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].

**[Bou]**N. Bourbaki,*Lie groups and Lie algebras*, Chapitres 1-3, Springer-Verlag, New York and Berlin, 1989. MR**979493 (89k:17001)****[Bou2]**-,*Groupes et algèbres de Lie*, Chapitres 4-6, Masson, Paris, 1981. [Hu] J. Humphreys,*Introduction to Lie algebras and representation theory*, Graduate Texts in Math., vol. 40, Springer-Verlag, New York and Berlin, 1972. MR**0323842 (48:2197)****[LR]**P. Lalonde and A. Ram,*Standard Lyndon bases of Lie algebras and enveloping algebras, extended abstract*, Posters and Software Demonstrations, Proc. 5th Conf. Formal Power Series and Algebraic Combinatorics, Florence, 1993, pp. 99-104.**[Lo]**M. Lothaire,*Combinatorics on words*, Encyclopedia of Math., vol. 17, Addison-Wesley, 1983. MR**675953 (84g:05002)****[Re]**C. Reutenauer,*Free Lie algebras*, London Math. Soc. Monos. (N.S.), No. 7, Oxford Univ. Press, 1993. MR**1231799 (94j:17002)**

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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