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 Mosc. Math. J., 2014, Volume 14, Number 1, Pages 83–119 (Mi mmj516)

The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for $\widehat{\mathfrak{gl}}_{m|n}$

A. I. Moleva, E. Ragoucyb

a School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
b LAPTH, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux cedex, France

Abstract: We prove an analogue of the MacMahon Master Theorem for the right quantum superalgebras. In particular, we obtain a new and simple proof of this theorem for the right quantum algebras. In the super case the theorem is then used to construct higher order Sugawara operators for the affine Lie superalgebra $\widehat{\mathfrak{gl}}_{m|n}$ in an explicit form. The operators are elements of a completed universal enveloping algebra of $\widehat{\mathfrak{gl}}_{m|n}$ at the critical level. They occur as the coefficients in the expansion of a noncommutative Berezinian and as the traces of powers of generator matrices. The same construction yields higher Hamiltonians for the Gaudin model associated with the Lie superalgebra $\mathfrak{gl}_{m|n}$. We also use the Sugawara operators to produce algebraically independent generators of the algebra of singular vectors of any generic Verma module at the critical level over the affine Lie superalgebra.

Key words and phrases: MacMahon Master Theorem, Manin matrix, Newton theorem, noncommutative Berezinian, Sugawara operators, higher Gaudin Hamiltonians, singular vectors, Verma modules.

DOI: https://doi.org/10.17323/1609-4514-2014-14-1-83-119

Full text: http://www.mathjournals.org/.../2014-014-001-005.html
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Bibliographic databases:

MSC: 17A70, 17B67, 17B69, 17B80
Received: June 29, 2010; in revised form June 13, 2013
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Citation: A. I. Molev, E. Ragoucy, “The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for $\widehat{\mathfrak{gl}}_{m|n}$”, Mosc. Math. J., 14:1 (2014), 83–119

Citation in format AMSBIB
\Bibitem{MolRag14}
\by A.~I.~Molev, E.~Ragoucy
\paper The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for $\widehat{\mathfrak{gl}}_{m|n}$
\jour Mosc. Math.~J.
\yr 2014
\vol 14
\issue 1
\pages 83--119
\mathnet{http://mi.mathnet.ru/mmj516}
\crossref{https://doi.org/10.17323/1609-4514-2014-14-1-83-119}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3221948}

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• http://mi.mathnet.ru/eng/mmj/v14/i1/p83

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. Chervov, G. Falqui, V. Rubtsov, A. Silantyev, “Algebraic properties of Manin matrices. II: $q$-analogues and integrable systems”, Adv. Appl. Math., 60 (2014), 25–89
2. A. I. Molev, E. E. Mukhin, “Invariants of the vacuum module associated with the Lie superalgebra $\mathfrak{gl}(1|1)$”, J. Phys. A, 48:31 (2015), 314001, 20 pp.
3. E. Mukhin, B. Vicedo, Ch. Young, “Gaudin models for $\mathfrak{gl}(m|n)$”, J. Math. Phys., 56:5 (2015), 051704, 30 pp.
4. V. Futorny, A. Molev, “Quantization of the shift of argument subalgebras in type $A$”, Adv. Math., 285 (2015), 1358–1375
5. L. Frappat, N. Jing, A. Molev, E. Ragoucy, “Higher Sugawara operators for the quantum affine algebras of type A”, Comm. Math. Phys., 345:2 (2016), 631–657
6. A. I. Molev, E. Ragoucy, N. Rozhkovskaya, “Segal–Sugawara vectors for the Lie algebra of type $G_2$”, J. Algebra, 455 (2016), 386–401
7. A. I. Molev, E. E. Mukhin, “Eigenvalues of Bethe vectors in the Gaudin model”, Theoret. and Math. Phys., 192:3 (2017), 1258–1281
8. D. Adamovic, “A note on the affine vertex algebra associated to $\mathfrak{gl}(1|1)$ at the critical level and its generalizations”, Rad Hrvat. Akad. Znan. Umjet., 21:532 (2017), 75–87
9. C. A. S. Young, B. Vicedo, “$({\mathfrak{gl}}_M, {\mathfrak{gl}}_N)$-Dualities in Gaudin Models with Irregular Singularities”, SIGMA, 14 (2018), 040, 28 pp.
10. N. Jing, S. Kozic, A. Molev, F. Yang, “Center of the quantum affine vertex algebra in type $A$”, J. Algebra, 496 (2018), 138–186