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Gröbner-Shirshov bases of the Lie algebra \(D^{+}_{n}\). (English. Russian original) Zbl 1244.17008
St. Petersbg. Math. J. 22, No. 4, 573-614 (2011); translation from Algebra Anal. 22, No. 4, 76-136 (2010).
The theory of Gröbner-Shirshov bases has its origin in a A. I. Shirshov’s work about free Lie algebras in 1962 [“Some algorithm problems for Lie algebras”, Sib. Mat. Zh. 3, 292–296 (1962; Zbl 0104.26004)], although remained unknown during years. Three years afterwards Buchberger, in his doctoral thesis [B. Buchberger, “An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal”, Ph. D Thesis (1965), see also J. Symb. Comput. 41, No. 3-4, 475–511 (2006; Zbl 1158.01307)], developed the same ideas for commutative polynomials, instead of Lie polynomials. In both contexts, this kind of bases has had a great amount of applications to theoretical questions as well as to computational algebra. Not only in both contexts, since new results in Gröbner-Shirshov bases have appeared in associative algebras, semisimple Lie superalgebras, irreducible modules, Kac-Moody algebras, Coxeter groups, conformal algebras, dialgebras, Leibniz algebras and so on. To know more, it is possible to consult the extensive bibliography about the topic contained in this paper.
In the 1990s, L. A. Bokut’ and A. A. Klein, computed in [“Serre relations and Gröbner Shirshov bases for simple Lie algebras. I, II. Int. J. Algebra Comput. 6, No. 4, 389–400 (1996; Zbl 0866.17007), ibid. 6, No. 4, 401–412 (1996; Zbl 0866.17008)] the reduced Gröbner-Shirshov bases of the classical simple Lie algebras, but only for a determined ordering of generators.
This paper is the last one of a series in which Koryukin tries to solve the problem of finding, for the classical simple Lie algebras over fields of characteristic 0, under an arbitrary ordering of generators, the reduced bases and describing them in terms of the root systems.
Concretely, in the present paper, he solves the problem for \(D_n^+\), the nilradical of the Borel subalgebra of \(D_n\), by using the reduced bases computed by himself in [A. N. Koryukin and K. P. Shum, “Reduced bases of the Lie algebra \(D_n^+\)”, Sib. Zh. Ind. Mat. 9, No. 4, 90–104 (2006; Zbl 1224.17016)].
After long computations, the author constructs the basis and describes the SR-words (split into subsets according to certain occurrences of letters) with the help of some graphs related to the corresponding root system. The main innovation is that this description of a reduced Gröbner-Shirshov basis of \(D_n^+\) is simultaneous for each of the \(n!\) possible orderings.
MSC:
17B22 Root systems
17B01 Identities, free Lie (super)algebras
17B70 Graded Lie (super)algebras
17B20 Simple, semisimple, reductive (super)algebras
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