A4,12

The Lie-algebra A_ (4, 12) is defined via the adjoint map (1,2)-tensor ad.

A12 = ad = Table2ad[2 Τ[({{0, 0, e_1, -e_2}, {0, 0, e_2, e_1}, {0, 0, 0, 0}, {0, 0, 0, 0}})]] ; [adJacobi[ad]] ;

The automorphism group of the Lie-algebra is generated by the following matrices:

M = Algebra[4, Τdα[ad, 1, #1] &, ] ;

ShowMat[EXP[λ #1] &/@M]

We reduce the scalar product: Due to the Lie-algebra automorphism α, we may assume b_2=0.

α = ( {{Cos[λ], -Sin[λ], 0, 0}, {Sin[λ], Cos[λ], 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}} ) ;

Τα[B, 0, α]//MF

Investigation: b_1≠0

Investigation: b_1=0


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