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Algebras defined by two quadratic relations. (Russian) Zbl 0566.16010

All algebras are supposed to be associative and finitely generated. For some years great interest arose by the hypothesis of Kostrikin-Shafarevich-Govorov about the rationality of the Hilbert series of algebras with finitely many homogeneous defining relations. The negative solution of this hypothesis was given simultaneously and independently by the author [Mat. Zametki 27, 21–32 (1980; Zbl 0432.16018)] and J. B. Shearer [J. Algebra 62, 228–231 (1980; Zbl 0436.16001)]. On the other hand, J. Backelin [C. R. Acad. Sci., Paris, Sér. A 287, 843–846 (1978; Zbl 0395.16025)], based on the result of V. N. Gerasimov [see Algebra Logika 15, 384–435 (1976; Zbl 0372.08001)], proved that every algebra with only one homogeneous relation has rational Hilbert series.
The main result of this paper is as follows: The Hilbert series is rational for any algebra with two homogeneous defining relations of degree two. Moreover, some classification of these algebras is given, and in all cases the Hilbert series is obtained and the basis.
The method used by the author has an interesting history. In the above paper the author rediscovered the method of constructing a full system of defining relations for any finitely presented algebra. Earlier, this method in an explicit form was given in the reviewer’s paper [Algebra Logika 15, 117–142 (1976; Zbl 0349.16007)] and by G. Bergman [Adv. Math. 29, 178–218 (1978; Zbl 0326.16019)]. There, the reviewer mentioned that for the case of Lie algebras (this case is more complicated than that of associative algebras) this method in the explicit form was proposed by A. I. Shirshov [see Sib. Mat. Zh. 3, 292–296 (1962; Zbl 0104.260)]. G. Bergman in his paper mentioned this method to be of folklore nature. In a more general context the same method is considered in the fundamental survey by B. Buchberger and R. Loos [Computing, Suppl. 4, 11–43 (1982; Zbl 0494.68045)]. [Note that this survey does not contain any publication on this theme in Russian.]

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16W50 Graded rings and modules (associative rings and algebras)
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
16Exx Homological methods in associative algebras
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