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 form a basis for the free Lie algebra generated by . Suppose that is a Lie algebra given by a generating set and a Lie ideal 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 . We show that a similar approach to the universal enveloping algebra naturally leads to a Poincaré-Birkhoff-Witt type basis of the enveloping algebra of . 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].

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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