Geodesic
$W$-algebras and higher analogues of the~Knizhnik--Zamolodchikov equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 182 (2015) no. 3, pp. 355-372

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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. 
@article{TMF_2015_182_3_a0,
     author = {D. V. Artamonov and V. A. Golubeva},
     title = {$W$-algebras and higher analogues of {the~Knizhnik--Zamolodchikov} equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {355--372},
     publisher = {mathdoc},
     volume = {182},
     number = {3},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2015_182_3_a0/}
}
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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/