International Journal of Mathematics and Mathematical Sciences

Research Article

Adjoint and Trivial Cohomologies of Nilpotent Complex Lie Algebras of Dimension

Table 4

Cohomology table for NLAs of dimension 7: rank 1.
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbers
𝔤 7 , 0 . 7 (1,11,51,113,148,112,45,7)(1,5,13,17,16,15,10,3)(1,3,4,4,4,4,3,1)
𝔤 7 , 0 . 8 (1,12,53,118,152,113,45,7)(1,6,16,24,25,20,11,3)(1,3,5,7,7,5,3,1)
𝔤 7 , 1 . 0 1 ( 𝑖 ) (1,11,49,114,148,110,44,7)(1,5,11,16,17,13,7,2)(1,2,3,5,5,3,2,1)
𝔤 7 , 1 . 0 1 ( 𝑖 𝑖 ) (1,11,49,115,148,110,44,7)(1,5,11,17,18,13,7,2)(1,2,3,4,4,3,2,1)
𝔤 7 , 1 . 0 2 generic(1,10,48,113,147,109,44,7)(1,4,9,14,15,11,6,2)(1,2,3,4,4,3,2,1)
𝔤 7 , 1 . 0 3 (1, 𝔤 7 , 1 . 1 ( 𝑖 𝜆 ) , 𝔤 7 , 1 . 1 ( 𝑖 𝜆 ) 𝜆 = 1 ,113,147,109,44,7)(1, 1 1 ,15,11,6,2)unchanged
4 9 (1,10,48, 5 , 1 1 , 1 5 ,147, 𝔤 7 , 1 . 1 ( 𝑖 𝜆 ) 𝜆 = 3 ,44,7)(1,4,9, 1 1 4 , 12,7,2)(1,2, 1 1 0 ,2,1)
1 5 , 1 6 (1,10,48,113, 4 , 5 , 5 , 4 ,109,44,7)(1,4,9,14, 𝔤 7 , 1 . 1 ( 𝑖 𝜆 ) 𝜆 ∈ { − 𝜔 , − 𝜔 2 } ,6,2)(1,2,3, 1 4 8 ,3,2,1)
1 6 , 1 2 (1,11,48,114,147,109,44,7)(1,5,10,15,16,11,6,2)(1,2,3,4,4,3,2,1)
5 , 5 (2,10,48,113,147,109,44,7)(2,5,9,14,15,11,6,2)(1,2,3,4,4,3,2,1)
𝔤 7 , 1 . 1 ( 𝑖 𝑖 ) (1,11,48,114,149,110,44,7)(1,5,10,15,18,14,7,2)(1,2,3,6,6,3,2,1)
𝔤 7 , 1 . 1 ( 𝑖 𝑖 𝑖 ) (1,10,50,113,148,112,45,7)(1,4,11,16,16,15,10,3)(1,3,4,4,4,4,3,1)
𝔤 7 , 1 . 1 ( 𝑖 𝑣 ) (1,11,50,115,149,112,45,7)(1,5,12,18,19,16,10,3)(1,3,4,4,4,4,3,1)
𝔤 7 , 1 . 1 ( 𝑣 ) ( 𝔤 7 , 1 . 1 ( 𝑣 𝑖 ) ) generic(1,12,53,121,154,114,45,7)(1,6,16,27,30,23,12,3)(1,3,6,7,7,6,3,1)
𝔤 7 , 1 . 2 ( 𝑖 𝜆 ) (1,12,53, 𝜆 ≠ − 1 ,114,45,7)(1,6,16, 𝔤 7 , 1 . 2 ( 𝑖 𝜆 ) 𝜆 ∈ { 1 , − 𝜔 , − 𝜔 2 } ,12,3)(1,3,6, 1 2 2 , 1 5 5 ,6,3,1)
2 8 , 3 2 , 2 4 (1,12,53,121,154,114,45,7)(1,6,16,27,30,23,12,3)(1,3,6,7,7,6,3,1)
8 , 8 (2,12,53,122,154,114,45,7)(2,7,16,28,31,23,12,3)(1,3,6,7,7,6,3,1)
𝔤 7 , 1 . 2 ( 𝑖 𝑖 ) (1,12,53,122,154,114,45,7)(1,6,16,28,31,23,12,3)(1,3,6,7,7,6,3,1)
𝔤 7 , 1 . 2 ( 𝑖 𝑖 𝑖 ) generic(1,13,55,122,155,113,45,7)(1,7,19,30,32,23,11,3)(1,3,5,7,7,5,3,1)
𝔤 7 , 1 . 2 ( 𝑖 𝑣 ) (1,13,55,122, 𝔤 7 , 1 . 3 ( 𝑖 𝜆 ) ,113,45,7)(1,7,19,30, 𝔤 7 , 1 . 3 ( 𝑖 𝜆 ) 𝜆 = 1 ,11,3)(1,3,5, 1 5 6 ,5,3,1)
3 3 , 2 4 (1,14,55,123,155,114,45,7)(1,8,20,31,33,24,12,3)(1,3,6,8,8,6,3,1)
8 , 8 (1,13,55,122,155,113,45,7)(1,7,19,30,32,23,11,3)(1,3,5,7,7,5,3,1)
𝔤 7 , 1 . 3 ( 𝑖 𝑖 ) (2,13,57,123,157,113,45,7)(2,8,21,33,35,25,11,3)(1,3,5,9,9,5,3,1)
𝔤 7 , 1 . 3 ( 𝑖 𝑖 𝑖 ) (1,13,56,123,155,117,46,7)(1,7,20,32,33,27,16,4)(1,4,6,7,7,6,4,1)
𝔤 7 , 1 . 3 ( 𝑖 𝑣 ) (1,12,50,116,149,110,44,7)(1,6,13,19,20,14,7,2)(1,2,4,6,6,4,2,1)
𝔤 7 , 1 . 3 ( 𝑣 ) (2,11,49,115,147,109,44,7)(2,6,11,17,17,11,6,2)(1,2,3,4,4,3,2,1)
𝔤 7 , 1 . 4 (1,12,52,118,149,110,44,7)(1,6,15,23,22,14,7,2)(1,2,4,6,6,4,2,1)
𝔤 7 , 1 . 5 ‡ (2,15,59,128,159,115,45,7)(2,10,25,40,42,29,13,3)(1,3,7,11,11,7,3,1)
𝔤 7 , 1 . 6 (1,11,53,120,152,113,45,7)(1,5,15,26,27,20,11,3)(1,3,5,6,6,5,3,1)
𝔤 7 , 1 . 7 (2,14,58,124,156,114,45,7)(2,9,23,35,35,25,12,3)(1,3,6,9,9,6,3,1)
𝔤 7 , 1 . 8 (1,11,50,116,150,111,44,7)(1,5,12,19,21,16,8,2)(1,2,4,7,7,4,2,1)
𝔤 7 , 1 . 9 (1,11,52,117,151,113,45,7)(1,5,14,22,23,19,11,3)(1,3,5,6,6,5,3,1)
𝔤 7 , 1 . 1 0 (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 , 1 . 1 1 (2,12,51,117,150,111,44,7)(2,7,14,21,22,16,8,2)(1,2,4,7,7,4,2,1)
𝔤 7 , 1 . 1 2 (2,11,49,115,148,110,44,7)(2,6,11,17,18,13,7,2)(1,2,3,4,4,3,2,1)
𝔤 7 , 1 . 1 3 (1,13,54,120,153,113,45,7)(1,7,18,27,28,21,11,3)(1,3,5,7,7,5,3,1)
𝔤 7 , 1 . 1 4 (2,15,59,128,158,115,45,7)(2,10,25,40,41,28,13,3)(1,3,7,10,10,7,3,1)
𝔤 7 , 1 . 1 5 (1,11,48,114,147,110,44,7)(1,5,10,15,16,12,7,2)(1,2,3,4,4,3,2,1)
𝔤 7 , 1 . 1 6 (2,13,56,123,155,114,45,7)(2,8,20,32,33,24,12,3)(1,3,6,8,8,6,3,1)
𝔤 7 , 1 . 1 7 (2,11,55,120,157,113,45,7)(2,6,17,28,32,25,11,3)(1,3,5,9,9,5,3,1)
𝔤 7 , 1 . 1 8 (1,11,50,114,148,112,45,7)(1,5,12,17,17,15,10,3)(1,3,4,4,4,4,3,1)
𝔤 7 , 1 . 1 9 (1,11,50,114,148,112,45,7)(1,5,12,17,17,15,10,3)(1,3,4,4,4,4,3,1)