Book Review
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MathSciNet review:
1567601
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Book Information:
Author:
Victor G. Kac
Title:
Infinite dimensional Lie algebras
Additional book information:
second edition, Cambridge University Press, 1985, xvii + 280 pp., $24.95. ISBN 0-521-32133-6.
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
I. B. Frenkel, J. Lepowsky, and A. Meurman, A natural representation of the Fischer-Griess Monster with the modular function $J$ as character, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 10, , Phys. Sci., 3256–3260. MR 747596, DOI 10.1073/pnas.81.10.3256
Ofer Gabber and Victor G. Kac, On defining relations of certain infinite-dimensional Lie algebras, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 185–189. MR 621889, DOI 10.1090/S0273-0979-1981-14940-5
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of ${\mathfrak {p}}$-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48. MR 185016
I. L. Kantor, Graded Lie algebras, Trudy Sem. Vektor. Tenzor. Anal. 15 (1970), 227–266 (Russian). MR 0297827
J. Lepowsky, Lie algebras and combinatorics, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 579–584. MR 562658
James Lepowsky and Robert Lee Wilson, A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities, Adv. in Math. 45 (1982), no. 1, 21–72. MR 663415, DOI 10.1016/S0001-8708(82)80012-1
James Lepowsky and Robert Lee Wilson, The structure of standard modules. I. Universal algebras and the Rogers-Ramanujan identities, Invent. Math. 77 (1984), no. 2, 199–290. MR 752821, DOI 10.1007/BF01388447
James Lepowsky and Robert Lee Wilson, The structure of standard modules. II. The case $A^{(1)}_1$, principal gradation, Invent. Math. 79 (1985), no. 3, 417–442. MR 782227, DOI 10.1007/BF01388515
R. V. Moody, Lie algebras associated with generalized Cartan matrices, Bull. Amer. Math. Soc. 73 (1967), 217–221. MR 207783, DOI 10.1090/S0002-9904-1967-11688-4
Robert V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211–230. MR 229687, DOI 10.1016/0021-8693(68)90096-3
Robert V. Moody, Euclidean Lie algebras, Canadian J. Math. 21 (1969), 1432–1454. MR 255627, DOI 10.4153/CJM-1969-158-2
Jean-Pierre Serre, Algèbres de Lie semi-simples complexes, W. A. Benjamin, Inc., New York-Amsterdam, 1966 (French). MR 0215886
I. M. Singer and Shlomo Sternberg, The infinite groups of Lie and Cartan. I. The transitive groups, J. Analyse Math. 15 (1965), 1–114. MR 217822, DOI 10.1007/BF02787690
Georgia Benkart, A Kac-Moody bibliography and some related references, Lie algebras and related topics (Windsor, Ont., 1984) CMS Conf. Proc., vol. 5, Amer. Math. Soc., Providence, RI, 1986, pp. 111–135. MR 832196
I. B. Frenkel, Representations of affine Lie algebras, Hecke modular forms and Korteweg-de Vries type equations, Lie algebras and related topics (New Brunswick, N.J., 1981) Lecture Notes in Math., vol. 933, Springer, Berlin-New York, 1982, pp. 71–110. MR 675108
Graeme Segal and George Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5–65. MR 783348
- 1.
- N. Bourbaki, Groupes et algèbres de Lie, Chaps. IV-VI, Paris, Hermann, 1968. MR 0240238
- 2.
- I. B. Frenkel, J. Lepowsky and A. Meurman, A natural representation of the Fischer-Griess monster with the modular function J as character, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), 3256-3260. MR 747596
- 3.
- O. Gabber and V. G. Kac, On defining relations of certain infinite-dimensional Lie algebras, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 185-189. MR 621889
- 4.
- J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, New York, 1972. MR 323842
- 5.
- N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 5-48. MR 185016
- 6.
- I. L. Kantor, Graded Lie algebras, Trudy Sem. Vekt. Tens. Anal. 15 (1970), 227-266. (Russian) An English translation of an announcement of these results appeared in Soviet Math. Dokl. 9 (1968), 409-412. MR 297827
- 7.
- J. Lepowsky, Lie algebras and combinatorics, Proc. Internat. Congress of Mathematicians, Helsinki, 1978, pp. 579-584. MR 562658
- 8.
- J. Lepowsky and R. L. Wilson, A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities, Adv. in Math. 45 (1982), 21-72. MR 663415
- 9.
- J. Lepowsky and R. L. Wilson, The structure of standard modules. I. Universal algebras and the Rogers-Ramanujan identities. Invent. Math. 77 (1984), 199-290. MR 752821
- 10.
- J. Lepowsky and R. L. Wilson, The structure of standard modules. II. The case A, Invent. Math. 79 (1985), 417-442. MR 782227
- 11.
- R. V. Moody, Lie algebras associated with generalized Cartan matrices, Bull. Amer. Math. Soc. 73 (1967), 217-230. MR 207783
- 12.
- R. V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211-230. MR 229687
- 13.
- R. V. Moody, Euclidean Lie algebras, Canad. J. Math. 21 (1969), 1432-1454. MR 255627
- 14.
- J.-P. Serre, Algèbres de Lie semi-simples complexes, Benjamin, New York and Amsterdam, 1966. MR 215886
- 15.
- I. Singer and S. Sternberg, On the infinite groups of Lie and Cartan, J. Analyse Math. 15 (1965), 1-114. MR 217822
- 16.
- G. M. Benkart, A Kac-Moody bibliography and some related references, Lie Algebras and Related Topics (D. J. Britten, F. W. Lemire, and R. V. Moody, eds.), Canadian Math. Soc. Conf. Proc. vol. 5, 1984, pp. 111-135, Amer. Math. Soc. Providence, R. I., 1986. MR 832196
- 17.
- I. B. Frenkel, Representations of affine Lie algebras, Hecke modular forms and KortewegdeVries type equations, Lie Algebras and Related Topics (D. J. Winter, ed.), Lecture Notes in Math., vol. 933, 1981, pp. 71-110. Springer-Verlag, Berlin-Heidelberg-New York, 1982. MR 675108
- 18.
- G. Segal and G. Wilson, Loop groups and equations of KdV type, Publ. Math. Inst. Hautes Études Sci. 61 (1985), 5-65. MR 783348
Review Information:
Reviewer:
George B. Seligman
Journal:
Bull. Amer. Math. Soc.
16 (1987), 144-149
DOI:
https://doi.org/10.1090/S0273-0979-1987-15492-9
