REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...
% test file for the Lie package
% 1. n-dimensional Lie algebras with dimL1=1
% n=6
array lienstrucin(6,6,6)$
lienstrucin(1,2,2):=lienstrucin(1,2,6):=lienstrucin(1,5,2):=-1$
lienstrucin(1,5,6):=lienstrucin(2,5,3):=lienstrucin(2,5,5):=-1$
lienstrucin(1,2,3):=lienstrucin(1,2,5):=lienstrucin(1,5,3):=1$
lienstrucin(1,5,5):=lienstrucin(2,5,2):=lienstrucin(2,5,6):=1$
liendimcom1(6);
{lie_algebra(2),commutative(4)}
% transformation matrix
lientrans;
[0 -1 1 0 1 -1]
[ ]
[0 1 0 0 0 0 ]
[ ]
[1 1 -1 0 -1 1 ]
[ ]
[0 0 0 1 0 0 ]
[ ]
[0 0 -1 0 0 1 ]
[ ]
[0 0 0 0 0 1 ]
clear lienstrucin$
% n=8
array lienstrucin(8,8,8)$
lienstrucin(1,2,2):=lienstrucin(1,5,2):=lienstrucin(2,4,3):=1$
lienstrucin(2,4,5):=lienstrucin(4,5,2):=1$
lienstrucin(1,2,3):=lienstrucin(1,2,5):=lienstrucin(1,5,3):=-1$
lienstrucin(1,5,5):=lienstrucin(2,4,2):=lienstrucin(4,5,3):=-1$
lienstrucin(4,5,5):=-1$
lienstrucin(1,2,6):=lienstrucin(1,5,6):=lienstrucin(4,5,6):=5$
lienstrucin(2,4,6):=-5$
liendimcom1(8);
{heisenberg(3),commutative(5)}
% same with verbose output
on tr_lie$
liendimcom1(8);
Your Lie algebra is the direct sum of the Lie algebra H(3)
and the 5-dimensional commutative Lie algebra, where
H(3) is 3-dimensional and there exists a basis
{X(1),...,X(3)} in H(3) with:
[X(2),X(3)]=[X(2*i),X(2*i+1)]=...=[X(2),X(3)]=X(1)
The transformation into this form is:
X(1):=5*y(6) - y(5) - y(3) + y(2)
X(2):=y(1)
X(3):=y(2)
X(4):=y(4) - y(1)
X(5):=y(5) - y(2)
X(6):=y(6)
X(7):=y(7)
X(8):=y(8)
{heisenberg(3),commutative(5)}
clear lienstrucin$
off tr_lie$
% 2. 4-dimensional Lie algebras
% Korteweg-de Vries Equation: u_t+u_{xxx}+uu_x=0
% symmetry algebra spanned by four vector fields:
% v_1=d_x, v_2=d_t, v_3=td_x+d_u, v_4=xd_x+3td_t-2ud_u
array liestrin(4,4,4)$
liestrin(1,4,1):=liestrin(2,3,1):=1$
liestrin(2,4,2):=3$
liestrin(3,4,3):=-2$
lieclass(4);
{liealg(4),comtab(16),5}
clear liestrin$
% dimL1=3, dimL2=3
array liestrin(4,4,4)$
liestrin(1,2,1):=-6$
liestrin(1,2,3):=-2$
liestrin(1,2,4):=6$
liestrin(1,3,1):=-1$
liestrin(1,3,2):=1$
liestrin(1,3,4):=1$
liestrin(2,3,1):=-3$
liestrin(2,3,4):=2$
liestrin(2,4,1):=6$
liestrin(2,4,3):=2$
liestrin(2,4,4):=-6$
liestrin(3,4,1):=1$
liestrin(3,4,2):=-1$
liestrin(3,4,4):=-1$
lieclass(4);
{liealg(4),comtab(21)}
% same with verbose output
on tr_lie$
lieclass(4);
[W,X]=Y, [W,Y]=-X, [X,Y]=W
{liealg(4),comtab(21)}
% transformation matrix
liemat;
[ 3 0 1 -3 ]
[ ]
[ - 3 2 ]
[--------- 0 0 ---------]
[ sqrt(2) sqrt(2) ]
[ ]
[ - 1 1 1 ]
[--------- --------- 0 ---------]
[ sqrt(2) sqrt(2) sqrt(2) ]
[ ]
[ -2 0 0 2 ]
clear liestrin$
off tr_lie$
end$
(TIME: lie 8010 8359)