Gröbner-Shirshov Bases for Algebras, Groups, and Semigroups

L. Bokut (), Y. Fong () and L.-S. Shiao ()
Additional contact information
L. Bokut: Institute of Mathematics
Y. Fong: National Cheng Kung University, Department of Mathematics
L.-S. Shiao: Chang Jung Christian University

A chapter in Proceedings of the Third International Algebra Conference, 2003, pp 17-32 from Springer

Abstract: Abstract What is now called by the Gröbner-Shirshov bases method had been invented in a highly original paper by A.I. Shirshov ([85], 1962) on Lie algebras or to be more precise on Lie polynomials (elements of a free Lie algebra). This very fact is widely accepted now (see below; see also a recent survey by V. Ufnarovsky [87]). As it happened, roughly speaking the same ideas had been discovered by B. Buchberger ([41], 1965) for ordinary polynomials in his Theses under supervision of W. Gröbner (published in ([42], 1970), see English translation in [45]). Let us have some parallels between papers [85] and [42]. The notion of composition of two Lie polynomials in [85] corresponds to the notion of s-polynomial in [42]. The process of constructing a set S* for any set S of Lie polynomials in [85] corresponds to the very Buchberger algorithm in [42]. A main lemma by Shirshov in [85] (that is called later by Composition lemma: If f ∈ S*, then % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WGMbaaaiabg2da9iaadggadaqdaaqaaiaadohaaaGaamOyaaaa!3ACC! ]]

Keywords: Word Problem; Associative Algebra; Braid Group; Coxeter Group; Vertex Algebra (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-94-017-0337-6_2

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