Vershik, A. M.; Shoikhet, B. B. Graded Lie algebras whose Cartan subalgebra is the algebra of polynomials in one variable. (English. Russian original) Zbl 1017.17029 Theor. Math. Phys. 123, No. 2, 701-707 (2000); translation from Teor. Mat. Fiz. 123, No. 2, 345-352 (2000). Summary: We define a class of infinite-dimensional Lie algebras that generalize the universal enveloping algebra of the algebra \(\text{sl}(2,\mathbb{C})\) regarded as a Lie algebra. These algebras are a special case of \(\mathbb{Z}\)-graded Lie algebras with a continuous root system, namely, their Cartan subalgebra is the algebra of polynomials in one variable. The continuous limit of these algebras defines new Poisson brackets on algebraic surfaces. MSC: 17B70 Graded Lie (super)algebras 17B65 Infinite-dimensional Lie (super)algebras Keywords:infinite-dimensional Lie algebras; \(\mathbb{Z}\)-graded Lie algebras; continuous root system; continuous limit; Poisson brackets × Cite Format Result Cite Review PDF Full Text: DOI References: [1] M. Saveliev and A. Vershik,Commun. Math. Phys.,126, 367–378 (1989). · Zbl 0691.17012 · doi:10.1007/BF02125130 [2] M. Saveliev and A. Vershik,Phys. Lett. A,143, No. 3, 121–128 (1990). · doi:10.1016/0375-9601(90)90662-8 [3] A. Vershik,Alg. Anal.,4, No. 6, 103–113 (1992). [4] A. Vershik, ”On the classification of Z-graded Lie algebras of constant growth which have the algebraC[h] as Cartan subalgebra,” in:Quantum Groups (Lect. Notes Math., Vol. 1510) (P. P. Kulish, ed.), Springer, Berlin (1992), pp. 395. · Zbl 0780.17030 [5] B. L. Feigin,Russ. Math. Surv.,43, 169–170 (1988). · Zbl 0667.17006 · doi:10.1070/RM1988v043n02ABEH001720 [6] B. Shoihet,J. Math. Sci.,92, 3764–3806 (1998). · Zbl 0916.17019 · doi:10.1007/BF02434006 [7] J. Dixmier,J. Algebra,24, 551–564 (1973). · Zbl 0252.17004 · doi:10.1016/0021-8693(73)90127-0 [8] A. Josef,Isr. J. Math.,28, No. 3, 177–192 (1977). · Zbl 0366.17006 · doi:10.1007/BF02759808 [9] P. Smith,Trans. Am. Math. Soc.,322, No. 1, 285–314 (1990). · Zbl 0732.16019 · doi:10.2307/2001532 [10] V. Kač and H. Radul,Commun. Math. Phys.,157, 429–457 (1993). · Zbl 0826.17027 · doi:10.1007/BF02096878 [11] V. Kač,Infinite-Dimensional Lie Algebras, Birkhäuser, Boston, Mass. (1983). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.