# Certain algorithmic problems for Lie algebras

@article{Shirshov1999CertainAP, title={Certain algorithmic problems for Lie algebras}, author={A. I. Shirshov}, journal={SIGSAM Bull.}, year={1999}, volume={33}, pages={3-6} }

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.

## 37 Citations

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

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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…

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### Gröbner-Shirshov Bases for Free Partially Commutative Lie Algebras

- Mathematics
- 2011

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,…

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- Mathematics
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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

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

- Mathematics
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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

- MathematicsInventiones 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…