Dear physicist colleagues, I'm a mathematician and I'd like to ask for some help on finding occurrences of certain mathematical objects in the Physics literature. I'm not sure if this question fits here, but if it doesn't, I beg you pardon and ask some moderator to promptly remove it.

I'm interested in non-semisimple Lie algebras occurring in Physics. As is well known, Lie groups such as ${\rm SU}(3)$, ${\rm SU}(2)$, ${\rm U}(1)$, and the corresponding Lie algebras, play an important role in Quantum Field Theory and related subjects. All these happen to be semisimple Lie groups, as are their Lie algebras. To put it simply, all the ${\rm SL}(n)$, ${\rm SO}(n)$, ${\rm Sp}(n)$, ${\rm SU}(n)$... and also the so-called 'exceptional' ones, for example, $E_8$, which appears in this paper, are all *semisimple* Lie groups.

I'd like to know *if* and specially *where* other types of Lie algebras, for instance the *nilpotent* or the *solvable* ones, occur in Physics. The word 'where' assumes here a double meaning: on the one hand, in what particular phenomena and branches of Physics; and on the other, if possible, in what papers, and under what name. Probably, if one never heard of such occurrences, a formal definition of these objects won't be of much use, but anyway, you can take a look, for example, here and here if you wish.

The reason why I'm looking for these occurrences is to find some relations between these types of Lie algebras and other mathematical objects that might appear in the same physical contexts as them, as well as to get in contact with the often striking intuition developed by physicists when dealing with mathematical objects in general.

Hope I don't bother too much! Thanks in advance. =]