Certain algorithmic problems for Lie algebras

  title={Certain algorithmic problems for Lie algebras},
  author={A. I. Shirshov},
  journal={SIGSAM Bull.},
This short paper describes work similar to that appearing in Buchberger's 1965 thesis inventing Gröbner bases, but in the context of Lie Algebras. Preceding Buchberger by only three years, this paper, along with the two cited references, are the original papers defining what have become known as Gröbner-Shirshov bases. 

Gröbner–Shirshov bases for Lie Ω-algebras and free Rota–Baxter Lie algebras

We generalize the Lyndon–Shirshov words to the Lyndon–Shirshov Ω-words on a set X and prove that the set of all the nonassociative Lyndon–Shirshov Ω-words forms a linear basis of the free Lie

Gröbner-Shirshov Bases of the Generalized Bruck-Reilly -Extension

In this paper we first define a presentation for the generalized Bruck-Reilly ∗-extension of a monoid and then we work on its Grobner-Shirshov bases.

Some new results on Grobner-Shirshov bases

In this survey article, we report some new results of Grobner-Shirshov bases, including new Composition-Diamond lemmas and some applications of some known Composition-Diamond lemmas.

Gröbner-Shirshov Bases for Free Partially Commutative Lie Algebras

In this article, by using Composition-Diamond lemma for Lie algebras, we give a Gröbner-Shirshov basis for free partially commutative Lie algebra over a commutative ring with unit. As an application,


In this paper, by using Gröbner–Shirshov bases, we show that in the following classes, each (respectively, countably generated) algebra can be embedded into a simple (respectively, two-generated)

Gröbner-Shirshov Bases for Braid Groups in Adyan-Thurston Generators

We give a Grobner-Shirshov basis of the braid group Bn+1 in Adyan-Thurston generators. We also deal with the braid group of type Bn. As results, a new algorithm for getting the Adyan-Thurston normal

Groebner-Shirshov Basis for the Chinese Monoid

In this paper, a Groebner-Shirshov basis for the Chinese monoid is obtained and an algorithm for the normal form of the Chinese monoid is given.

Gröbner bases for coloured operads

In this work, we develop the machinery of Gröbner bases for coloured operads, which allows us to establish a useful criterion of Koszulness of a coloured operad. Among the examples for which we show

Generalized anti-commutative Gröbner-Shirshov basis theory and free Sabinin algebras

Abstract E. Chibrikov defined regular monomials (here called Chibrikov words) and proved that they form a linear basis of a free Sabinin algebra. In this paper, we introduce the notion of a

Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups

  • R. Gray
  • Mathematics
    Inventiones mathematicae
  • 2019
We prove the following results: (1) There is a one-relator inverse monoid $$\mathrm {Inv}\langle A\,|\,w=1 \rangle $$ Inv ⟨ A | w = 1 ⟩ with undecidable word problem; and (2) There are one-relator