Geodesic
$W$-algebras and higher analogues of the Knizhnik–Zamolodchikov equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 182 (2015) no. 3, pp. 355-372 Cet article a éte moissonné depuis la source Math-Net.Ru

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The key role in the derivation of the Knizhnik–Zamolodchikov equations in the Wess–Zumino–Witten model is played by the energy–momentum tensor, which is constructed from a second-order Casimir element in the universal enveloping algebra of the corresponding Lie algebra. We investigate the possibility of constructing analogues of Knizhnik–Zamolodchikov equations using higher-order central elements. We consider the Casimir element of the third order for the Lie algebra $\mathfrak{sl}_N$ and of the fourth order for $\mathfrak{o}_N$. The construction is impossible in the first case, but we succeed in obtaining the sought equation in the second case.
Mots-clés : Casimir element
Keywords: $W$-algebra, Kniznik–Zamolodchikov equation, commutative Pfaffian. 
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D. V. Artamonov; V. A. Golubeva. $W$-algebras and higher analogues of the Knizhnik–Zamolodchikov equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 182 (2015) no. 3, pp. 355-372. http://geodesic.mathdoc.fr/item/TMF_2015_182_3_a0/

[1] P. Di Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory, Springer, New York, 1997 | DOI | MR | Zbl

[2] F. A. Bais, P. Bouwknegt, M. Surridge, K. Schoutens, Nucl. Phys. B, 304 (1988), 348–370 | DOI | MR | Zbl

[3] P. Bouwknegt, K. Schoutens, Phys. Rep., 223:4 (1993), 183–276, arXiv: hep-th/9210010 | DOI | MR

[4] A. I. Molev, M. N. Nazarov, Math. Ann., 313:2 (1999), 315–357 | DOI | MR | Zbl

[5] M. Itoh, T. Umeda, Compositio Math., 127:3 (2001), 333–359 | DOI | MR | Zbl

[6] M. Itoh, J. Lie Theory, 10:2 (2000), 463–489 | MR | Zbl

[7] D. V. Artamonov, V. A. Golubeva, Matem. sb., 203:12 (2012), 5–34 | DOI | DOI | MR | Zbl

[8] V. G. Knizhnik, A. B. Zamolodchikov, Nucl. Phys. B., 247:1 (1984), 83–103 | DOI | MR | Zbl

[9] J. A. Fuchs, Affine Lie algebras and Quantum Groups. An Introduction with Application in Conformal Field Theory, Cambridge Univ. Press, Cambridge, 1995 | MR

[10] P. Etingof, I. B. Frenkel, A. A. Kirillov, Jr., Lectures on Representation Theory and Knizhnik–Zamolodchikov Equations, Mathematical Surveys and Monographs, 58, AMS, Providence, RI, 1998 | DOI | MR

[11] V. G. Drinfeld, Algebra i analiz, 2:4 (1990), 149–181 | MR | Zbl

[12] B. L. Feigin, E. Frenkel, N. Reshetikhin, Commun. Math. Phys., 166:1 (1994), 27–62, arXiv: hep-th/9402022 | DOI | MR | Zbl

[13] G. M. T. Watts, “$W$-algebras and their representations”, Conformal Field Theories and Integrable Models (Budapest, Hungary, August 13–18, 1996), Lecture Notes in Physics, 498, eds. Z. Horvath, L. Palla, Springer, Berlin, 1997, 55–84 | DOI | MR | Zbl

[14] S. Ribault, JHEP, 10 (2009), 002, 29 pp., arXiv: 0811.4587 | DOI | MR

[15] A. Chervov, L. Rybnikov, D. Talalaev, Rational Lax operators and their quantization, arXiv: hep-th/0404106

[16] D. V. Artamonov, V. A. Golubeva, W-algebras and higher analogs of the Knizhnik–Zamolodchikov equations, arXiv: 1401.4960