International Journal of Mathematics and Mathematical Sciences

Research Article

Adjoint and Trivial Cohomologies of Nilpotent Complex Lie Algebras of Dimension

Table 6

Cohomology table for NLAs of dimension 7: rank 3.
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbers
𝔤 6 , 1 9 × ℂ generic(1,15,59,130,159,116,45,7)(1,9,25,42,44,30,14,3)(1,3,7,9,9,7,3,1)
𝔤 6 , 2 0 × ℂ ( 𝔤 7 , 3 . 1 ( 𝑖 𝜆 ) ,15,59,130, 𝔤 7 , 3 . 1 ( 𝑖 𝜆 ) 𝜆 = 0 , 1 ,116,45,7)( 2 , 1 6 1 ,25,42, 2 ,14,3)(1,3,7, 1 0 ,7,3,1)
4 6 , 3 2 (1, 1 1 , 1 1 ,59, 𝔤 7 , 3 . 1 ( 𝑖 𝜆 ) 1 𝜆 = − 1 , 2 , 2 ,159, 1 7 ,45,7)(1, 1 3 1 ,3)(1,3, 1 1 7 , 1 1 , 2 7 , 4 3 , 4 5 , 3 1 , 1 5 , 8 ,3,1)
1 0 , 1 0 (1,15,59,130, 8 ,116,45,7)(1,9,25,42, 𝔤 7 , 3 . 1 ( 𝑖 𝜆 ) 𝜆 ∈ { − 𝜔 , − 𝜔 2 } ,14,3)(1,3,7, 1 6 0 ,7,3,1)
4 5 , 3 1 (1,15,63,130,161,119,46,7)(1,9,29,46,46,35,18,4)(1,4,8,9,9,8,4,1)
1 0 , 1 0 (2,17,62,130,158,115,45,7)(2,12,30,45,43,28,13,3)(1,3,7,10,10,7,3,1)
𝔤 7 , 3 . 1 ( 𝑖 𝑖 𝑖 ) (2,15,56,126,156,115,45,7)(2,10,22,35,37,26,13,3)(1,3,7,9,9,7,3,1)
𝔤 7 , 3 . 2 (2,13,55,124,157,114,45,7)(2,8,19,32,36,26,12,3)(1,3,6,10,10,6,3,1)
𝔤 7 , 3 . 3 (2,14,56,125,157,114,45,7)(2,9,21,34,37,26,12,3)(1,3,6,10,10,6,3,1)
𝔤 7 , 3 . 4 (3,18,64,134,163,117,45,7)(3,14,33,51,52,35,15,3)(1,3,8,14,14,8,3,1)
𝔤 7 , 3 . 5 (2,15,61,131,164,119,46,7)(2,10,27,45,50,38,18,4)(1,4,8,11,11,8,4,1)
𝔤 7 , 3 . 6 (2,19,68,139,169,120,46,7)(2,14,38,60,63,44,19,4)(1,4,9,14,14,9,4,1)
𝔤 7 , 3 . 7 (2,18,65,137,168,120,46,7)(2,13,34,55,60,43,19,4)(1,4,9,13,13,9,4,1)
𝔤 7 , 3 . 8 (1,15,58,124,157,118,46,7)(1,9,24,35,36,30,17,4)(1,4,7,8,8,7,4,1)
𝔤 7 , 3 . 9 (2,18,65,136,166,119,46,7)(2,13,34,54,57,40,18,4)(1,4,8,13,13,8,4,1)
𝔤 7 , 3 . 1 0 (3,19,66,147,172,122,46,7)(3,15,36,66,74,49,21,4)(1,4,11,14,14,11,4,1)
𝔤 7 , 3 . 1 1 (2,16,59,127,159,117,46,7)(2,11,26,39,41,31,16,4)(1,4,6,9,9,6,4,1)
𝔤 7 , 3 . 1 2 (2,18,65,135,166,120,46,7)(2,13,34,53,56,41,19,4)(1,4,9,12,12,9,4,1)
𝔤 7 , 3 . 1 3 (2,17,63,133,164,119,46,7)(2,12,31,49,52,38,18,4)(1,4,8,11,11,8,4,1)
𝔤 7 , 3 . 1 4 (1,14,58,123,157,118,46,7)(1,8,23,34,35,30,17,4)(1,4,7,8,8,7,4,1)
𝔤 7 , 3 . 1 5 (1,16,56,127,157,117,46,7)(1,10,23,36,39,29,16,4)(1,4,6,9,9,6,4,1)
𝔤 7 , 3 . 1 6 (1,19,70,135,170,125,47,7)(1,13,40,58,60,50,25,5)(1,5,10,11,11,10,5,1)
𝔤 7 , 3 . 1 7 (2,19,70,148,171,124,47,7)(2,14,40,71,74,50,24,5)(1,5,9,15,15,9,5,1)
𝔤 7 , 3 . 1 8 (2,19,66,133,163,115,45,7)(2,14,36,52,51,33,13,3)(1,3,7,13,13,7,3,1)
𝔤 7 , 3 . 1 9 (2,15,58,127,159,114,45,7)(2,10,24,38,41,28,12,3)(1,3,6,10,10,6,3,1)
𝔤 7 , 3 . 2 0 (2,15,57,124,157,114,45,7)(2,10,23,34,36,26,12,3)(1,3,6,10,10,6,3,1)
𝔤 7 , 3 . 2 1 (3,17,63,132,159,116,45,7)(3,13,31,48,46,30,14,3)(1,3,7,11,11,7,3,1)
𝔤 7 , 3 . 2 2 (3,22,75,150,175,122,46,7)(3,18,48,78,80,52,21,4)(1,4,11,17,17,11,4,1)
𝔤 7 , 3 . 2 3 (3,18,65,135,165,119,46,7)(3,14,34,53,55,39,18,4)(1,4,8,11,11,8,4,1)
𝔤 7 , 3 . 2 4 (3,20,70,142,170,120,46,7)(3,16,41,65,67,45,19,4)(1,4,9,14,14,9,4,1)
𝔤 6 , 5 × ℂ (3,19,66,135,165,119,46,7)(3,15,36,54,55,39,18,4)(1,4,8,11,11,8,4,1)
𝔤 6 , 7 × ℂ (2,17,64,134,165,119,46,7)(2,12,32,51,54,39,18,4)(1,4,8,11,11,8,4,1)
𝔤 6 , 8 × ℂ (2,17,63,132,164,119,46,7)(2,12,31,48,51,38,18,4)(1,4,8,11,11,8,4,1)
𝔤 6 , 1 0 × ℂ (2,15,58,125,159,118,46,7)(2,10,24,36,39,32,17,4)(1,4,7,8,8,7,4,1)
𝔤 6 , 1 1 × ℂ (3,16,58,126,158,115,45,7)(3,12,25,37,39,28,13,3)(1,3,6,10,10,6,3,1)
𝔤 6 , 1 3 × ℂ (2,14,56,124,157,115,45,7)(2,9,21,33,36,27,13,3)(1,3,6,10,10,6,3,1)
𝔤 6 , 1 4 × ℂ (2,15,57,123,154,113,45,7)(2,10,23,33,32,22,11,3)(1,3,5,7,7,5,3,1)
𝔤 6 , 1 5 × ℂ (2,13,51,115,149,112,45,7)(2,8,15,19,19,16,10,3)(1,3,4,4,4,4,3,1)
𝔤 6 , 1 6 × ℂ (3,18,63,132,164,119,46,7)(3,14,32,48,51,38,18,4)(1,4,8,11,11,8,4,1)