147-154, 36th International Symposium on Symbolic and Algebraic Computation, ISSAC 2011, San Jose, CA, United States, 6/8/11. https://doi.org/10.1145/1993886.1993912

**On Rota's problem for linear operators in associative algebras.** / Guo, Li; Sit, William Y.; Zhang, Ronghua.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

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

AU - Guo, Li

AU - Sit, William Y.

AU - Zhang, Ronghua

PY - 2011/7/1

Y1 - 2011/7/1

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.

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 - https://doi.org/10.1145/1993886.1993912

DO - https://doi.org/10.1145/1993886.1993912

M3 - Conference contribution

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

ER -