Beyond Borcherds Lie algebras and inside

Berman, Stephen; Jurisich, Elizabeth; Tan, Shaobin

doi:10.1090/S0002-9947-00-02582-4

Beyond Borcherds Lie algebras and inside
HTML articles powered by AMS MathViewer

by Stephen Berman, Elizabeth Jurisich and Shaobin Tan PDF

Trans. Amer. Math. Soc. 353 (2001), 1183-1219 Request permission

Abstract:

We give a definition for a new class of Lie algebras by generators and relations which simultaneously generalize the Borcherds Lie algebras and the Slodowy G.I.M. Lie algebras. After proving these algebras are always subalgebras of Borcherds Lie algebras, as well as some other basic properties, we give a vertex operator representation for a factor of them. We need to develop a highly non-trivial generalization of the square length two cut off theorem of Goddard and Olive to do this.

References

S. Berman, On generators and relations for certain involutory subalgebras of Kac-Moody Lie algebras, Comm. Algebra 17 (1989), no. 12, 3165–3185. MR 1030614, DOI 10.1080/00927878908823899
S. Berman and R. V. Moody, Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy, Invent. Math. 108 (1992), no. 2, 323–347. MR 1161095, DOI 10.1007/BF02100608
Richard E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068–3071. MR 843307, DOI 10.1073/pnas.83.10.3068
Richard Borcherds, Generalized Kac-Moody algebras, J. Algebra 115 (1988), no. 2, 501–512. MR 943273, DOI 10.1016/0021-8693(88)90275-X
Richard E. Borcherds, Central extensions of generalized Kac-Moody algebras, J. Algebra 140 (1991), no. 2, 330–335. MR 1120425, DOI 10.1016/0021-8693(91)90158-5
Georgia Benkart and Efim Zelmanov, Lie algebras graded by finite root systems and intersection matrix algebras, Invent. Math. 126 (1996), no. 1, 1–45. MR 1408554, DOI 10.1007/s002220050087
S. Eswara Rao, R. V. Moody, and T. Yokonuma, Lie algebras and Weyl groups arising from vertex operator representations, Nova J. Algebra Geom. 1 (1992), no. 1, 15–57. MR 1163780
Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR 996026
I. B. Frenkel, Representations of Kac-Moody algebras and dual resonance models, Applications of group theory in physics and mathematical physics (Chicago, 1982) Lectures in Appl. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1985, pp. 325–353. MR 789298
P. Goddard and D. Olive, Algebras, lattices and strings, Vertex operators in mathematics and physics (Berkeley, Calif., 1983) Math. Sci. Res. Inst. Publ., vol. 3, Springer, New York, 1985, pp. 51–96. MR 781374, DOI 10.1007/978-1-4613-9550-8_{5}
Elizabeth Jurisich, An exposition of generalized Kac-Moody algebras, Lie algebras and their representations (Seoul, 1995) Contemp. Math., vol. 194, Amer. Math. Soc., Providence, RI, 1996, pp. 121–159. MR 1395597, DOI 10.1090/conm/194/02391
Elizabeth Jurisich, Generalized Kac-Moody Lie algebras, free Lie algebras and the structure of the Monster Lie algebra, J. Pure Appl. Algebra 126 (1998), no. 1-3, 233–266. MR 1600542, DOI 10.1016/S0022-4049(96)00142-9
E. Jurisich, J. Lepowsky, and R. L. Wilson, Realizations of the Monster Lie algebra, Selecta Math. (N.S.) 1 (1995), no. 1, 129–161. MR 1327230, DOI 10.1007/BF01614075
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Hermann Kober, Transformationen von algebraischem Typ, Ann. of Math. (2) 40 (1939), 549–559 (German). MR 96, DOI 10.2307/1968939
P. Slodowy, Beyond Kac-Moody algebras and inside, Can. Math. Soc. Conf. Proc. 5(1986), 361-371.
P. Slodowy, Kac-Moody algebras, assoziiert Gruppen und Verallgemeinerugen, Habiliation-sschrift, Universitat Bonn, 1984.