Abstract
Noncommutative algebras, defined by the generators and relations, are considered. The definition and main results connected with the Gröbner bases, Hilbert series and Anick’s resolution are formulated. Special attention is paid to universal enveloping algebras. Four main examples illustrate the main concepts and ideas. Algorithmic problems arising in the calculation of the Hilbert series are investigated. The existence of finite state automata, defining the behavior of the Hilbert series is discussed.
Two programs for calculating Gröbner bases and Anick’s resolution are considered.
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Bibliography
Anick, D.: Non-commutative graded algebras and their Hilbert series, J.Algebra, 78, No 1, (1982) 120–140.
Anick, D.: Diophantine equations Hilbert series and undecidable spaces, Ann Math.,II Ser. 122, (1985) 87–112.
Anick, D.: On the homology of associative algebras, Trans. Am. Math. Soc., 296, No 2, (1986), 641–659.
Backelin, J., Bergman: Computer algebra program, available by ftp to ftp://matematik.su.se
Lichtman, A., Ufnarovski V.: On Growth of Lie Algebras, Algebra Colloquium 2, 1 (1985), 45–49.
Petrogradsky V.M.: On some types of intermediate growth in Lie algebras, Uspechi Mat.Nauk, 1993, 48, 5, 181–182.
Roos J.E.: A computer-aided study of the graded Lie algebra of a local commutative noetherian ring,Journal of Pure and Applied Algebra 91, (1994).
Ufnarovskij V.: Poincaré series of graded algebras, Mat.Zametki 27, NO 1 (1980), 21–32. English transl: Math. Notes 27, (1980), 12–18.
Ufnarovski, V.: Combinatorial and Asymptotic Methods of Algebra, in: ‘Algebra-VI’ (A.I. Kostrikin and I.R. Shafarevich, Eds), Encyclopaedia of Mathematical Sciences, Vol. 57, Springer,(1995), 5–196.
Ufnarovski V.: Calculations of growth and Hilbert series by computer, Lect.Notes re Appl. Math, 151, (1993) 247–256
Ufnarovskij V.: A growth criterion for graphs and algebras defined by words, Mat Zametki 31, NO 3 (1980), 465–472. English transl: Math. Notes 31, (1982), 238–241.
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Cojocaru, S., Podoplelov, A., Ufnarovski, V. (1999). Non-Commutative Gröbner Bases and Anick’s Resolution. In: Dräxler, P., Ringel, C.M., Michler, G.O. (eds) Computational Methods for Representations of Groups and Algebras. Progress in Mathematics, vol 173. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8716-8_7
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DOI: https://doi.org/10.1007/978-3-0348-8716-8_7
Publisher Name: Birkhäuser, Basel
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