Dimension formula for graded Lie algebras and its applications

Kang, Seok-Jin; Kim, Myung-Hwan

doi:10.1090/S0002-9947-99-02239-4

Dimension formula for graded Lie algebras and its applications
HTML articles powered by AMS MathViewer

by Seok-Jin Kang and Myung-Hwan Kim PDF

Trans. Amer. Math. Soc. 351 (1999), 4281-4336 Request permission

Abstract:

In this paper, we investigate the structure of infinite dimensional Lie algebras $L=\bigoplus _{\alpha \in \Gamma } L_{\alpha }$ graded by a countable abelian semigroup $\Gamma$ satisfying a certain finiteness condition. The Euler-Poincaré principle yields the denominator identities for the $\Gamma$-graded Lie algebras, from which we derive a dimension formula for the homogeneous subspaces $L_{\alpha }$ $(\alpha \in \Gamma )$. Our dimension formula enables us to study the structure of the $\Gamma$-graded Lie algebras in a unified way. We will discuss some interesting applications of our dimension formula to the various classes of graded Lie algebras such as free Lie algebras, Kac-Moody algebras, and generalized Kac-Moody algebras. We will also discuss the relation of graded Lie algebras and the product identities for formal power series.

References

G. M. Benkart, D. J. Britten, and F. W. Lemire, Stability in modules for classical Lie algebras—a constructive approach, Mem. Amer. Math. Soc. 85 (1990), no. 430, vi+165. MR 1010997, DOI 10.1090/memo/0430
Georgia Benkart, Seok-Jin Kang, and Kailash C. Misra, Graded Lie algebras of Kac-Moody type, Adv. Math. 97 (1993), no. 2, 154–190. MR 1201842, DOI 10.1006/aima.1993.1005
Georgia Benkart, Seok-Jin Kang, and Kailash C. Misra, Indefinite Kac-Moody algebras of classical type, Adv. Math. 105 (1994), no. 1, 76–110. MR 1275194, DOI 10.1006/aima.1994.1040
Jie Du, $q$-Schur algebras, asymptotic forms, and quantum $\textrm {SL}_n$, J. Algebra 177 (1995), no. 2, 385–408. MR 1355207, DOI 10.1006/jabr.1995.1304
S. Berman and R. V. Moody, Lie algebra multiplicities, Proc. Amer. Math. Soc. 76 (1979), no. 2, 223–228. MR 537078, DOI 10.1090/S0002-9939-1979-0537078-X
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, The monster Lie algebra, Adv. Math. 83 (1990), no. 1, 30–47. MR 1069386, DOI 10.1016/0001-8708(90)90067-W
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
Richard E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), no. 2, 405–444. MR 1172696, DOI 10.1007/BF01232032
Richard E. Borcherds, A characterization of generalized Kac-Moody algebras, J. Algebra 174 (1995), no. 3, 1073–1079. MR 1337185, DOI 10.1006/jabr.1995.1167
Richard E. Borcherds, Automorphic forms on $\textrm {O}_{s+2,2}(\textbf {R})$ and infinite products, Invent. Math. 120 (1995), no. 1, 161–213. MR 1323986, DOI 10.1007/BF01241126
Borcherds, R. E., Automorphic forms and Lie algebras, in Current Developments in Mathematics, International Press (1996), 1–27.
Bourbaki, N., Lie Groups and Lie Algebras, Part 1, Hermann, Paris, 1975.
J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308–339. MR 554399, DOI 10.1112/blms/11.3.308
Alex Jay Feingold, A hyperbolic GCM Lie algebra and the Fibonacci numbers, Proc. Amer. Math. Soc. 80 (1980), no. 3, 379–385. MR 580988, DOI 10.1090/S0002-9939-1980-0580988-6
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
Fischer, B., Livingstone, D., Thorne, M. P., The characters of the “Monster” simple group, Birmingham, 1978.
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
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
Howard Garland and James Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976), no. 1, 37–76. MR 414645, DOI 10.1007/BF01418970
Reinhold W. Gebert, Introduction to vertex algebras, Borcherds algebras and the Monster Lie algebra, Internat. J. Modern Phys. A 8 (1993), no. 31, 5441–5503. MR 1248070, DOI 10.1142/S0217751X93002162
Reinhold W. Gebert and Jörg Teschner, On the fundamental representation of Borcherds algebras with one imaginary simple root, Lett. Math. Phys. 31 (1994), no. 4, 327–334. MR 1293497, DOI 10.1007/BF00762796
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR 1303592, DOI 10.1090/surv/040.1
Harada, K., Miyamoto, M., Yamada, H., A generalization of Kac-Moody Lie algebras, J. Algebra 180 (1996), 631–651.
Jin Hong, Kyeonghoon Jeong, and Jae-Hoon Kwon, Integral points on hyperbolas, J. Korean Math. Soc. 34 (1997), no. 1, 149–157. MR 1443345
Loo Keng Hua, Introduction to number theory, Springer-Verlag, Berlin-New York, 1982. Translated from the Chinese by Peter Shiu. MR 665428
Nathan Jacobson, Lie algebras, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR 559927
Jurisich, E., Generalized Kac-Moody Lie algebras, free Lie algebras, and the structure of the monster Lie algebra, J. Pure and Applied Algebra 126 (1998), 233–266.
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
Jurisich, E., Wilson, R. L., A generalization of Lazard’s theorem, preprint.
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
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
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Victor G. Kac and Seok-Jin Kang, Trace formula for graded Lie algebras and Monstrous Moonshine, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 141–154. MR 1357198
V. G. Kac, R. V. Moody, and M. Wakimoto, On $E_{10}$, Differential geometrical methods in theoretical physics (Como, 1987) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 250, Kluwer Acad. Publ., Dordrecht, 1988, pp. 109–128. MR 981374
Kang, S.-J., Gradations and structure of Kac-Moody Lie algebras, Yale University Ph. D. dissertation (1990).
Seok-Jin Kang, Kac-Moody Lie algebras, spectral sequences, and the Witt formula, Trans. Amer. Math. Soc. 339 (1993), no. 2, 463–493. MR 1102889, DOI 10.1090/S0002-9947-1993-1102889-0
Seok-Jin Kang, Root multiplicities of the hyperbolic Kac-Moody Lie algebra $HA^{(1)}_1$, J. Algebra 160 (1993), no. 2, 492–523. MR 1244925, DOI 10.1006/jabr.1993.1198
Stephen Slebarski, On the characters of affine Kac-Moody groups, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 2, 179–195. MR 1221043, DOI 10.1017/S0013091500018320
Seok-Jin Kang, Root multiplicities of Kac-Moody algebras, Duke Math. J. 74 (1994), no. 3, 635–666. MR 1277948, DOI 10.1215/S0012-7094-94-07423-1
Seok-Jin Kang, Generalized Kac-Moody algebras and the modular function $j$, Math. Ann. 298 (1994), no. 2, 373–384. MR 1256622, DOI 10.1007/BF01459740
Seok-Jin Kang, Root multiplicities of graded Lie algebras, Lie algebras and their representations (Seoul, 1995) Contemp. Math., vol. 194, Amer. Math. Soc., Providence, RI, 1996, pp. 161–176. MR 1395598, DOI 10.1090/conm/194/02392
Kang, S.-J., Graded Lie superalgebras and the superdimension formula, J. Algebra 204 (1998), 597–655.
Seok-Jin Kang and Myung-Hwan Kim, Free Lie algebras, generalized Witt formula, and the denominator identity, J. Algebra 183 (1996), no. 2, 560–594. MR 1399040, DOI 10.1006/jabr.1996.0233
Kang, S.-J., Kim, M.-H., Borcherds superalgebras and a Monstrous Lie superalgebra, Math. Ann. 307 (1997), 677–694.
Seok-Jin Kang, Masaki Kashiwara, Kailash C. Misra, Tetsuji Miwa, Toshiki Nakashima, and Atsushi Nakayashiki, Affine crystals and vertex models, Infinite analysis, Part A, B (Kyoto, 1991) Adv. Ser. Math. Phys., vol. 16, World Sci. Publ., River Edge, NJ, 1992, pp. 449–484. MR 1187560, DOI 10.1142/s0217751x92003896
Seok-Jin Kang and Duncan J. Melville, Root multiplicities of the Kac-Moody algebras $HA^{(1)}_n$, J. Algebra 170 (1994), no. 1, 277–299. MR 1302841, DOI 10.1006/jabr.1994.1338
Seok-Jin Kang and Duncan J. Melville, Rank $2$ symmetric hyperbolic Kac-Moody algebras, Nagoya Math. J. 140 (1995), 41–75. MR 1369479, DOI 10.1017/S0027763000005419
Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 142696, DOI 10.2307/1970237
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
Bong H. Lian and Gregg J. Zuckerman, Moonshine cohomology, Sūrikaisekikenkyūsho K\B{o}kyūroku 904 (1995), 87–115. Moonshine and vertex operator algebra (Japanese) (Kyoto, 1994). MR 1363974
Li Shi Liu, Kostant’s formula for Kac-Moody Lie algebras, J. Algebra 149 (1992), no. 1, 155–178. MR 1165205, DOI 10.1016/0021-8693(92)90010-J
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, Root systems of hyperbolic type, Adv. in Math. 33 (1979), no. 2, 144–160. MR 544847, DOI 10.1016/S0001-8708(79)80003-1
Satoshi Naito, The strong Bernstein-Gel′fand-Gel′fand resolution for generalized Kac-Moody algebras. I. The existence of the resolution, Publ. Res. Inst. Math. Sci. 29 (1993), no. 4, 709–730. MR 1245445, DOI 10.2977/prims/1195166746
Peterson, D. H., Freudenthal-type formulas for root and weight multiplicities, preprint (1983).
Jean-Pierre Serre, Lie algebras and Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1965. Lectures given at Harvard University, 1964. MR 0218496
J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216
N. Sthanumoorthy and A. Uma Maheswari, Purely imaginary roots of Kac-Moody algebras, Comm. Algebra 24 (1996), no. 2, 677–693. MR 1373498, DOI 10.1080/00927879608825591
N. Sthanumoorthy and A. Uma Maheswari, Root multiplicities of extended-hyperbolic Kac-Moody algebras, Comm. Algebra 24 (1996), no. 14, 4495–4512. MR 1421202, DOI 10.1080/00927879608825828
J. G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979), no. 3, 352–353. MR 554402, DOI 10.1112/blms/11.3.352