Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gr\"obner-Shirshov bases for Lie $\Omega$-algebras and free Rota-Baxter
  Lie algebras   In this paper, we generalize the Lyndon-Shirshov words to Lyndon-Shirshov
$\Omega$-words on a set $X$ and prove that the set of all non-associative
Lyndon-Shirshov $\Omega$-words forms a linear basis of the free Lie
$\Omega$-algebra on the set $X$. From this, we establish Gr\"{o}bner-Shirshov
bases theory for Lie $\Omega$-algebras. As applications, we give
Gr\"{o}bner-Shirshov bases for free $\lambda$-Rota-Baxter Lie algebras, free
modified $\lambda$-Rota-Baxter Lie algebras and free Nijenhuis Lie algebras and
then linear bases of such three free algebras are obtained.

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The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
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The categories of this paper are: eess
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: solv-int
3
The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: q-alg