All real 3-dimensional Lie algebras

The following list was taken from the paper "Invariants of real low dimension Lie algebras" [Patera et al.]. Algebraic sums of lower dimensional Lie algebras are omitted

AppendTo[ad3, Table2ad[({{0, e_1, -2e_2}, {0, 0, e_3}, {0, 0, 0}})]] ;

AppendTo[ad3, Table2ad[({{0, e_3, -e_2}, {0, 0, e_1}, {0, 0, 0}})]] ;

The following list summarizes the commutator tables. The basis is e_1,e_2,e_3. The coefficient a ranges over special intervals. First 0<a<1, secondly 0<a.

ShowAd/@ad3

The killing form reveals, which of the algebras are solvable, and which are semi-simple.

ShowMat[adKilling/@ad3]

According to a Theorem, κ([g,g],g)=0 is equivalent to solvability. The first 7 algebras are solvable. We confirm for instance

 = NS[NS[Join @@ T[ad = ad3[[7]], {3, 1, 2}]]] ;

E0[ . adKilling[ad]]

{True}

The characteristic polynomials with roots of the 7 algebras are

{χ = adCharacter[#1, x, λ], Solve[E0[χ], λ]} &/@ad3//MF

The chapter on Lie groups, we derive Lie group actions that induce all algebras listed here. _3 is an exception.

Created by Mathematica  (September 30, 2006) Valid XHTML 1.1!