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Gröbner–Shirshov bases of the Lie algebra $D^{+}_{n}$

A. N. Koryukin

2011
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St. Petersburg Mathematical Journal
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Over a field of characteristic 0, the reduced Gröbner-Shirshov bases (RGShB) are computed in the positive part D + n of the simple finite-dimensional Lie algebra D n for the canonical generators corresponding to simple roots, under an arbitrary ordering of these generators (i.e., an aritrary basis among the n! bases is fixed and analyzed). In this setting, the RGShBs were previously computed by the author for the Lie algebras A + n , B + n , and C + n . For one ordering of the generators, the
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... e generators, the RGShBs of these algebras were calculated by Bokut and Klein (1996) . 2010 Mathematics Subject Classification. Primary 17B22. 573 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 574 A. N. KORYUKIN these authors). As a consequence, it became possible to apply computational techniques to many problems. For example, if a GShB of an arbitrary Lie algebra is recursive, then this algebra contains a linear recursive basis, and the question as to whether an arbitrary Lie polynomial belongs to an ideal of relations is solved algorithmically. Thus, a GShB is an important object of study; an active investigation of this object began in the 1990s. Bokut and Klein found GShBs of the simple finite-dimensional Lie algebras A n , B n , C n , D n , E 6 , E 7 , E 8 , F 4 , G 2 ; see [7]- [10] . Moreover, together with Malcolmson, they described the GShBs for quantum universal enveloping algebras of type A n [11] and proved that a set of Lie polynomials is a GShB of the corresponding Lie algebra if and only if it is a GShB in the associative sense [12] . The idea of a GShB, suggested in the cases of commutative algebras and Lie algebras, has received further development. The recognition of some properties of algebras given with the help of GShBs was the subject of the paper [13] by Gateva-Ivanova and Latyshev. In the 1980s, Mikhalev extended the techniques of compositions to the case of superalgebras by proving the composition lemma for color superalgebras; see [14]-[16]. In [17] , Bokut , Kang, Lee, and Malcolmson found GShBs for the simple finite-dimensional Lie superalgebras of type A n , B n , C n , or D n . In [18, 19] , Kang and Lee constructed the theory of GShBs for modules over Lie algebras and found GShBs (which were called Gröbner-Shirshov pairs in this case) for the irreducible sl n -modules. Chibrikov [20] simplified significantly the notion of a GShB for modules: it became not a pair but a single set in a free module over a free algebra. Moreover, the theory of GShBs was employed in the cases of conformal and vertex algebras. In the papers [21, 22] , Roitman proved the existence of free conformal algebras (this fact is not a consequence of theorems in universal algebra, because conformal algebras do not form a variety). In [23], Bokut , Fong, and Ke obtained the results of Roitman in a different way. Those authors applied ideas and techniques related to GShBs to associative conformal algebras [24] . Vasil iev and Pavlov [25] listed all monomial orders (in the commutative case), and they plan to review all GBs of a finitely generated commutative algebra. The history of the theory of GShBs was presented in [26, 27] . In the papers [7, 8] , for a certain ordering of generators, Bokut and Klein computed the reduced GShBs (BGShBs) for series of simple Lie algebras (the RGShBs are distinguished by the fact that they are uniquely determined by generators and the order on generators). For affine nontwisted Kac-Moody algebras, Poroshenko [28, 29] computed first a certain basis of a Lie algebra as a linear space (this basis was uniquely determined by the order on the Lie monomials that correspond to regular words, a so-called reduced basis), and then he found the RGShB uniquely determined by that basis. A conjecture was stated that, for the classical simple Lie algebras, the reduced basis, which is uniquely determined by the root system and the order on the set of monomials, can be described in terms of the root system. Later, this conjecture was confirmed for fields of characteristic 0. So far, the RGShBs have been computed only for one ordering of the generators. Bokut proposed to compute RGShBs (for the order deg-lex) under an arbitrary ordering of generators. Thus, an entirely new problem arose: for classical simple Lie algebras, under an arbitrary ordering of generators, to describe the reduced bases and then the RGShBs uniquely determined by them in terms of the root system. In this wording, the generators of the Lie algebra are fixed, but the order on them is arbitrary, and the n! RGShBs are analyzed simultaneously (n is the number of generators). This entirely new problem was solved by the author for the Lie algebras A + n , B + n , C + n over a field of characteristic 0 in the papers [30]- [32] , and for the Lie algebra D + n in the present License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GRÖBNER-SHIRSHOV BASES OF THE LIE ALGEBRA D + n 575 paper. This is a new step in the development of the theory of GShBs. For this paper, the reduced bases of the Lie algebra D + n were computed earlier in [33] . Before that, the reduced bases for the algebra D + n had been found in [34, 8] , but for only one ordering of generators.

doi:10.1090/s1061-0022-2011-01159-8
fatcat:drgvgkuqgzavfazniax2qlrfce