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Standard Lyndon bases of Lie algebras and enveloping algebras


Authors: Pierre Lalonde and Arun Ram
Journal: Trans. Amer. Math. Soc. 347 (1995), 1821-1830
MSC: Primary 17B35; Secondary 16S30, 17B01
DOI: https://doi.org/10.1090/S0002-9947-1995-1273505-4
MathSciNet review: 1273505
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Abstract: It is well known that the standard bracketings of Lyndon words in an alphabet $ A$ form a basis for the free Lie algebra $ {\text{Lie}}(A)$ generated by $ A$. Suppose that $ \mathfrak{g} \cong {\text{Lie}}(A)/J$ is a Lie algebra given by a generating set $ A$ and a Lie ideal $ J$ of relations. Using a Gröbner basis type approach we define a set of "standard" Lyndon words, a subset of the set Lyndon words, such that the standard bracketings of these words form a basis of the Lie algebra $ \mathfrak{g}$. We show that a similar approach to the universal enveloping algebra $ \mathfrak{g}$ naturally leads to a Poincaré-Birkhoff-Witt type basis of the enveloping algebra of $ \mathfrak{g}$. We prove that the standard words satisfy the property that any factor of a standard word is again standard. Given root tables, this property is nearly sufficient to determine the standard Lyndon words for the complex finite-dimensional simple Lie algebras. We give an inductive procedure for computing the standard Lyndon words and give a complete list of the standard Lyndon words for the complex finite-dimensional simple Lie algebras. These results were announced in [LR].


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1273505-4
Keywords: Lie algebras, Lyndon words, Serre relations, Gröbner bases
Article copyright: © Copyright 1995 American Mathematical Society