docu5_lie-groups.nb

All real 3-dimensional Lie groups

The following list was taken from the paper "Invariants of real low dimension Lie algebras" [Patera et al.].

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

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

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

The first 7 Lie algebras in the list are semi-direct products determined by a certain derivation δ. The groups are all diffeomorphic to . We list the actions G×G→G in coordinates.

$#1 {x , x , x }◦{ y , y , y } = {x_1 + y_1 - x_3 y_2, x_2 + y_2, x_3 + y_3} 1 2 3 1 2 3$

$#2 {x , x , x }◦{ y , y , y } = {x_1 + ^(-x_3) (y_1 - x_3 y_2), x_2 + ^(-x_3) y_2, x_3 + y_3} 1 2 3 1 2 3$

$#3 {x , x , x }◦{ y , y , y } = {x_1 + ^(-x_3) y_1, x_2 + ^(-x_3) y_2, x_3 + y_3} 1 2 3 1 2 3$

$#4 {x , x , x }◦{ y , y , y } = {x_1 + ^(-x_3) y_1, x_2 + ^x_3 y_2, x_3 + y_3} 1 2 3 1 2 3$

$#5 {x , x , x }◦{ y , y , y } = {x_1 + ^(-x_3) y_1, x_2 + ^(-a x_3) y_2, x_3 + y_3} 1 2 3 1 2 3$

$#6 {x , x , x }◦{ y , y , y } = {x_1 + Cos[x_3] y_1 - Sin[x_3] y_2, x_2 + Sin[x_3] y_1 + Cos[x_3] y_2, x_3 + y_3} 1 2 3 1 2 3$

The computer determines the group action by exponentiating δ. The derivation δ is part of the commutator table.

$ShowMat[EXP[#1[[{1, 2}, {1, 2}, 3]] x_3] &/@ad3[[Range[7]]]]$

Lets show all figures that come with Lie algebra #7.

$jnf ( {{- x_3 + a x_3, 0}, {0,  x_3 + a x_3}} )$

$- exp ( {{^(a x_3) Cos[x_3], ^(a x_3) Sin[x_3]}, {-^(a x_3) Sin[x_3], ^(a x_3) Cos[x_3]}} )$

${{x_1 + y_1, x_2 + y_2, x_3 + y_3}, {0, 0, 0}, {-x_1, -x_2, -x_3}}$

$act {x_1 + ^(-a x_3) (Cos[x_3] y_1 - Sin[x_3] y_2), x_2 + ^(-a x_3) (Sin[x_3] y_1 + Cos[x_3] y_2), x_3 + y_3}$

$inv {-^(a x_3) (Cos[x_3] x_1 + Sin[x_3] x_2), ^(a x_3) (Sin[x_3] x_1 - Cos[x_3] x_2), -x_3}$


$( {{^(-a x_3) Cos[x_3], -^(-a x_3) Sin[x_3], 0}, {^(-a x_3) Sin[x_3], ^(-a x_3) Cos[x_3], 0}, {0, 0, 1}} )$	$( {{^(-a x_3) Cos[x_3], -^(-a x_3) Sin[x_3], a x_1 + x_2}, {^(-a x_3) Sin[x_3], ^(-a x_3) Cos[x_3], -x_1 + a x_2}, {0, 0, 1}} )$

The last two algebras are simple. Lie algebra #8 is R and is treated in the introduction. Lie algebra #9 is R and treated in a separate section. So far, we have ommitted the 3-dimensional abelian group. Lets state a non trivial action ×→ as follows