International Journal of Mathematics and Mathematical Sciences

Research Article

Adjoint and Trivial Cohomologies of Nilpotent Complex Lie Algebras of Dimension

Table 2

Pairs of 7-dimensional NLAs having same adjoint cohomology.
#PairAdjoint cohomology
1 𝔤 7 , 3 . 1 ( 𝑖 𝜆 ) 1 𝜆 = 0 , 1 , 𝜆 = − 1 , 2 , 2 , 𝜆 ∈ { − 𝜔 , − 𝜔 2 }
2 ( 𝔤 7 , 0 . 4 ( 𝜆 ) , 𝔤 7 , 0 . 5 ) ( 1 , 4 , 9 , 1 5 , 1 6 , 1 2 , 7 , 2 )
3 ( 𝔤 7 , 1 . 1 ( 𝑖 𝜆 ) ( 𝜆 = − 𝜔 ) , 𝔤 7 , 1 . 1 ( 𝑖 𝜆 ) ( 𝜆 = − 𝜔 2 ) ) ( 1 , 4 , 9 , 1 4 , 1 6 , 1 2 , 6 , 2 )
4 ( 𝔤 7 , 1 . 2 ( 𝑖 𝜆 ) ( g e n e r i c ) , 𝔤 7 , 1 . 2 ( 𝑖 𝑖 ) ) ( 1 , 6 , 1 6 , 2 7 , 3 0 , 2 3 , 1 2 , 3 )
5 ( 𝔤 7 , 1 . 3 ( 𝑖 𝜆 ) ( g e n e r i c ) , 𝔤 7 , 1 . 3 ( 𝑖 𝑖 𝑖 ) ) ( 1 , 7 , 1 9 , 3 0 , 3 2 , 2 3 , 1 1 , 3 )
6 ( 𝔤 7 , 1 . 2 0 , 𝔤 7 , 1 . 2 1 ) ( 1 , 5 , 1 2 , 1 7 , 1 7 , 1 5 , 1 0 , 3 )
7 ( 𝔤 7 , 2 . 1 ( 𝑖 𝜆 ) ( g e n e r i c ) , 𝔤 7 , 1 . 3 ( 𝑖 𝑖 ) ) ( 1 , 8 , 2 0 , 3 1 , 3 3 , 2 4 , 1 2 , 3 )
8 ( 𝔤 7 , 2 . 1 ( 𝑖 𝜆 ) ( 𝜆 = 0 ) , 𝔤 7 , 2 . 1 1 ) ( 2 , 9 , 2 1 , 3 3 , 3 4 , 2 4 , 1 2 , 3 )
9 ( 𝔤 7 , 2 . 1 ( 𝑖 𝜆 ) ( 𝜆 = − 𝜔 ) , 𝔤 7 , 2 . 1 ( 𝑖 𝜆 ) ( 𝜆 = − 𝜔 2 ) ) ( 1 , 8 , 2 0 , 3 1 , 3 4 , 2 5 , 1 2 , 3 )
10 ( 𝔤 7 , 2 . 2 6 , 𝔤 7 , 3 . 4 ) ( 2 , 8 , 1 9 , 3 2 , 3 6 , 2 6 , 1 2 , 3 )
11 ( 𝔤 7 , 2 . 3 1 , 𝔤 7 , 2 . 4 1 ) ( 1 , 7 , 1 8 , 2 9 , 3 2 , 2 4 , 1 2 , 3 )
12 ( 𝔤 7 , 2 . 3 4 , 𝔤 7 , 2 . 3 5 ) ( 2 , 7 , 1 8 , 2 9 , 3 3 , 2 5 , 1 1 , 3 )
13 ( 𝔤 7 , 2 . 3 6 , 𝔤 6 , 1 2 × ℂ ) ( 2 , 1 1 , 2 9 , 4 7 , 5 1 , 3 8 , 1 8 , 4 )