# 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 𝔤𝔩̂_{m|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

**Summary:**

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 𝔤𝔩̂_{m|n} in an explicit form. The
operators are elements of a completed universal enveloping algebra of
𝔤𝔩̂_{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 𝔤𝔩̂_{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.

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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