Bulletin of the American Mathematical Society

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Book Information:

Author: J. F. Cornwell
Title: Group theory in physics, volume III, Supersymmetries and infinite-dimensional algebras
Additional book information: Techniques of Physics (N. H. Marsh, ed.), Academic Press, New York, 1989, 615 pp., $55.00. ISBN 0-12-189805-9.

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

Denis Bernard and Jean Thierry-Mieg, Level one representations of the simple affine Kac-Moody algebras in their homogeneous gradations, Comm. Math. Phys. 111 (1987), no. 2, 181–246. MR 899850

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

Alex J. Feingold and Igor B. Frenkel, A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus $2$, Math. Ann. 263 (1983), no. 1, 87–144. MR 697333, DOI 10.1007/BF01457086

Alex J. Feingold and Igor B. Frenkel, Classical affine algebras, Adv. in Math. 56 (1985), no. 2, 117–172. MR 788937, DOI 10.1016/0001-8708(85)90027-1

Alex J. Feingold, Igor B. Frenkel, and John F. X. Ries, Spinor construction of vertex operator algebras, triality, and $E^{(1)}_8$, Contemporary Mathematics, vol. 121, American Mathematical Society, Providence, RI, 1991. MR 1123265, DOI 10.1090/conm/121

Alex J. Feingold and James Lepowsky, The Weyl-Kac character formula and power series identities, Adv. in Math. 29 (1978), no. 3, 271–309. MR 509801, DOI 10.1016/0001-8708(78)90020-8

I. B. Frenkel, Spinor representations of affine Lie algebras, Proc. Nat. Acad. Sci. U.S.A. 77 (1980), no. 11, 6303–6306. MR 592548, DOI 10.1073/pnas.77.11.6303

I. B. Frenkel, Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory, J. Functional Analysis 44 (1981), no. 3, 259–327. MR 643037, DOI 10.1016/0022-1236(81)90012-4

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

I. B. Frenkel and V. G. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980/81), no. 1, 23–66. MR 595581, DOI 10.1007/BF01391662

[FLM] I. B. Frenkel, J. Lepowsky, and A. Meurman, Vertex operator algebras and the monster, Pure Appl. Math., vol. 134, Academic Press, Boston, 1989.

Peter Goddard and David Olive, Kac-Moody and Virasoro algebras in relation to quantum physics, Internat. J. Modern Phys. A 1 (1986), no. 2, 303–414. MR 864165, DOI 10.1142/S0217751X86000149

P. Goddard, W. Nahm, D. Olive, and A. Schwimmer, Vertex operators for non-simply-laced algebras, Comm. Math. Phys. 107 (1986), no. 2, 179–212. MR 863639

V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403

V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1323–1367 (Russian). MR 0259961

V. G. Kac, Infinite-dimensional Lie algebras, and the Dedekind $\eta$-function, Funkcional. Anal. i Priložen. 8 (1974), no. 1, 77–78 (Russian). MR 0374210

V. G. Kac, Infinite-dimensional algebras, Dedekind’s $\eta$-function, classical Möbius function and the very strange formula, Adv. in Math. 30 (1978), no. 2, 85–136. MR 513845, DOI 10.1016/0001-8708(78)90033-6

V. G. Kac, Infinite-dimensional algebras, Dedekind’s $\eta$-function, classical Möbius function and the very strange formula, Adv. in Math. 30 (1978), no. 2, 85–136. MR 513845, DOI 10.1016/0001-8708(78)90033-6

V. G. Kac, D. A. Kazhdan, J. Lepowsky, and R. L. Wilson, Realization of the basic representations of the Euclidean Lie algebras, Adv. in Math. 42 (1981), no. 1, 83–112. MR 633784, DOI 10.1016/0001-8708(81)90053-0

V. G. Kac and D. H. Peterson, Affine Lie algebras and Hecke modular forms, Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 3, 1057–1061. MR 585190, DOI 10.1090/S0273-0979-1980-14854-5

Victor G. Kac and Dale H. Peterson, Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 6, 3308–3312. MR 619827, DOI 10.1073/pnas.78.6.3308

Victor G. Kac and Dale H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. in Math. 53 (1984), no. 2, 125–264. MR 750341, DOI 10.1016/0001-8708(84)90032-X

Victor G. Kac and Dale H. Peterson, $112$ constructions of the basic representation of the loop group of $E_8$, Symposium on anomalies, geometry, topology (Chicago, Ill., 1985) World Sci. Publishing, Singapore, 1985, pp. 276–298. MR 850863

Victor G. Kac and Minoru Wakimoto, Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), no. 14, 4956–4960. MR 949675, DOI 10.1073/pnas.85.14.4956

Victor G. Kac and Minoru Wakimoto, Modular and conformal invariance constraints in representation theory of affine algebras, Adv. in Math. 70 (1988), no. 2, 156–236. MR 954660, DOI 10.1016/0001-8708(88)90055-2

J. Lepowsky, Macdonald-type identities, Advances in Math. 27 (1978), no. 3, 230–234. MR 554353, DOI 10.1016/0001-8708(78)90099-3

J. Lepowsky, Generalized Verma modules, loop space cohomology and MacDonald-type identities, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 2, 169–234. MR 543216

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

J. Lepowsky, Euclidean Lie algebras and the modular function $j$, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 567–570. MR 604635

J. Lepowsky, Application of the numerator formula to $k$-rowed plane partitions, Adv. in Math. 35 (1980), no. 2, 179–194. MR 560134, DOI 10.1016/0001-8708(80)90047-X

J. Lepowsky, Calculus of twisted vertex operators, Proc. Nat. Acad. Sci. U.S.A. 82 (1985), no. 24, 8295–8299. MR 820716, DOI 10.1073/pnas.82.24.8295

J. Lepowsky and S. Milne, Lie algebraic approaches to classical partition identities, Adv. in Math. 29 (1978), no. 1, 15–59. MR 501091, DOI 10.1016/0001-8708(78)90004-X

James Lepowsky and Robert V. Moody, Hyperbolic Lie algebras and quasiregular cusps on Hilbert modular surfaces, Math. Ann. 245 (1979), no. 1, 63–88. MR 552580, DOI 10.1007/BF01420431

James Lepowsky and Robert Lee Wilson, Construction of the affine Lie algebra $A_{1}^{{}}(1)$, Comm. Math. Phys. 62 (1978), no. 1, 43–53. MR 573075

James Lepowsky and Robert Lee Wilson, The Rogers-Ramanujan identities: Lie theoretic interpretation and proof, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 2, 699–701. MR 605423, DOI 10.1073/pnas.78.2.699

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, A new family of algebras underlying the Rogers-Ramanujan identities and generalizations, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 12, 7254–7258. MR 638674, DOI 10.1073/pnas.78.12.7254

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

I. G. Macdonald, Affine root systems and Dedekind’s $\eta$-function, Invent. Math. 15 (1972), 91–143. MR 357528, DOI 10.1007/BF01418931

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

R. V. Moody, Macdonald identities and Euclidean Lie algebras, Proc. Amer. Math. Soc. 48 (1975), 43–52. MR 442048, DOI 10.1090/S0002-9939-1975-0442048-2

Akihiro Tsuchiya and Yukihiro Kanie, Vertex operators in conformal field theory on $\textbf {P}^1$ and monodromy representations of braid group, Conformal field theory and solvable lattice models (Kyoto, 1986) Adv. Stud. Pure Math., vol. 16, Academic Press, Boston, MA, 1988, pp. 297–372. MR 972998, DOI 10.2969/aspm/01610297

Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772

C. N. Yang and M. L. Ge (eds.), Braid group, knot theory and statistical mechanics, Advanced Series in Mathematical Physics, vol. 9, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. MR 1062420

Masaaki Yoshida, Discrete reflection groups in a parabolic subgroup of $\textrm {Sp}(2,\,\textbf {R})$ and symmetrizable hyperbolic generalized Cartan matrices of rank $3$, J. Math. Soc. Japan 36 (1984), no. 2, 243–258. MR 740316, DOI 10.2969/jmsj/03620243

[Ben] G. Benkart, A Kac-Moody bibliography and some related references, Lie Algebras and Related Topics, Canadian Mathematical Society Conference Proceedings (D. J. Britten, F. W. Lemire, and R. V. Moody, eds. ), vol. 5, Amer. Math. Soc. Providence, RI, 1986, pp. 111-135. MR 0832196




[BT] D. Bernard and J. Thierry-Mieg, Level one representations of the simple affine Kac-Moody algebras in their homogeneous gradations, Comm. Math. Phys. 111 (1987), 181-246. MR 899850




[Bo] R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), 3068-3071. MR 843307




[FF1] A. J. Feingold and I. B. Frenkel, A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus 2, Math. Ann. 263 (1983), 87-144. MR 697333




[FF2] A. J. Feingold and I. B. Frenkel, Classical affine algebras, Adv. in Math. 56 (1985), 117-172. MR 788937




[FFR] A. J. Feingold, I. B. Frenkel, and J. F. X. Ries, Spinor construction of vertex operator algebras, triality and E₈⁽¹⁾, Contemp. Math., Amer. Math. Soc. Providence, RI, 1991 (to appear). MR 1123265




[FL] A. J. Feingold and J. Lepowsky, The Weyl-Kac character formula and power series identities, Adv. in Math. 29 (1978), 271-309. MR 509801




[Fr1] I. Frenkel, Spinor representations of affine Lie algebras, Proc. Nat. Acad. Sci. U.S.A. 77 (11) (1980), 6303-6306. MR 592548




[Fr2] I. Frenkel, Two constructions of affine Lie algebra representations and bosonfermion correspondence in quantum field theory, J. Funct. Anal. 44 (3) (1981), 259-327. MR 643037




[Fr3] I. Frenkel, Representations of affine Lie algebras, Hecke modular forms and Korteweg-de Vries type equations, Proceedings of the 1981 Rutgers Conference on Lie Algebras and Related Topics, Lecture Notes in Math., vol. 933, Springer-Verlag, Berlin and New York, 1982, pp. 71-110. MR 675108




[FK] I. Frenkel and V. G. Kac, Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62 (1980), 23-66. MR 595581




[FLM] I. B. Frenkel, J. Lepowsky, and A. Meurman, Vertex operator algebras and the monster, Pure Appl. Math., vol. 134, Academic Press, Boston, 1989.




[GO] P. Goddard and D. I. Olive, Kac-Moody and Virasoro algebras in relation to quantum physics, Internat. J. Modern Phys. A 1 (2) (1986), 303-414. MR 864165




[GNOS] P. Goddard, W. Nahm, D. I. Olive, and A. Schwimmer, Vertex operators for non-simply-laced algebras, Comm. Math. Phys. 107 (1986), 179-212. MR 863639




[J] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), 335-388. MR 908150




[K1] V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Izv. Akad. Nauk SSSR 32 (1968), 1323-1367, (Russian); Math. USSR-Izv. 2 (1968), 1271-1311, (English transl. ). MR 259961




[K2] V. G. Kac, Infinite-dimensional Lie algebras and Dedekind's η-function, Funkt. Anal. i Ego Prilozheniya 8 (1974), 77-78, (Russian); Functional Anal. Appl. 8 (1974), 68-70, (English transl. ). MR 374210




[K3] V. G. Kac, Infinite-dimensional algebras, Dedekind's η-function, classical Mobius function and the very strange formula, Adv. in Math. 30 (1978), 85-136. MR 513845




[K4] V. G. Kac, An elucidation of "Infinite-dimensional algebras... and the very strange formula". E₈⁽¹⁾ and the cube root of the modular invariant j, Adv. in Math. 35 (1980), 264-273. MR 563927




[KKLW] V. G. Kac, D. A. Kazhdan, J. Lepowsky, and R.L. Wilson, Realization of the basic representations of the Euclidean Lie algebras, Adv. in Math. 42 (1981), 83-112. MR 633784




[KP1] V. G. Kac and D. H. Peterson, Affine Lie algebras and Hecke modular forms, Bull. Amer. Math. Soc. (N.S.) 3 (3) (1980), 1057-1061. MR 585190




[KP2] V. G. Kac and D. H. Peterson, Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Nat. Acad. Sci. U.S.A. 78 (6) (1981), 3308-3312. MR 619827




[KP3] V. G. Kac and D. H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. in Math. 53 (1984), 125-264. MR 750341




[KP4] V. G. Kac and D. H. Peterson, 112 Constructions of the basic representations of the loop group of E₈, Proc. of the Conf. Anomalies, Geometry, Topology, Argonne 1985, World Scientific, 1985, pp. 276-298. MR 850863




[KW1] V. Kac and M. Wakimoto, Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. U. S. A. 85 (1988), 4956-4960. MR 949675




[KW2] V. Kac and M. Wakimoto, Modular and conformal invariance constraints in representation theory of affine algebras, Adv. in Math. 70 (2) (1988), 156-236. MR 954660




[L1] J. Lepowsky, Macdonald-type identities, Adv. in Math. 27 (1978), 230-234. MR 554353




[L2] J. Lepowsky, Generalized Verma modules, loop space cohomology and Macdonald-type identities, Ann. Sci. École Norm. Sup. (4) 12 (1979), 169-234. MR 543216




[L3] J. Lepowsky, Lie algebras and combinatorics, Proc. Internat. Congr. Mathematicians, Helsinki, 1978, part 2, Academia Scientiarum Fennica, Helsinki, 1980, pp. 579-584. MR 562658




[L4] J. Lepowsky, Euclidean Lie algebras and the modular function j, Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, RI, 1980, pp. 567-570. MR 604635




[L5] J. Lepowsky, Application of the numerator formula to k-rowed plane partitions, Adv. in Math. 35 (1980), 179-194. MR 560134




[L6] J. Lepowsky, Calculus of twisted vertex operators, Proc. Nat. Acad. Sci. U.S.A. 82 (1985), 8295-8299. MR 820716




[LMi] J. Lepowsky and S. Milne, Lie algebraic approaches to classical partition identities, Adv. in Math. 29 (1978), 15-59. MR 501091




[LMo] J. Lepowsky and R. V. Moody, Hyperbolic Lie algebras and quasi-regular cusps on Hilbert modular surfaces, Math. Ann. 245 (1979), 63-88. MR 552580




[LW1] J. Lepowsky and R. L. Wilson, Construction of the affine Lie algebra A₁⁽¹⁾, Comm. Math. Phys. 62 (1978), 43-53. MR 573075




[LW2] J. Lepowsky and R. L. Wilson, The Rogers-Ramanujan identities: Lie theoretic interpretation and proof, Proc. Nat. Acad. Sci. U.S.A. 78 (2) (1981), 699-701. MR 605423




[LW3] 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




[LW4] J. Lepowsky and R. L. Wilson, A new family of algebras underlying the Rogers-Ramanujan identities and generalizations, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 7254-7258. MR 638674




[LW5] 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




[LW6] J. Lepowsky and R. L. Wilson, The structure of standard modules. II. The case A₁⁽¹⁾, principal gradation, Invent. Math. 79 (1985), 417-442. MR 782227




[Mac] I. G. Macdonald, Affine root systems and Dedekind's η-function, Invent. Math. 15 (1972), 91-143. MR 357528




[Mo1] R. V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 210-230. MR 229687




[Mo2] R. V. Moody, Euclidean Lie algebras, Canad. J. Math. 21 (1969), 1432-1454. MR 255627




[Mo3] R. V. Moody, Macdonald identities and Euclidean Lie algebras, Proc. Amer. Math. Soc. 48 (1975), 43-52. MR 442048




[TK] A. Tsuchiya and Y. Kanie, Vertex operators in conformal field theory on $P\sp 1$ and monodromy representations of braid group, Conformal Field Theory and Solvable Lattice Models, Ad. Stud. Pure Math., vol. 16, Academic Press, NY, 1988, pp. 297-372. MR 972998




[W] E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351-399. MR 990772




[YG] C. N. Yang and M. L. Ge (eds.), Braid group, knot theory and statistical mechanics, Adv. Ser. Math. Phys., vol. 9, World Scientific, Singapore, 1989. MR 1062420




[Yo] M. Yoshida, Discrete reflection groups in a parabolic subgroup of ${\rm Sp}(2,\,R)$ and symmetrizable hyperbolic generalized Cartan matrices of rank 3, J. Math. Soc. Japan 36 (2) (1984), 243-258. MR 740316