The n-Dimensional Quadratic Heisenberg Algebra as a “Non–Commutative” sl(2,C)

We prove that the commutation relations among the generators of the quadratic Heisenberg algebra of dimension $n\in\mathbb{N}$, look like a kind of non-commutative extension of sl(2,C) (more precisely of its unique 1– dimensional central extension), denoted $heis_{2;C}(n)$ and called the complex n–dimensional quadratic Boson algebra. This non-commutativity has a dif- ferent nature from the one considered in quantum groups. We prove the exponentiability of these algebras (for any n) in the Fock representation. We obtain the group multiplication law, in coordinates of the first and second kind, for the quadratic Boson group and we show that, in the case of the adjoint representation, these multiplication laws can be expressed in terms of a generalization of the Jordan multiplication. We investigate the connections between these two types of coordinates (disentangling formulas). From this we deduce a new proof of the expression of the vacuum characteristic function of homogeneous quadratic boson fields.