Capelli Identities for Lie Superalgebras

The Capelli identity [1] is one of the best exploited results of the classical invariant theory. It provides a set of distinguished generators C1, . . . ,CN for the centre of the enveloping algebra U(glN ) of the general linear Lie algebra. For any non-negative integer M consider the natural action of the Lie algebra glN in the vector space CN⊗CM . Extend it to the action of the algebra U(glN ) in the space of polynomial functions on C ⊗ C . The resulting representation of U(glN ) by differential operators on C ⊗ C with polynomial coefficients is faithful when M > N . The image of the centre Z(glN ) of the algebra U(glN ) under this representation coincides with the ring J of glN×glM -invariant differential operators on CN⊗CM with polynomial coefficients. The latter ring has a distinguished set of generators I1, . . . ,IL with L = min (M,N) which are called the Cayley operators [2 ]. If xib with i = 1, . . . ,N and b = 1, . . . ,M are the standard coordinates on C ⊗ C and ∂ib are the corresponding partial derivations, then In equals