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  Are there examples of non-semisimple Lie algebras occurring in Physics?

+ 1 like - 0 dislike

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. =]

asked Aug 2, 2020 in Mathematics by Dry Bones [ no revision ]
recategorized Aug 4, 2020 by Dilaton

The Lie algebra of the Poincare group is not semisimple but plays a very important role in relativistic physics. 

@ArnoldNeumaier Indeed, it's the semi-direct product of a semisimple Lie algebra (corresponding to Lorentz transformations) by an Abelian one (corresponding to translations in Minkowsky spacetime). I'd be glad to know if there are analogous examples, maybe in condensed matter?, of semi-direct products involving other groups, e.g. the compact ones that appear in QFT. But still, any examples of nilpotent or solvable Lie algebras occurring in Physics would be of special interest for me. Thanks for the comment!

The Euclidean group of rigid moments in $R^d$, a semidirect product of the translation group and $SO(d)$, and its Lie algebra are also important in physics, though less often in explicit form. For $d=2$ it is solvable.

1 Answer

+ 2 like - 0 dislike

The Heisenberg algebra

$[x,p]=z, [z,x]=[z,p]=0$

is nilpotent. It appears in quantum mechanics.

answered Aug 9, 2020 by alex S Arvanitakis [ no revision ]

So is its generalization to higher-dimensional $x$ and $p$.

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