We propose a new realization of the elliptic quantum group equipped with the H-Hopf algebroid structure on the basis of the elliptic algebra Uq,p(ŝl2). The algebra Uq,p(ŝl2) has a constructive definition in terms of the Drinfeld generators of the quantum affine algebra Uq(ŝl2) and a Heisenberg algebra. This yields a systematic construction of both finite and infinite-dimensional dynamical repre...

In this article the full set of irreducible unitary holomorphic representations of U (p, q) is determined. For a general Hermitian symmetric space G/K of the noncompact type the set of irreducible unitary representations of G on scalar valued holomorphic functions has been found by Wallach [14] and by Rossi and Vergne [11]. For the groups Mp(n, ~) and SU(n, n) unitary irreducible representation...

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at q = 1. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for adjoint ⊗ adjoint → adjoint. We present a practical method for the determination of these quantum...

The theory of apolarity was first developed by Clebsch, Lasker, Richw x mond, Sylvester, and Wakeford 10, 17, 20 . They were first interested in studying homogeneous polynomials of degree p and in q variables, and in expressing them as sums of pth powers of linear terms. The problem is to minimize the number of pth powers which are required in such a sum. For instance, a result due to Sylvester...