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In this paper, we firstly construct free Lie $Omega$-superalgebras by the super-Lyndon-Shirshov $Omega$-monomials. Secondly, we establish Gr{o}bner-Shirshov bases theory for Lie $Omega$-superalgebras. Thirdly, as an application, we give a linear basis of a free Lie Rota-Baxter superalgebra on a $mathbb{Z}_2$-graded set.
In this paper, we obtain respectively some new linear bases of free unitary (modified) weighted differential algebras and free nonunitary (modified) Rota-Baxter algebras, in terms of the method of Gr{o}bner-Shirshov bases.
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
In this paper, we establish a local Lie theory for relative Rota-Baxter operators of weight $1$. First we recall the category of relative Rota-Baxter operators of weight $1$ on Lie algebras and construct a cohomology theory for them. We use the secon
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.
In this paper, first we introduce the notion of a twisted Rota-Baxter operator on a 3-Lie algebra $g$ with a representation on $V$. We show that a twisted Rota-Baxter operator induces a 3-Lie algebra structure on $V$, which represents on $g$. By this