On the universal central extension of hyperelliptic current algebras

Cox, Ben

doi:10.1090/proc/13057

On the universal central extension of hyperelliptic current algebras
HTML articles powered by AMS MathViewer

by Ben Cox PDF

Proc. Amer. Math. Soc. 144 (2016), 2825-2835 Request permission

Abstract:

Let $p(t)\in \mathbb C[t]$ be a polynomial with distinct roots and nonzero constant term. We describe, using Faà de Bruno’s formula and Bell polynomials, the universal central extension in terms of generators and relations for the hyperelliptic current Lie algebras $\mathfrak g\otimes R$ whose coordinate ring is of the form $R=\mathbb C[t,t^{-1},u | u^2=p(t)]$.

References

L. F. A Arbogast, Du Calcul des Derivations, 1800.
Hidetoshi Awata, Akihiro Tsuchiya, and Yasuhiko Yamada, Integral formulas for the WZNW correlation functions, Nuclear Phys. B 365 (1991), no. 3, 680–696. MR 1136712, DOI 10.1016/0550-3213(91)90515-Y
André Bueno, Ben Cox, and Vyacheslav Futorny, Free field realizations of the elliptic affine Lie algebra $\mathfrak {sl}(2,\textbf {R})\oplus (\Omega _R/d\textrm {R})$, J. Geom. Phys. 59 (2009), no. 9, 1258–1270. MR 2541818, DOI 10.1016/j.geomphys.2009.06.007
E. T. Bell, Partition polynomials, Ann. of Math. (2) 29 (1927/28), no. 1-4, 38–46. MR 1502817, DOI 10.2307/1967979
Murray Bremner, Generalized affine Kac-Moody Lie algebras over localizations of the polynomial ring in one variable, Canad. Math. Bull. 37 (1994), no. 1, 21–28. MR 1261553, DOI 10.4153/CMB-1994-004-8
Murray Bremner, Universal central extensions of elliptic affine Lie algebras, J. Math. Phys. 35 (1994), no. 12, 6685–6692. MR 1303073, DOI 10.1063/1.530700
Murray Bremner, Four-point affine Lie algebras, Proc. Amer. Math. Soc. 123 (1995), no. 7, 1981–1989. MR 1249871, DOI 10.1090/S0002-9939-1995-1249871-8
Ben Cox and Vyacheslav Futorny, DJKM algebras I: their universal central extension, Proc. Amer. Math. Soc. 139 (2011), no. 10, 3451–3460. MR 2813377, DOI 10.1090/S0002-9939-2011-10906-7
Ben Cox, Vyacheslav Futorny, and Juan A. Tirao, DJKM algebras and non-classical orthogonal polynomials, J. Differential Equations 255 (2013), no. 9, 2846–2870. MR 3090080, DOI 10.1016/j.jde.2013.07.020
Ben Cox, Xiangqian Guo, Rencai Lu, and Kaiming Zhao, $n$-point Virasoro algebras and their modules of densities, Commun. Contemp. Math. 16 (2014), no. 3, 1350047, 27. MR 3211093, DOI 10.1142/S0219199713500478
Ben Cox, Xianquan Guo, Rencai Lu, and Kaiming Zhao, Simple superelliptic Lie algebras, Math arXiv:1412.7777, 2015.
Ben Cox and Elizabeth Jurisich, Realizations of the three-point Lie algebra $\mathfrak {sl}(2,{\scr R})\oplus (\Omega _{{\scr R}}/d{\scr R})$, Pacific J. Math. 270 (2014), no. 1, 27–47. MR 3245847, DOI 10.2140/pjm.2014.270.27
Ben Cox, Realizations of the four point affine Lie algebra $\mathfrak {sl}(2,R)\oplus (\Omega _R/dR)$, Pacific J. Math. 234 (2008), no. 2, 261–289. MR 2373448, DOI 10.2140/pjm.2008.234.261
M. Faà. de Bruno, Sur une nouvelle formule de calcul differentiel, The Quarterly Journal of Pure and Applied Mathematics, 1:359–360, 1857.
Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, Mathematical Surveys and Monographs, vol. 58, American Mathematical Society, Providence, RI, 1998. MR 1629472, DOI 10.1090/surv/058
Boris Feigin, Edward Frenkel, and Nikolai Reshetikhin, Gaudin model, Bethe ansatz and critical level, Comm. Math. Phys. 166 (1994), no. 1, 27–62. MR 1309540, DOI 10.1007/BF02099300
Edward Frenkel, Wakimoto modules, opers and the center at the critical level, Adv. Math. 195 (2005), no. 2, 297–404. MR 2146349, DOI 10.1016/j.aim.2004.08.002
Christian Kassel, Kähler differentials and coverings of complex simple Lie algebras extended over a commutative algebra, Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983), 1984, pp. 265–275. MR 772062, DOI 10.1016/0022-4049(84)90040-9
C. Kassel and J.-L. Loday, Extensions centrales d’algèbres de Lie, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 4, 119–142 (1983) (French, with English summary). MR 694130, DOI 10.5802/aif.896
Igor Moiseevich Krichever and S. P. Novikov, Algebras of Virasoro type, Riemann surfaces and strings in Minkowski space, Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 47–61, 96 (Russian). MR 925072
Igor Moiseevich Krichever and S. P. Novikov, Algebras of Virasoro type, Riemann surfaces and the structures of soliton theory, Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 46–63 (Russian). MR 902293
Igor Moiseevich Krichever and S. P. Novikov, Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces, Funktsional. Anal. i Prilozhen. 23 (1989), no. 1, 24–40 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 1, 19–33. MR 998426, DOI 10.1007/BF01078570
Gen Kuroki, Fock space representations of affine Lie algebras and integral representations in the Wess-Zumino-Witten models, Comm. Math. Phys. 142 (1991), no. 3, 511–542. MR 1138049, DOI 10.1007/BF02099099
Martin Schlichenmaier, Krichever-Novikov type algebras, De Gruyter Studies in Mathematics, vol. 53, De Gruyter, Berlin, 2014. Theory and applications. MR 3237422, DOI 10.1515/9783110279641
Oleg K. Sheinman, Current algebras on Riemann surfaces, De Gruyter Expositions in Mathematics, vol. 58, Walter de Gruyter GmbH & Co. KG, Berlin, 2012. New results and applications. MR 2985911, DOI 10.1515/9783110264524
V. V. Schechtman and A. N. Varchenko, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990), no. 4, 279–283. MR 1077959, DOI 10.1007/BF00626523