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

This article is cited in 10 scientific papers (total in 10 papers)

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.


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MSC: 17A70, 17B67, 17B69, 17B80
Received: June 29, 2010; in revised form June 13, 2013

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
\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

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    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  crossref  zmath  isi  elib  scopus
    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.  crossref  mathscinet  zmath  isi  elib  scopus
    3. E. Mukhin, B. Vicedo, Ch. Young, “Gaudin models for $\mathfrak{gl}(m|n)$”, J. Math. Phys., 56:5 (2015), 051704, 30 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    4. V. Futorny, A. Molev, “Quantization of the shift of argument subalgebras in type $A$”, Adv. Math., 285 (2015), 1358–1375  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    7. A. I. Molev, E. E. Mukhin, “Eigenvalues of Bethe vectors in the Gaudin model”, Theoret. and Math. Phys., 192:3 (2017), 1258–1281  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    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.  mathnet  crossref
    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  crossref  mathscinet  zmath  isi  scopus
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