Geodesic
Twisted de Rham Complex on Line and Singular Vectors in $\widehat{{\mathfrak{sl}_2}}$ Verma Modules
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider two complexes. The first complex is the twisted de Rham complex of scalar meromorphic differential forms on projective line, holomorphic on the complement to a finite set of points. The second complex is the chain complex of the Lie algebra of $\mathfrak{sl}_2$-valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra $\widehat{{\mathfrak{sl}_2}}$. In [Schechtman V., Varchenko A., Mosc. Math. J. 17 (2017), 787–802] a construction of a monomorphism of the first complex to the second was suggested and it was indicated that under this monomorphism the existence of singular vectors in the Verma modules (the Malikov–Feigin–Fuchs singular vectors) is reflected in the relations between the cohomology classes of the de Rham complex. In this paper we prove these results.
Keywords: twisted de Rham complex, logarithmic differential forms
Mots-clés : $\widehat{{\mathfrak{sl}_2}}$-modules, Lie algebra chain complexes. 
@article{SIGMA_2019_15_a74,
     author = {Alexey Slinkin and Alexander Varchenko},
     title = {Twisted de {Rham} {Complex} on {Line} and {Singular} {Vectors} in $\widehat{{\mathfrak{sl}_2}}$ {Verma} {Modules}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2019},
     volume = {15},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a74/}
}
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Alexey Slinkin; Alexander Varchenko. Twisted de Rham Complex on Line and Singular Vectors in $\widehat{{\mathfrak{sl}_2}}$ Verma Modules. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a74/

[1] Feigin B., Schechtman V., Varchenko A., “On algebraic equations satisfied by hypergeometric correlators in WZW models. I”, Comm. Math. Phys., 163 (1994), 173–184 | DOI | MR | Zbl

[2] Feigin B., Schechtman V., Varchenko A., “On algebraic equations satisfied by hypergeometric correlators in WZW models. II”, Comm. Math. Phys., 170 (1995), 219–247 | DOI | MR | Zbl

[3] Kac V. G., Kazhdan D. A., “Structure of representations with highest weight of infinite-dimensional Lie algebras”, Adv. Math., 34 (1979), 97–108 | DOI | MR | Zbl

[4] Khoroshkin S., Schechtman V., “Factorizable ${\mathcal D}$-modules”, Math. Res. Lett., 4 (1997), 239–257, arXiv: q-alg/9611018 | DOI | MR | Zbl

[5] Khoroshkin S., Varchenko A., “Quiver $\mathcal D$-modules and homology of local systems over an arrangement of hyperplanes”, IInt. Math. Res. Pap., 2006 (2006), 69590, 116 pp., arXiv: math.QA/0510451 | DOI | MR | Zbl

[6] Malikov F., Feigin B., Fuks D., “Singular vectors in Verma modules over Kac–Moody algebras”, Funct. Anal. Appl., 20 (1986), 103–113 | DOI | MR | Zbl

[7] Schechtman V., Terao H., Varchenko A., “Local systems over complements of hyperplanes and the Kac–Kazhdan conditions for singular vectors”, J. Pure Appl. Algebra, 100 (1995), 93–102, arXiv: hep-th/9411083 | DOI | MR | Zbl

[8] Schechtman V., Varchenko A., “Arrangements of hyperplanes and Lie algebra homology”, Invent. Math., 106 (1991), 139–194 | DOI | MR | Zbl

[9] Schechtman V., Varchenko A., “Rational differential forms on the line and singular vectors in Verma modules over $\widehat {\mathfrak{sl}}_2$”, Mosc. Math. J., 17 (2017), 787–802, arXiv: 1511.09014 | DOI | MR | Zbl