Crystal bases, dilogarithm identities and torsion in algebraic $K$-theory
HTML articles powered by AMS MathViewer
- by Edward Frenkel and András Szenes
- J. Amer. Math. Soc. 8 (1995), 629-664
- DOI: https://doi.org/10.1090/S0894-0347-1995-1266736-4
- PDF | Request permission
References
- George E. Andrews, R. J. Baxter, and P. J. Forrester, Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Statist. Phys. 35 (1984), no. 3-4, 193–266. MR 748075, DOI 10.1007/BF01014383
- A. A. Beĭlinson, Higher regulators and values of $L$-functions of curves, Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 46–47 (Russian). MR 575206 S. Bloch, Higher regulators, algebraic $K$-theory and values of zeta-functions of elliptic curves, Irvine Lecture Notes, 1978.
- Spencer Bloch, The dilogarithm and extensions of Lie algebras, Algebraic $K$-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980) Lecture Notes in Math., vol. 854, Springer, Berlin, 1981, pp. 1–23. MR 618298, DOI 10.1007/BFb0089515
- Robert F. Coleman, Dilogarithms, regulators and $p$-adic $L$-functions, Invent. Math. 69 (1982), no. 2, 171–208. MR 674400, DOI 10.1007/BF01399500
- E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, One-dimensional configuration sums in vertex models and affine Lie algebra characters, Lett. Math. Phys. 17 (1989), no. 1, 69–77. MR 990586, DOI 10.1007/BF00420017
- Chih Han Sah, Scissors congruences. I. The Gauss-Bonnet map, Math. Scand. 49 (1981), no. 2, 181–210 (1982). MR 661890, DOI 10.7146/math.scand.a-11930
- W. Nahm, A. Recknagel, and M. Terhoeven, Dilogarithm identities in conformal field theory, Modern Phys. Lett. A 8 (1993), no. 19, 1835–1847. MR 1228370, DOI 10.1142/S0217732393001562
- Edward Frenkel and András Szenes, Dilogarithm identities, $q$-difference equations, and the Virasoro algebra, Internat. Math. Res. Notices 2 (1993), 53–60. MR 1203254, DOI 10.1155/S1073792893000054
- A. B. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995), no. 2, 197–318. MR 1348706, DOI 10.1006/aima.1995.1045
- Suresh Govindachar, Explicit weight two motivic cohomology complexes and algebraic $K$-theory, $K$-Theory 6 (1992), no. 5, 387–430. MR 1194842, DOI 10.1007/BF00961337
- Michio Jimbo, Kailash C. Misra, Tetsuji Miwa, and Masato Okado, Combinatorics of representations of $U_q(\widehat {{\mathfrak {s}}{\mathfrak {l}}}(n))$ at $q=0$, Comm. Math. Phys. 136 (1991), no. 3, 543–566. MR 1099695
- 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 —, Infinite-dimensional Lie algebras, 3rd ed., Cambridge Univ. Press, London and New York, 1990.
- 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, Masaki Kashiwara, Kailash C. Misra, Tetsuji Miwa, Toshiki Nakashima, and Atsushi Nakayashiki, Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68 (1992), no. 3, 499–607. MR 1194953, DOI 10.1215/S0012-7094-92-06821-9
- M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
- A. N. Kirillov and N. Yu. Reshetikhin, Exact solution of the $XXZ$ Heisenberg model of spin $S$, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 145 (1985), no. Voprosy Kvant. Teor. Polya i Statist. Fiz. 5, 109–133, 191, 195 (Russian). MR 857965
- A. N. Kirillov, Identities for the Rogers dilogarithmic function connected with simple Lie algebras, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 164 (1987), no. Differentsial′naya Geom. Gruppy Li i Mekh. IX, 121–133, 198 (Russian, with English summary); English transl., J. Soviet Math. 47 (1989), no. 2, 2450–2459. MR 947332, DOI 10.1007/BF01840426 —, Spectra in conformal field theories and dilogarithm identities I, Preprint, December 1992. R. Lee and R. H. Szcarba, The group ${K_3}(Z)$ is cyclic of order 48, Ann. of Math. (2) 104 (1976), 31-60.
- Marc Levine, The indecomposable $K_3$ of fields, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 2, 255–344. MR 1005161, DOI 10.24033/asens.1585
- Leonard Lewin, Polylogarithms and associated functions, North-Holland Publishing Co., New York-Amsterdam, 1981. With a foreword by A. J. Van der Poorten. MR 618278
- Stephen Lichtenbaum, Groups related to scissors-congruence groups, Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 151–157. MR 991980, DOI 10.1090/conm/083/991980
- G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
- A. S. Merkur′ev and A. A. Suslin, The group $K_3$ for a field, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 3, 522–545 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 3, 541–565. MR 1072694
- W. Nahm, A. Recknagel, and M. Terhoeven, Dilogarithm identities in conformal field theory, Modern Phys. Lett. A 8 (1993), no. 19, 1835–1847. MR 1228370, DOI 10.1142/S0217732393001562
- Walter Parry and Chih-Han Sah, Third homology of $\textrm {SL}(2,\,\textbf {R})$ made discrete, J. Pure Appl. Algebra 30 (1983), no. 2, 181–209. MR 722372, DOI 10.1016/0022-4049(83)90054-3
- Dinakar Ramakrishnan, Regulators, algebraic cycles, and values of $L$-functions, Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 183–310. MR 991982, DOI 10.1090/conm/083/991982
- Bruce Richmond and George Szekeres, Some formulas related to dilogarithms, the zeta function and the Andrews-Gordon identities, J. Austral. Math. Soc. Ser. A 31 (1981), no. 3, 362–373. MR 633444, DOI 10.1017/S1446788700019492
- Chih-Han Sah, Homology of classical Lie groups made discrete. III, J. Pure Appl. Algebra 56 (1989), no. 3, 269–312. MR 982639, DOI 10.1016/0022-4049(89)90061-3 A. A. Suslin, ${K_3}$ of a field and the Bloch group, Proc. Steklov Inst. Math. 4 (1991), 217-239.
- Charles A. Weibel, Mennicke-type symbols for relative $K_{2}$, Algebraic $K$-theory, number theory, geometry and analysis (Bielefeld, 1982) Lecture Notes in Math., vol. 1046, Springer, Berlin, 1984, pp. 451–464. MR 750695, DOI 10.1007/BFb0072036
- Don Zagier, Polylogarithms, Dedekind zeta functions and the algebraic $K$-theory of fields, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 391–430. MR 1085270, DOI 10.1007/978-1-4612-0457-2_{1}9
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: J. Amer. Math. Soc. 8 (1995), 629-664
- MSC: Primary 17B67; Secondary 11G99, 11R70, 17B10, 19F27, 33B10
- DOI: https://doi.org/10.1090/S0894-0347-1995-1266736-4
- MathSciNet review: 1266736