Rota's program on algebraic operators, rewriting systems and Gröbner-Shirshov bases

Abstract

Many years ago, Rota proposed a program on determining algebraic identities that can be satisfied by linear operators. After an extended period of dormant, progress on this program picked up speed in recent years, thanks to perspectives from operated algebras and Gröbner-Shirshov bases. These advances were achieved in a series of papers from special cases to more general situations. These perspectives also indicate that Rota's insight can be manifested very broadly, for other algebraic structures such as Lie algebras, and further in the context of operads. This paper gives a survey on the motivation, early developments and recent advances on Rota's program, for linear operators on associative algebras and Lie algebras. Emphasis will be given to the applications of rewriting systems and Gröbner-Shirshov bases. Problems, old and new, are proposed throughout the paper to prompt further developments on Rota's program.

arXiv comment29 pages

References in corpus (6)

• Hopf Algebras, Renormalization and Noncommutative Geometry
• Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras
• Quantum Bi-Hamiltonian Systems
• Integration and geometrization of Rota-Baxter Lie algebras
• Differential Type Operators and Gröbner-Shirshov Bases
• Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases