Squared white noise and other non-Gaussian noises as Levy processes on real Lie algebras

It is shown how the relations of the renormalized squared white noise defined by Accardi, Lu, and Volovich \cite{accardi+lu+volovich99} can be realized as factorizable current representations or L\'evy processes on the real Lie algebra $\eufrak{sl}_2$. This allows to obtain its It\^o table, which turns out to be infinite-dimensional. The linear white noise without or with number operator is shown to be a L\'evy process on the Heisenberg-Weyl Lie algebra or the oscillator Lie algebra. Furthermore, a joint realization of the linear and quadratic white noise relations is constructed, but it is proved that no such realizations exist with a vacuum that is an eigenvector of the central element and the annihilator. Classical L\'evy processes are shown to arise as components of L\'evy process on real Lie algebras and their distributions are characterized.