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Gröbner-Shirshov bases, conformal algebras, and pseudo-algebras. (English. Russian original) Zbl 1150.16020

J. Math. Sci., New York 131, No. 5, 5962-6003 (2005); translation from Sovrem. Mat. Prilozh. 13, 92-130 (2004).
From the introduction: This paper continues our papers [L. A. Bokut’, P. S. Kolesnikov, J. Math. Sci., New York 116, No. 1, 2894-2916 (2003); translation from Zap. Nauchn. Semin. POMI 272, 26-67 (2000; Zbl 1069.16026) and L. A. Bokut’, Y. Fong, W.-F. Ke, P. S. Kolesnikov, Fundam. Prikl. Mat. 6, No. 3, 669-706 (2000; Zbl 0990.17007)].
Just as the papers mentioned above, this work is distinctly divided into two parts. In the first part, we consider “classical” algebras; the second part is devoted to conformal algebras and pseudo-algebras. In the present paper, we substantially restrict the classical case to the class of associative algebras, more precisely, to (semi)group algebras, and still further to (semi)groups. The fact is that the Gröbner-Shirshov basis (GSB) of a (semi)group is the same as the GSB of its (semi)group algebra (over an arbitrary field). By virtue of the composition-diamond lemma (CD-lemma), finding of the GSB of a (semi)group leads to normal forms of words of the initial (semi)group; this form is called the PBW-form since it arises from considerations close to the Poincaré-Birkhoff-Witt theorem. We use the ring-theoretical terminology (CD-lemma, PBW-form) for (semi)groups in order to emphasize the fact that we are speaking, in essence, about applying the ring-theoretical method for studying (semi)groups presented by generators and defining relations.
In the second part, we proceed with the study of conformal algebras (and more general pseudo-algebras) started in [loc. cit.]. We find a clear correspondence between identities on pseudo-algebras and varieties of conformal algebras. This fact gives us a foundation for introducing varieties of pseudo-algebras.
The notion of pseudo-algebra allows us to get a common presentation of some questions in the theory of usual and conformal algebras. In fact, the high level of similarity between properties of usual and conformal algebras is due to the fact that they are particular cases of pseudo-algebras. For example, the Noetherianity of any associative enveloping conformal algebra of a finite-type, Lie conformal algebra holds even in the general case of pseudo-algebras. Moreover, for Jordan pseudo-algebras of finite type, it is possible to construct an analogue of the Tits-Kantor-Koecher (TKK) construction of embedding them into Lie pseudo-algebras. In particular, this leads us to a classification of simple Jordan pseudo-algebras of finite type.
As in the case of usual algebras, one of the main applications of the CD-lemma for associative conformal algebras is to explore the structure of universal associative enveloping of Lie conformal algebras. Similarly, we apply the CD-lemma to obtain a GSB of the “minimal” universal enveloping conformal algebra of simple conformal Lie superalgebras of type \(W_N\).

MSC:

16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
17B01 Identities, free Lie (super)algebras
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
17A30 Nonassociative algebras satisfying other identities
17B69 Vertex operators; vertex operator algebras and related structures
20M25 Semigroup rings, multiplicative semigroups of rings
16S34 Group rings
17B35 Universal enveloping (super)algebras
16S30 Universal enveloping algebras of Lie algebras
16Z05 Computational aspects of associative rings (general theory)
68W30 Symbolic computation and algebraic computation
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