More from the Lie-algebraic realm:
simple Lie algebra: of course, the ubiquitous sl(2)
solvable Lie algebra: < x,y | [x,y]=x >
nilpotent Lie algebra: < x,y,z | [x,y]=z, [x,z]=[y,z]=0 >
simple Lie algebra in characteristic p: < e_i, i\in Z/pZ | [e_i,e_j]=(j-i)e_{i+j} > (Witt algebra)
semisimple Lie algebra in characteristic p: sl(2)\otimes K[t]/(t^p) + Kd/dt
Kac-Moody algebra: sl(2)\otimes C[t,t^{-1}] + Ctd/dt + Cz
associative commutative algebra: K[t]/(t^n)
algebraic group: SL_2(K)
an algebraic object given by generators and relations: the corresponding 2-generated free object (e.g., 2-generated free group, 2-generated free Lie algebra, etc.)
