Finite-dimensional Lie algebras have a long history going back to Lie, Killing, Elie Cartan, ... I will start with a short survey of their theory. Then I will pass to the main subject of my talk, namely those infinite-dimensional positively graded graded Lie algebras that come up in
Commutative ring theory.
For a long time it seemed impossible to say something in these three cases if you were inspired by the finite-dimensional case. But this has now changed and I will give explicit examples illustrationg this. The phenomena that I will describe have in some cases been found and inspired by computer calculations using a programme liedim which has been written by Clas Löfwall. My main emphasis will be case 1) above, but the precolloquium talk 14.15-15.00 by Alexander Berglund will put more emphasis on the other cases.