International Journal of Mathematics and Mathematical Sciences

Research Article

Adjoint and Trivial Cohomologies of Nilpotent Complex Lie Algebras of Dimension

Table 5

Cohomology table for NLAs of dimension 7: rank 2. (continues on next page).
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbers
𝔤 7 , 1 . 2 0 generic(1,14,55,123,155,114,45,7)(1,8,20,31,33,24,12,3)(1,3,6,8,8,6,3,1)
𝔤 7 , 1 . 2 1 ( 𝔤 7 , 2 . 1 ( 𝑖 𝜆 ) ,14, 𝔤 7 , 2 . 1 ( 𝑖 𝜆 ) 𝜆 = 0 ,155,114,45,7)( 2 ,24,12,3)unchanged
5 6 , 1 2 4 (1, 2 , 9 , 2 1 , 3 3 , 3 4 ,55,123,155,114,45,7)(1, 𝔤 7 , 2 . 1 ( 𝑖 𝜆 ) 1 𝜆 = 2 ,31,33,24,12,3)unchanged
1 5 (1,14,55, 9 , 2 1 ,45,7)(1,8,20, 𝔤 7 , 2 . 1 ( 𝑖 𝜆 ) 𝜆 = 2 ,3)(1,3, 1 2 5 , 1 5 7 , 1 1 5 ,3,1)
3 3 , 3 7 , 2 7 , 1 3 (1,14,55,123, 7 , 9 , 9 , 7 ,114,45,7)(1,8,20,31, 𝔤 7 , 2 . 1 ( 𝑖 𝜆 ) 𝜆 ∈ { − 𝜔 , − 𝜔 2 } ,12,3)(1,3,6, 1 5 6 ,6,3,1)
3 4 , 2 5 (1,14,56,123,155,114,45,7)(1,8,21,32,33,24,12,3)(1,3,6,8,8,6,3,1)
9 , 9 (1,14,58,125,159,118,46,7)(1,8,23,36,39,32,17,4)(1,4,7,8,8,7,4,1)
𝔤 7 , 2 . 1 ( 𝑖 𝑖 ) (1,14,58,124,157,118,46,7)(1,8,23,35,36,30,17,4)(1,4,7,8,8,7,4,1)
𝔤 7 , 2 . 1 ( 𝑖 𝑖 𝑖 ) (2,14,57,124,157,114,45,7)(2,9,22,34,36,26,12,3)(1,3,6,10,10,6,3,1)
𝔤 7 , 2 . 1 ( 𝑖 𝑣 ) (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 7 , 2 . 1 ( 𝑣 ) (1,13,53,119,149,110,44,7)(1,7,17,25,23,14,7,2)(1,2,4,6,6,4,2,1)
𝔤 7 , 2 . 2 (2,12,49,115,147,110,44,7)(2,7,12,17,17,12,7,2)(1,2,3,4,4,3,2,1)
𝔤 7 , 2 . 3 (1,12,49,114,149,110,44,7)(1,6,12,16,18,14,7,2)(1,2,3,6,6,3,2,1)
𝔤 7 , 2 . 4 ‡ ■ (2,12,49,115,148,110,44,7)(2,7,12,17,18,13,7,2)(1,2,3,4,4,3,2,1)
𝔤 7 , 2 . 5 (2,13,52,120,150,111,44,7)(2,8,16,25,25,16,8,2)(1,2,4,7,7,4,2,1)
𝔤 7 , 2 . 6 (2,13,51,119,152,112,44,7)(2,8,15,23,26,19,9,2)(1,2,5,9,9,5,2,1)
𝔤 7 , 2 . 7 ‡ (2,12,50,118,151,112,44,7)(2,7,13,21,24,18,9,2)(1,2,5,8,8,5,2,1)
𝔤 7 , 2 . 8 (1,12,51,115,149,112,45,7)(1,6,14,19,19,16,10,3)(1,3,4,4,4,4,3,1)
𝔤 7 , 2 . 9 (2,14,56,124,155,114,45,7)(2,9,21,33,34,24,12,3)(1,3,6,8,8,6,3,1)
𝔤 7 , 2 . 1 0 (2,14,57,123,157,113,45,7)(2,9,22,33,35,25,11,3)(1,3,5,9,9,5,3,1)
𝔤 7 , 2 . 1 1 (1,12,50,114,148,112,45,7)(1,6,13,17,17,15,10,3)(1,3,4,4,4,4,3,1)
𝔤 7 , 2 . 1 2 (1,12,51,113,148,112,45,7)(1,6,14,17,16,15,10,3)(1,3,4,4,4,4,3,1)
𝔤 7 , 2 . 1 3 (1,13,54,119,152,113,45,7)(1,7,18,26,26,20,11,3)(1,3,5,7,7,5,3,1)
𝔤 7 , 2 . 1 4 (1,14,55,120,153,113,45,7)(1,8,20,28,28,21,11,3)(1,3,5,7,7,5,3,1)
𝔤 7 , 2 . 1 5 (2,13,53,122,154,114,45,7)(2,8,17,28,31,23,12,3)(1,3,6,7,7,6,3,1)
𝔤 7 , 2 . 1 6 (2,15,58,125,157,115,45,7)(2,10,24,36,37,27,13,3)(1,3,6,10,10,6,3,1)
𝔤 7 , 2 . 1 7 (2,15,58,125,156,114,45,7)(2,10,24,36,36,25,12,3)(1,3,6,9,9,6,3,1)
𝔤 7 , 2 . 1 8 (2,16,59,128,158,115,45,7)(2,11,26,40,41,28,13,3)(1,3,7,10,10,7,3,1)
𝔤 7 , 2 . 1 9 (2,16,60,129,158,115,45,7)(2,11,27,42,42,28,13,3)(1,3,7,10,10,7,3,1)
𝔤 7 , 2 . 2 0 (2,14,56,125,156,115,45,7)(2,9,21,34,36,26,13,3)(1,3,7,9,9,7,3,1)
𝔤 7 , 2 . 2 1 (1,13,58,123,156,118,46,7)(1,7,22,34,34,29,17,4)(1,4,7,7,7,7,4,1)
𝔤 7 , 2 . 2 2 (2,13,54,123,155,114,45,7)(2,8,18,30,33,24,12,3)(1,3,6,8,8,6,3,1)
𝔤 7 , 2 . 2 3 (1,14,56,123,155,117,46,7)(1,8,21,32,33,27,16,4)(1,4,6,7,7,6,4,1)
𝔤 7 , 2 . 2 4 (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 , 2 . 2 5 (2,17,65,132,164,119,46,7)(2,12,33,50,51,38,18,4)(1,4,8,11,11,8,4,1)
𝔤 7 , 2 . 2 6 (1,16,63,131,163,119,46,7)(1,10,30,47,49,37,18,4)(1,4,8,10,10,8,4,1)
𝔤 7 , 2 . 2 7 (2,14,59,126,159,117,46,7)(2,9,24,38,40,31,16,4)(1,4,6,9,9,6,4,1)
𝔤 7 , 2 . 2 8 (1,15,56,125,156,117,46,7)(1,9,22,34,36,28,16,4)(1,4,6,8,8,6,4,1)
𝔤 7 , 2 . 2 9 (1,13,54,122,155,114,45,7)(1,7,18,29,32,24,12,3)(1,3,6,8,8,6,3,1)
𝔤 7 , 2 . 3 0 (1,14,55,122,156,113,45,7)(1,8,20,30,33,24,11,3)(1,3,5,8,8,5,3,1)
𝔤 7 , 2 . 3 1 (1,12,53,120,153,113,45,7)(1,6,16,26,28,21,11,3)(1,3,5,7,7,5,3,1)
𝔤 7 , 2 . 3 2 (2,12,55,121,157,113,45,7)(2,7,18,29,33,25,11,3)(1,3,5,9,9,5,3,1)
𝔤 7 , 2 . 3 3 (2,12,55,121,157,113,45,7)(2,7,18,29,33,25,11,3)(1,3,5,9,9,5,3,1)
𝔤 7 , 2 . 3 4 (2,16,62,132,164,119,46,7)(2,11,29,47,51,38,18,4)(1,4,8,11,11,8,4,1)
𝔤 7 , 2 . 3 5 (1,13,53,121,154,114,45,7)(1,7,17,27,30,23,12,3)(1,3,6,7,7,6,3,1)
𝔤 7 , 2 . 3 6 (2,16,65,131,164,119,46,7)(2,11,32,49,50,38,18,4)(1,4,8,11,11,8,4,1)
𝔤 7 , 2 . 3 7 (2,17,63,131,163,115,45,7)(2,12,31,47,49,33,13,3)(1,3,7,13,13,7,3,1)
𝔤 7 , 2 . 3 8 (3,16,63,132,159,115,45,7)(3,12,30,48,46,29,13,3)(1,3,7,11,11,7,3,1)
𝔤 7 , 2 . 3 9 (1,13,54,122,155,114,45,7)(1,7,18,29,32,24,12,3)(1,3,6,8,8,6,3,1)
𝔤 7 , 2 . 4 0 (2,14,58,126,159,114,45,7)(2,9,23,37,40,28,12,3)(1,3,6,10,10,6,3,1)
𝔤 7 , 2 . 4 1 (2,16,60,129,159,115,45,7)(2,11,27,42,43,29,13,3)(1,3,7,11,11,7,3,1)
𝔤 7 , 2 . 4 2 (2,16,60,130,161,116,45,7)(2,11,27,43,46,32,14,3)(1,3,7,11,11,7,3,1)
𝔤 7 , 2 . 4 3 (2,17,65,134,165,119,46,7)(2,12,33,52,54,39,18,4)(1,4,8,12,12,8,4,1)
𝔤 7 , 2 . 4 4 (2,16,62,132,164,119,46,7)(2,11,29,47,51,38,18,4)(1,4,8,11,11,8,4,1)
𝔤 7 , 2 . 4 5 (2,14,55,122,154,113,45,7)(2,9,20,30,31,22,11,3)(1,3,5,7,7,5,3,1)
𝔤 6 , 1 2 × ℂ (2,13,53,120,153,113,45,7)(2,8,17,26,28,21,11,3)(1,3,5,7,7,5,3,1)
𝔤 6 , 1 7 × ℂ (2,12,50,115,149,112,45,7)(2,7,13,18,19,16,10,3)(1,3,4,4,4,4,3,1)