Abstract
In this paper, the history and the main results of the theory of Gröbner–Shirshov bases are given for commutative, noncommutative, Lie, and conformal algebras from the beginning (1962) to the present time. The problem of constructing a base of a free Lie algebra is considered, as well as the problem of studying the structure of free products of Lie algebras, the word problem for Lie algebras, and the problem of embedding an arbitrary Lie algebra into an algebraically closed one. The modern form of the composition-diamond lemma (the CD lemma) is presented. The rewriting systems for groups are considered from the point of view of Gröbner–Shirshov bases. The important role of conformal algebras is treated, the statement of the CD lemma for associative conformal algebras is given, and some examples are considered. An analog of the Hilbert basis theorem for commutative conformalalgebras is stated. Bibliography: 173 titles.
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REFERENCES
S. I. Adyan, “Defining relations and algorithmic problems for semigroups and groups," Trudy Steklov Mat. Inst., 85 (1966).
S. I. Adyan and G. U. Oganesyan, “On the problems of equality and divisibility in semigroups with one defining relation," Izv. Akad. Nauk SSSR, Ser. Mat., 42, 219-225 (1978).
V. Ya. Belyaev, “Subrings of finitely presented associative rings," Algebra Logika, 17, 627-638 (1978).
L. A. Bokut', “The embedding of Lie algebras into algebraically closed Lie algebras," Algebra Logika, 1, 47-53 (1962).
L. A. Bokut', “The embedding of algebras into algebraically closed algebras," Dokl. Akad. Nauk SSSR, 154, 963-964 (1962).
L. A. Bokut', “A base of the free polynilpotent Lie algebra," Algebra Logika, 2, 13-19 (1963).
L. A. Bokut', “On a property of Boone's groups. I," Algebra Logika, 5, 5-23 (1966).
L. A. Bokut', “On a property of Boone's groups, II," Algebra Logika, 6, 15-24 (1967).
L. A. Bokut', “On Novikov groups," Algebra Logika, 6, 25-38 (1967).
L. A. Bokut', “On the embedding of rings into skew fields," Dokl Akad. Nauk SSSR, 175, 755-758 (1967).
L. A. Bokut', “Groups with a relative standard base," Sib. Mat. Zh., 9, 755-758 (1968).
L. A. Bokut' “The degrees of undecidability of the conjugacy problem for finitely presented groups," Algebra Logika, 7, Nos. 5-6 (1968).
L. A. Bokut', “The groups of fractions of the multiplicative semigroups of some rings. I, II, III," Sib. Mat. Zh., 10, No. 2, 246-286 (1969); 10, No. 4, 744-799 (1969); 10, No. 4, 800-819 (1969).
L. A. Bokut' “On a Maltsev problem," Sib. Mat. Zh. 10, No. 5, 965-1005 (1969).
L. A. Bokut' “Some problems in group theory and ring theory," Abstract Doctoral Thesis, Novosibirsk (1969).
L. A. Bokut', “Unsolvability of the equality problem and subalgebras of finitely presented Lie algebras," Izv. Akad. Nauk SSSR, Ser. Mat., 36, 1173-1219 (1972).
L. A. Bokut', “The embeddings into simple associative algebras," Algebra Logika, 15, 117-142 (1976).
L. A. Bokut', “On algebraically closed and simple Lie algebras, Trudy Steklov Mat. Inst., 148, 30-42 (1978).
L. A. Bokut', “A remark on the Borisov-Boone group," Sib. Mat. Zh., 26, 43-46 (1985).
S. D. Brodskii, “Equations over groups and groups with one defining relation," Sib. Mat. Zh., 25, 84-103 (1984).
A. T. Gainov, “Free commutative and free anticommutative products of algebras," Sib. Mat. Zh., 3, 805-833 (1962).
V. N. Gerasimov, “Distributive lattices of subspaces and the equality problem for algebras with one rela-tion," Algebra Logika, 15, 384-435 (1976).
M. M. Glukhov, “On free products and algorithmic problems in R-varieties of universal algebras," Dokl. Akad. Nauk SSSR, 193, 514-517 (1970).
M. M. Glukhov, “R-varieties of quasigroups and loops," in: Problems in the Theory of Quasigroups and Loops, Kishinev (1970), pp. 37-47.
M. M. Glukhov, “Free decompositions and algorithmic problems in R-varieties of universal algebras," Mat. Sb., 85, 307-338 (1971).
E. S. Golod, “Standard bases and homology. II," Trudy Steklov Mat. Inst., 208, 106-110 (1995).
E. S. Golod, “The homology algebra of the Shafarevich complex of a free algebra," Fund. Prikl. Mat., 5, No. 1 (1999).
E. S. Golod, “The homology of the Shafarevich complex and noncommutative complete intersections," Fund. Prikl. Mat., 5, No. 1 (1999).
E. S. Golod, “The Shafarevich complex and its applications," Abstract Doctoral Thesis, Moscow State University, Moscow (1999).
E. I. Zel'manov, “A solution of the weakened Burnside problem for groups of odd exponent," Izv. Akad. Nauk SSSR, Ser. Mat., 54, No. 1, 42-59 (1990).
E. I. Zel'manov, “A solution of the weakened Burnside problem for 2-groups," Mat. Sb., 182, No. 4, 568-592 (1991).
A. R. Kemer, “A property of finite bases for identities of an associative algebra," Algebra Logika, 26, 597-641 (1987).
A. R. Kemer, “A solution of the problem on a finite base for identities of associative algebras," Dokl. Akad. Nauk SSSR, 30, No. 6, 68-74 (1989).
P. S. Kolesnikov, “Makar-Limanov's algebraically closed skew fields," Algebra Logika, 2000 (to appear).
P. S. Kolesnikov, “On commutative conformal algebras," Master's thesis, Novosibirsk State University (2000).
P. S. Kolesnikov, “A base for a free associative commutative conformal algebra," in: The 4th Siberian Congress in Applied and Industrial Mathematics, Abstracts, part IV (2000), p. 108.
G. P. Kukin, “On the Cartesian subalgebra of the free Lie sum of Lie algebras," Algebra Logika, 9, No. 6, 701-713 (1970).
G. P. Kukin, “On the equality problem for Lie algebras," Sib. Mat. Zh., 18, No. 5, 1194-1197 (1977).
G. P. Kukin, “Subalgebras of finitely presented Lie algebras," Algebra Logika, 18, No. 3, 311-327 (1978).
A. G. Kurosh, “Nonassociative free algebras and free products of algebras," Mat. Sb., 20, No. 2, 239-262 (1947).
V. N. Latyshev, “On the algorithm of identity in Lie nilpotent associative algebras," Vestn. Kiev Univ., Ser. Mat., Mekh., 27, 67 (1985).
V. N. Latyshev, Combinatorial Ring Theory. Standard Bases [in Russian], Moscow State University, Moscow (1988).
V. N. Latyshev, E. V. Pankrat'ev, and A. V. Mikhalev, “The construction of a canonical simplificator in modules over the ring of polynomials," Vestn. Kiev Univ., Ser. Mat., Mekh., 27, 65-67 (1985).
A. I. Mal'tsev, Selected Works [in Russian], Vol. 1, Nauka, Moscow (1976).
Yu. V. Matiyasevich, “Enumerable sets are Diophantine," Dokl. Akad. Nauk SSSR, 191, No. 2, 279-282 (1970).
A. A. Mikhalev, “Subalgebras of free color Lie superalgebras," Mat. Zametki, 37, No. 5, 653-661 (1985).
A. A. Mikhalev, “Subalgebras of free Lie p-superalgebras," Mat. Zametki, 43, No. 2, 178-191 (1988).
A. A. Mikhalev “The composition lemma and the equality problem for color Lie superalgebras," Vest. Mosk. Univ., Ser. Mat., Mekh., 44, No. 5, 88-91 (1989).
A. A. Mikhalev, “The techniques of A. I. Shirshov's composition in Lie superalgebras (noncommutative Gröbner bases)," in: Proceedings of I. G. Petrovskii Seminar, 18 (1995), pp. 277-289.
A. V. Mikhalev and E. V. Pankrat'ev, Calculations in Difierential and Difierence Algebra [in Russian], Moscow State University, Moscow (1989).
A. A. Mikhalev and E. A. Vasil'eva, “Free left symmetric superalgebras," Fund. Prikl. Mat., 2, No2, 623-626 (1996).
A. A. Mikhalev and A. A. Zolotykh, “A complex of algorithms for computations in Lie superalgebras," Program. Comput. Software, 23, No. 1, 8-16 (1997).
P. S. Novikov, “On the algorithmic undecidability of the problem on the identity of words in groups," Trudy Steklov Mat. Inst., 44 (1955).
E. N. Poroshenko, “Gröbner-Shirshov bases of Kac-Moody algebras," Master's thesis Novosibirsk State University (1999).
E. N. Poroshenko, “Gröbner-Shirshov bases of Kac-Moody algebras B (1) n, in: Proceedings of the XXXVIII International Student Conference, Novosibirsk (2000).
E. N. Poroshenko, “Gröbner-Shirshov bases of Kac-Moody algebras B (1) n," in: The 4th Siberian Congress in Applied and Industrial Mathematics, Abstracts, part IV (2000), p. 113.
D. M. Puga, “Gröbner-Shirshov bases of Boone's groups," Bachelor's thesis, Novosibirsk State University (1998).
V. A. Ufnarovskii, “On the growth of algebras," Vestn. Mosk. Univ., Ser. Mat., Mekh., No. 4, 59-65 (1978).
V. A. Ufnarovskii, “On Poincarfie series of graded algebras," Mat. Zametki, 27, No. 1, 21-32 (1980).
V. A. Ufnarovskii, “A criterion of the growth of graphs and algebras defined by words," Mat. Zametki, 31, No. 3, 465-472 (1982).
V. A. Ufnarovskii, “Algebras defined by two quadratic relations. Studies in the theory of rings, algebras and modules," Mat. Issled., No. 78, 148-172 (1984).
V. A. Ufnarovskii, “Independence theorem and its consequences," Mat. Sb., 128, No. 1, 124-132 (1985).
V. A. Ufnarovskii, “On a finite presentation of the Hamilton algebras. Strictly regular algebras and topolo-gies," Mat. Issled., 103-112 (1987).
V. A. Ufnarovskii, “On the use of graphs for calculation of the base, growth, and Hilbert series of associative algebras, Mat. Sb., 180, No. 11, 1548-1560 (1989).
V. A. Ufnarovskii, “Combinatorial and asymptotic methods in algebra," Itogi Nauki Tekhniki, 57 (1990).
A. A. Fridman, “The relationship between the equality problem and the conjugacy problem for finitely presented groups," Trudy Moskov. Mat. Ob., 9, 329-356 (1960).
A. A. Fridman, “The degrees of undecidability of the equality problem for finitely presented groups," Dokl. Akad. Nauk SSSR, 147, No. 4, 805-808 (1962).
A. A. Fridman, The Degrees of Undecidability of the Equality Problem for Finitely Presented Groups [in Russian], Nauka, Moscow (1967).
O. G. Kharlampovich, “Finitely presented solvable groups and Lie algebras with unsolvable equality prob-lem," Mat. Zametki, 46, No. 3, 80-92 (1989).
D. L. Chubarov, “An application of the composition lemma to the construction of a universal enveloping conformal Lie algebra," Master's thesis, Novosibirsk State University (2000).
L. V. Shabunin, “The decidability of the elementary theory of finitely presented quasigroups," Mat. Za-metki, 47, No. 4, 138-146 (1990).
L. V. Shabunin, “The decidability of the theories of finitely presented quasigroups from R-varieties of quasigroups," Sib. Mat. Zh., 32, No. 3, 201-211 (1991).
L. V. Shabunin, “On the elementary equivalence of free quasigroups," Fund. Prikl. Mat., 5, No 3, 885-901 (1999).
L. V. Shabunin, “Free and finitely presented algebras of varieties of quasigroups and Cantor varieties," Abstract Doctoral Thesis, Mathematical Institute, Novosibirsk (2000).
A. I. Shirshov, “Some problems in the theory of nonassociative rings and algebras," Abstract Ph.D. Thesis, Moscow State University, Moscow (1953).
A. I. Shirshov, “Subalgebras of free Lie algebras," Mat. Sb., 33(75), No. 2, 441-452 (1953).
A. I. Shirshov, “On the rings with identity relations," Mat. Sb., 43, No. 2, 277-283 (1957).
A. I. Shirshov, “On free Lie rings," Mat. Sb., 45(87), 113-122 (1958).
A. I. Shirshov, “On bases for free Lie algebras," Algebra Logika, 1, 14-19 (1962).
A. I. Shirshov, “Some algorithmic problems for Lie algebras," Sib. Mat. Zh., 3, 292-296 (1962).
A. I. Shirshov, “On a hypothesis in the theory of Lie algebras," Sib. Mat. Zh., 3, 297-301 (1962).
A. d'Andrea and V. G. Kac, “Structure theory of finite conformal algebras, Sel. Math., New Ser., 4, 377-418 (1998).
D. J. Anick, “Noncommutative graded algebras and their Hilbert series," J. Algebra, 78, No. 1, 120-140 (1982).
D. Anick and C. Löfwall, “Hilbert series of finitely presented algebras. Algebra, algebraic topology and their interactions," Lect. Notes Math., 1183, 32-55 (1986).
D. I. Anick, “On the cohomology of associative algebras," Trans. Am. Math. Soc., 296, 641-659 (1986).
D. J. Anick “Recent progress in Hilbert and Poincare series," Lect. Notes Math., 1318, 1-25 (1988).
Yu. A. Bakhturin, A. A. Mikhalev, V. M. Petrogradsky, and M. V. Zaitsev, “Infinite-dimensional Lie superalgebras," De Gruyter Exp. Math., 7 (1992).
J. Backelin, “La serie de Poincarfie-Betti d'une algebre gradufiee de type fini a une relation est rationelle," C. R. Acad. Sci., Ser., 287, 843-846 (1978).
K. I. Beidar, W. S. Martindale, and A. V. Mikhalev, “Rings with generalized identities," Pure Appl. Math., 196 (1996).
A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory," Nuclear Phys., 241, 333-380 (1984).
A. Ya. Belov, V. V. Borisenko, and V. N. Latyshev, “Monomial algebras," J. Math. Sci., 87, No.3, 3463-3575 (1997).
A. Belov and T. Gateva-Ivanova, “Radicals of monomial algebras," in: First International Tainan-Moscow Algebra Workshop. Proceedings of the International Conference, de Gruyter, Berlin (1996), pp. 159-169.
G. M. Bergman, “The diamond lemma for ring theory," Adv. Math., 29, 178-218 (1978).
L. A. Bokut', “Theorems of embedding in the theory of algebras," Colloq. Math., 14, 349-353 (1966).
L. A. Bokut', “The method of Gröbner-Shirshov bases," Sib. Adv. Math., 9, No. 3, 1-16 (1999).
L. A. Bokut', Y. Fong, and W.-F. Ke, “Gröbner-Shirshov bases and the composition lemma for associative conformal algebras: an example," Contemporary Math. (to appear).
L. A. Bokut', Y. Fong, and W.-F. Ke, “Composition-diamond lemma for associative conformal algebras," Preprint.
L. A. Bokut', Y. Fong, and W.-F. Ke, “Free associative conformal algebras," in: Proceedings of the 2 nd Tainan-Moscow Algebra and Combinatorics Workshop, Tainan (1997), pp. 13-25.
L. A. Bokut', S.-J. Kang, K.-H. Lee, and P. Malcolmson, “Gröbner-Shirshov bases for Lie superalgebras and their universal enveloping algebras," J. Algebra, 217, No. 2, 461-495 (1999).
L. A. Bokut' and G. P. Kukin, Algorithmic and Combinatorial Algebra. Mathematics and Its Applications, Kluwer Academic, Dordrecht (1994).
L. A. Bokut' and A. A. Klein, “Serre relations and Gröbner-Shirshov bases for simple Lie algebras. I, II," Internat. J. Algebra Comput., 6, 389-400, 401-412 (1996).
L. A. Bokut' and A. A. Klein, “Gröbner-Shirshov bases for exceptional Lie algebras. I," in: Ring Theory. Selected Papers from the Conference Held in Miskolc, July 15-20, 1996, Amsterdam (1998), pp. 51-57.
L. A. Bokut' and A. A. Klein, “Gröbner-Shirshov bases for the exceptional Lie algebras E6, E7, and E8," in: Algebras and Combinatorics, Springer-Verlag, Singapore (1999), pp. 37-46.
L. A. Bokut' and P. Malcolmson, “Gröbner-Shirshov bases for quantum enveloping algebras," Israel J. Math., 96, 97-113 (1996).
L. A. Bokut' and P. Malcolmson, “Gröbner-Shirshov bases for relations of a Lie algebra and its enveloping algebra," in: Algebras and Combinatorics, Springer-Verlag, Singapore (1999), pp. 47-54.
W. W. Boone, “The word problem," Ann. Math., 70, 207-265 (1959).
W. W. Boone, “Finitely presented group whose word problem has the same degree as that of an arbitrary given Thue system (an application of methods of Britton)," Proc. Nat. Acad. Sci. USA, 53, No. 2, 265-269 (1965).
R. E. Borcherds, “Vertex algebras, Kac-Moody algebras, and the Monster," Proc. Natl. Acad. Sci. USA, 83, 3068-3071 (1986).
A. J. Bowtell, “On a question of Maltsev," J. Algebra, 7, 126-139 (1967).
B. Buchberger, “An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal" [in German], Ph.D. thesis, University of Innsbruck, Austria (1965).
B. Buchberger, “An algorithmical criteria for the solvability of algebraic systems of equations" [in German], Aequationes Math., 4, 374-383 (1970).
Griöbner Bases and Applications, (eds. B. Buchberger and F. Winkler), London Math. Soc. Lect. Notes, 251 (1998).
K.-T. Chen, R. H. Fox, and R. C. Lyndon, “Free difierential calculus. IV. The quotient groups of the lower central series," Ann. Math., 68, 81-95 (1958).
C. R. L. Clapham, “Finitely presented groups with word problems of arbitrary degrees of unsolvability," Proc. Lond. Math. Soc., 14, 633-676 (1964).
D. J. Collins, “Recursively enumerable degrees and the conjugacy problem," J. Symb. Logic, 32, No.3, 432-433 (1967).
P. M. Cohn, “Embedding problems for rings and semigroups. Universal algebra and its links with logic, algebra, combinatorics and computer science," in: Proceedings of the 25th School on Universal Algebra, Darmstadt (1984), pp. 115-126.
D. Eisenbud, I. Peeva, and B. Sturmfels, “Non-commutative Gröbner bases for commutative algebras," Proc. Amer. Math. Soc., 126, No. 3, 687-691 (1998).
T. Evans, “The word problem for abstract algebras," J. Lond. Math. Soc., 26, No. 1, 64-71 (1951).
T. Evans, “On multiplicative systems defined by generators and relations. I. Normal form theorems," Proc. Cambridge Phil. Soc., 47, No 4, 637-645 (1951).
I. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster, Academic Press, London (1988).
T. Gateva-Ivanova, “Skew polynomial rings with binomial relations," J. Algebra, 185, No. 3, 710-753 (1996).
T. Gateva-Ivanova and V. Latyshev, “On recognizable properties of associative algebras. Computational aspects of commutative algebra," J. Symbolic Comput., 6, No. 2-3, 371-388 (1988).
V. P. Gerdt and V. V. Kornyak, “A program for constructing a complete system of relations, basis ele-ments and tables of commutators of finitely presented Lie algebras and superalgebras," Program. Comput. Software, 23, No. 3, 164-172 (1997).
E. S. Golod, “Standard bases and homology," Lect. Notes Math., 1352, 88-95 (1988).
E. Green, T. Mora, and V. Ufnarovski, “The noncommutative Gröbner freaks," Progr. Comput. Sci. Appl. Logic, 15, 93-104 (1998).
E. Green, “An introduction to noncommutative Groebner bases," Lect. Notes Pure Appl. Math., 151, 167-190 (1993).
M. Hall, “A basis for free Lie rings and higher commutators in free groups," Proc. Am. Math. Soc., 1, 575-581 (1950).
P. Hall, “On the embedding of a group in a join of given groups," J. Austral. Math. Soc., 14, No. 4, 434-495 (1974).
G. Hermann, “The question of finitely many steps in the theory of polynomial ideals, Math. Ann., 95, 736-788 (1926).
G. Higman, “Subgroups of finitely presented groups," Proc. Roy. Soc., Ser. A., 262, 455-475 (1961).
H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II," Ann. Math., 79, 109-203, 205-326 (1964).
N. Jacobson, “Some recent developments in the theory of algebras with polynomial identities," Lect. Notes Math., 697, 8-46 (1978).
V. G. Kac, “Formal distribution algebras and conformal algebras," in: The XIIth International Congress in Mathematical Physics, Cambridge (1999), pp. 80-97.
V. Kac, Vertex Algebras for Beginners, Amer. Math. Soc., Providence, Rhode Island (1998).
A. Kandry-Rody and V. Weipfenning, “Noncommutative Greubner bases in algebras of solvable type," J. Symbolic Comput., 9, 1-26 (1990).
S.-J. Kang and K.-H. Lee, “Gröbner-Shirshov bases for representation theory," J. Korean Math. Soc., 37, 55-72 (2000).
S.-J. Kang and K.-H. Lee, “Gröbner-Shirshov bases for irreducible sln+1-modules," J. Algebra (to appear).
S.-J. Kang, I.-S. Lee, K.-H. Lee, and H. Oh, “Gröbner-Shirshov pairs for Specht modules of Hecke algebras," Preprint.
O. G. Kharlampovich and M. V. Sapir, “Algorithmic problems in varieties," Inter. J. Algebra Comput., 5, No. 4, 5, 379-602 (1995).
A. A. Klein, “Rings nonembeddable in fields with multiplicative semigroups embeddable in groups," J. Algebra, 7, 100-125 (1967).
N. Koblitz, Algebraic Aspects of Cryptography, Springer-Verlag, Berlin (1998).
P. S. Kolesnikov, “The Makar-Limanov construction of an algebraically closed skew field via Mal'cev-Neumann series," in: Formal Power Series and Algebraic Combinatorics, 12 th International Conference, Springer-Verlag, Berlin (2000), pp. 454-460.
M. V. Kondratjeva, A. B. Levin, A. V. Mikhalev, and E. V. Pankratjev, Difierential and Difierence Di-mension Polynomials, Kluwer Academic, Dordrecht (1999).
V. N. Latyshev, “Lie nilpotency: recognition and word problems," in: First International Tainan-Moscow Algebra Workshop, de Gruyter, Berlin (1996), pp. 237-239.
V. N. Latyshev, “Canonization and standard bases of filtered structures," in: Lie Algebras, Rings and Related Topics, Springer-Verlag, Hong Kong (2000), pp. 61-79.
V. N. Latyshev, “An improved version of standard bases," in: Formal Power Series and Algebraic Combi-natorics, 12th International Conference, Springer-Verlag, Berlin (2000), pp. 496 505.
J. A. de Loera, B. Sturmfels, and R. R. Thomas, “Groebner bases and triangulations of the second hyper-simplex," Combinatorica, 15, No. 3, 409-424 (1995).
M. Lothaire, Combinatorics on Words. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1997).
R. C. Lyndon, “On Burnside's problem. I," Trans. Am. Math. Soc., 77, 202-215 (1954).
R. C. Lyndon, “On Burnside's problem. II," Trans. Am. Math. Soc., 78, 329-332 (1955).
L. G. Makar-Limanov, “Algebraically closed skew fields," J. Algebra, 93, 117-135 (1985).
A. I. Mal'cev, “On the immersion of an algebraic ring into a field," Math. Ann., 113, 689-691 (1937).
A. A. Mikhalev, “The composition lemma for color Lie superalgebras and for Lie p-superalgebras," Con-temp. Math., 131, No. 2, 91-104 (1992).
A. A. Mikhalev and A. A. Zolotykh, Combinatorial Aspects of Lie Superalgebras, CRC Press, Boca Raton (1995).
A. A. Mikhalev and A. A. Zolotykh, “Standard Gröbner-Shirshov bases of free algebras over rings. I. Free associative algebras," Intern. J. Algebra Comput., 8, No. 6, 689-726 (1998).
A. A. Mikhalev and A. V. Sereguine, “Standard bases of one-sided ideals of algebras of P-type," in: Algebras and Combinatorics. An International Congress, Springer-Verlag, Singapore (1999), pp. 345-351.
A. A. Mikhalev and E. A. Vasilieva, “Standard bases of ideals of free supercommutative polynomial alge-bras (?-Gröbner bases)," in: Lie Algebras, Rings and Related Topics, Springer-Verlag, Hong Kong (2000), pp. 108-125.
T. Mora, “Gröbner bases for nocommutative polynomial rings," Lect. Notes Comput. Sci., 229, 353-362 (1986).
T. Mora, “Gröbner bases in noncommutative algebras," Lect. Notes Comput. Sci., 358, 150-161 (1988).
T. Mora, “An introduction to commutative and noncommutative Grobner bases," Theoret. Comput. Sci., 134, No. 1, 131-173 (1994).
M. H. A. Newman, “On theory with combinatorial definition of equivalence," Ann. Math., 43, 233-243 (1942).
E. Poroshenko, “Gröbner-Shirshov bases for Kac-Moody algebras A (1) n and B (1) n," in: Formal Power Series and Algebraic Combinatorics, 12th International Conference, Springer-Verlag, Berlin (2000), pp. 552-563.
C. Procesi, “Rings with polynomial identities," Pure Appl. Math., 17 (1973).
C. Reutenauer, “Dimensions and characters of the derived series of the free Lie algebra," in: Mfielanges Ofierts a M.-P. Schützenberger, Paris (1990), pp. 171-184.
C. Reutenauer, Free Lie Algebras, London Mathematical Society Monographs. New Series, 7, New York (1993).
M. Roitman, “On free conformal and vertex algebras," J. Algebra, 217, 496-527 (1999).
M. Roitman, “Universal enveloping conformal algebras, Selecta Math. (to appear).
M. Roitman, “A criterion for embedding Lie conformal algebras into associative conformal algebras," Preprint (1999).
V. A. Ufnarovskij, “Calculations of growth and Hilbert series by computer," Lect. Notes Pure Appl. Math., 151, 247-256 (1994).
V. Ufnarovski, “Introduction to noncommutative Gröbner bases theory. Gröbner bases and applications," London Math. Soc. Lect. Notes Ser., 251, 259-280 (1998).
X. G. Viennot, “Algfiebres de Lie libres et monoides libres," Lect. Notes Math., 691 (1978).
E. Witt, “Die Unterringe der freien Lieschen Ringe," Math. Zeit., 64, 195-216 (1956).
A. A. Zolotykh and A. A. Mikhalev, “Algorithms for constructing standard Gröbner-Shirshov bases of ideals of free algebras over commutative rings,” Programmirovanie, No. 6, 10-11 (1998).
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Bokut', L.A., Kolesnikov, P.S. Gröbner–Shirshov Bases: From their Incipiency to the Present. Journal of Mathematical Sciences 116, 2894–2916 (2003). https://doi.org/10.1023/A:1023490323855
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DOI: https://doi.org/10.1023/A:1023490323855