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

A. I. Molev and E. Ragoucy

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 ${\mathfrak{gl}}_{m|n}$