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Moscow Mathematical Journal

Volume 14, Issue 1, January–March 2014  pp. 83–119.

The MacMahon Master Theorem for Right Quantum Superalgebras and Higher Sugawara Operators for 𝔤𝔩̂m|n

Authors A. I. Molev (1) and E. Ragoucy (2)
Author institution: (1) School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
(2) LAPTH, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux cedex, France


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 𝔤𝔩̂m|n in an explicit form. The operators are elements of a completed universal enveloping algebra of 𝔤𝔩̂m|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 𝔤𝔩̂m|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.

2010 Mathematics Subject Classification. 17A70, 17B67, 17B69, 17B80.

Keywords:  MacMahon master theorem, Manin matrix, Newton theorem, noncommutative Berezinian, Sugawara operators, higher Gaudin Hamiltonians, singular vectors, Verma modules.

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