Cohomology of a flag variety as a Bethe algebra. (English. Russian original) Zbl 1271.14072
Funct. Anal. Appl. 45, No. 4, 252-264 (2011); translation from Funkts. Anal. Prilozh. 45, No. 4, 16-31 (2011).
Summary: We interpret the equivariant cohomology \(H_{GL_n}^\ast(\mathcal F_{\boldsymbol\lambda},\mathbb C)\) of a partial flag variety \(\mathcal F_{\boldsymbol\lambda}\) parametrizing chains of subspaces \(0=F_0\subset F_1\subset\cdots\subset F_N=\mathbb C^n\), \(\dim F_i/F_{i-1}=\lambda_i\), as the Bethe algebra \(\mathcal B^\infty(\mathcal V^\pm_{\boldsymbol\lambda})\) of the \(\mathfrak{gl}_N\)-weight subspace \(\mathcal V^\pm_{\boldsymbol\lambda}\) of a \(\mathfrak{gl}_N[t]\)-module \(\mathcal V^\pm\).
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 81R99 | Groups and algebras in quantum theory |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 32C36 | Local cohomology of analytic spaces |
References:
| [1] | M. F. Atiyah and R. Bott, ”The moment map and equivariant cohomology,” Topology, 23:1 (1984), 1–28. · Zbl 0521.58025 · doi:10.1016/0040-9383(84)90021-1 |
| [2] | A. Braverman, D. Maulik, and A. Okounkov, Quantum cohomology of the Springer resolution, http://arxiv.org/abs/1001.0056 . · Zbl 1226.14069 |
| [3] | V. Chari and S. Loktev, ”Weyl, Demazure and fusion modules for the current algebra of sl r+1,” Adv. Math., 207:2 (2006), 928–960. · Zbl 1161.17318 · doi:10.1016/j.aim.2006.01.012 |
| [4] | V. Chari and A. Pressley, ”Weyl modules for classical and quantum affine algebras,” Represent. Theory (electronic), 5 (2001), 191–223. · Zbl 0989.17019 · doi:10.1090/S1088-4165-01-00115-7 |
| [5] | A. Chervov and D. Talalaev, Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, http://arxiv.org/abs/hep-th/0604128 . · Zbl 1179.82052 |
| [6] | M. Gaudin, ”Diagonalisation d’une classe d’Hamiltoniens de spin,” J. Physique, 37:10 (1976), 1089–1098. |
| [7] | M. Gaudin, La fonction d’onde de Bethe, Collection du Commissariat à l’Énergie Atomique: Série Scientifique, Masson, Paris, 1983. |
| [8] | E. Mukhin, V. Tarasov, and A. Varchenko, ”Bethe eigenvectors of higher transfer matrices,” J. Stat. Mech. Theory (electronic), No. 8 (2006), P08002. |
| [9] | E. Mukhin, V. Tarasov, and A. Varchenko, ”Schubert calculus and representations of general linear group,” J. Amer. Math. Soc., 22:4 (2009), 909–940. · Zbl 1205.17026 · doi:10.1090/S0894-0347-09-00640-7 |
| [10] | E. Mukhin, V. Tarasov, and A. Varchenko, ”Spaces of quasi-exponentials and representations of glN,” J. Phys. A, 41:19, 194017 (2008). · Zbl 1142.81011 · doi:10.1088/1751-8113/41/19/194017 |
| [11] | E. Mukhin and A. Varchenko, ”Critical points of master functions and flag varieties,” Commun. Contemp. Math., 6:1 (2004), 111–163. · Zbl 1050.17022 · doi:10.1142/S0219199704001288 |
| [12] | A. Okounkov, ”Quantum groups and quantum cohomology,” in: Lectures at the 15th Midrasha Mathematicae on Derived Categories of Algebro-Geometric Origin and Integrable Systems, December 19–24, Jerusalem, 2010. |
| [13] | R. Rimányi and A. Varchenko, Conformal blocks in the tensor product of vector representations and localization formulas, http://arxiv.org/abs/0911.3253 . · Zbl 1228.81250 |
| [14] | R. Rimányi, V. Schechtman, and A. Varchenko, Conformal blocks and equivariant cohomology, http://arxiv.org/abs/1007.3155 . · Zbl 1259.81080 |
| [15] | D. Talalaev, Quantization of the Gaudin system, http://arxiv.org/abs/hep-th/0404153 . · Zbl 1111.82015 |
| [16] | A. Varchenko, ”A Selberg integral type formula for an sl2 one-dimensional space of conformal blocks,” Mosc. Math. J., 10:2 (2010), 469–475. · Zbl 1222.81259 |
| [17] | E. Vasserot, ”Représentations de groupes quantiques et permutations,” Ann. Sci. École Norm. Sup., 26:6 (1993), 747–773. · Zbl 0822.17005 |
| [18] | E. Vasserot, ”Affine quantum groups and equivariant K-theory,” Transformation Groups, 3:3 (1998), 269–299. · Zbl 0969.17009 · doi:10.1007/BF01236876 |
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