Narrow Results

For M a finite dimensional complex vector space and A a certain type of (unital) subalgebra of End(M) (including some specific types of physical significance in the field of quantum spin chains) we give an algorithm for constructing the centraliser or commutant B of A on M. We give examples, and discuss the conditions for centralising to be an involution, i.e. A, B a dual pair, and for B and A to be Morita equivalent. A special case of one example shows that H_n(q), U_q(sl₂) act as a dual pair on the tensored vector representation for all q.

We construct a dual pair associated to the Hamiltonian geometric formulation of perfect fluids with free boundaries. This dual pair is defined on the cotangent bundle of the space of volume preserving embeddings of a manifold with boundary into a boundaryless manifold of the same dimension. The dual pair properties are rigorously verified in the infinite-dimensional Fréchet manifold setting. It provides an example of a dual pair associated to actions that are not completely mutually orthogonal.

Combining the twisted Jacobi structure [Twisted Jacobi manifolds, twisted Dirac–Jacobi structures and quasi-Jacobi bialgebroids, J. Phys. A: Math. Gen.39(33) (2006) 10449–10475] with that of a Poisson structure with a 3-form background [Poisson geometry with a 3-form background, Prog. Theor. Phys. Suppl.144 (2001) 145–154], alias twisted Poisson, we propose and analyze a new structure, called Jacobi structure with background. The background is a pair consisting of a $3$-form and a $2$-form. We describe their characteristic leaves. For twisted contact dual pairs, we show the correspondence of characteristic leaves of the two Jacobi manifolds with background.

We get several identities of differential operators in determinantal form. These identities are non-commutative versions of the formula of Cauchy-Binet or Laplace expansions of determinants, and if we take principal symbols, they are reduced to such classical formulas. These identities are naturally arising from the generators of the rings of invariant differential operators over symmetric spaces, and have strong resemblance to the classical Capelli identities. Thus we call those identities the Capelli identities for symmetric pairs.