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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Standard Lyndon bases of Lie algebras and enveloping algebras
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by Pierre Lalonde and Arun Ram PDF
Trans. Amer. Math. Soc. 347 (1995), 1821-1830 Request permission

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
  • Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1975 edition. MR 979493
  • James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842, DOI 10.1007/978-1-4612-6398-2
  • 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.
  • M. Lothaire, Combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 17, Addison-Wesley Publishing Co., Reading, Mass., 1983. A collective work by Dominique Perrin, Jean Berstel, Christian Choffrut, Robert Cori, Dominique Foata, Jean Eric Pin, Guiseppe Pirillo, Christophe Reutenauer, Marcel-P. Schützenberger, Jacques Sakarovitch and Imre Simon; With a foreword by Roger Lyndon; Edited and with a preface by Perrin. MR 675953
  • Christophe Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New Series, vol. 7, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1231799
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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