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Register a new userRotas program on algebraic operators, rewriting systems and Grobner-Shirshov bases
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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 Grobner-Shirshov bases. These advances were achieved in a series of papers from special cases to more general situations. These perspectives also indicate that Rotas 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 Rotas program, for linear operators on associative algebras and Lie algebras. Emphasis will be given to the applications of rewriting systems and Grobner-Shirshov bases. Problems, old and new, are proposed throughout the paper to prompt further developments on Rotas program.
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This paper investigates algebraic objects equipped with an operator, such as operated monoids, operated algebras etc. Various free object functors in these operated contexts are explicitly constructed. For operated algebras whose operator satisfies a set $Phi$ of relations (usually called operated polynomial identities (aka. OPIs)), Guo defined free objects, called free $Phi$-algebras, via universal algebra. Free $Phi$-algebras over algebras are studied in details. A mild sufficient condition is found such that $Phi$ together with a Grobner-Shirshov basis of an algebra $A$ form a Grobner-Shirshov basis of the free $Phi$-algebra over algebra $A$ in the sense of Guo et al.. Ample examples for which this condition holds are provided, such as all Rota-Baxter type OPIs, a class of differential type OPIs, averaging OPIs and Reynolds OPI.
We develop a theory of parafree augmented algebras similar to the theory of parafree groups and explore some questions related to the Parafree Conjecture. We provide an example of finitely generated parafree augmented algebra of infinite cohomological dimension. Motivated by this example, we prove a version of the Composition-Diamond lemma for complete augmented algebras and provide a sufficient condition for augmented algebra to be residually nilpotent on the language of its relations.
In this paper, we define the Grobner-Shirshov basis for a dialgebra. The Composition-Diamond lemma for dialgebras is given then. As results, we give Grobner-Shirshov bases for the universal enveloping algebra of a Leibniz algebra, the bar extension of a dialgebra, the free product of two dialgebras, and Clifford dialgebra. We obtain some normal forms for algebras mentioned the above.
In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Groebner-Shirshov bases of free Rota-Baxter algebra, $lambda$-differential algebra and $lambda$-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to those obtained by Ebrahimi-Fard and Guo, and Guo and Keigher recently by using other methods.
In this paper, we review Shirshovs method for free Lie algebras invented by him in 1962 which is now called the Groebner-Shirshov bases theory.
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