2 Cohomology
2.1 First cohomology
2.1-1 LieOneCohomology
‣ LieOneCohomology ( L, B, M ) | ( function ) |
‣ LieOneCobounds ( L, B, M ) | ( function ) |
‣ LieOneCocycles ( L, B, M ) | ( function ) |
These functions take as input a Lie algebra L, a basis B of L and a list of matrices M such that the map mapping the elements of B to the elements of M induces a representation of L.
Let B={ b_1, ..., b_n } and V be the natural L-module defined by the given representation. Then the map Z^1(L,V) → V^n: φ ↦ (b_1^φ, ..., b_n^ϕ) is a faithful representation of Z^1(L,V).
The functions LieOneCobounds and LieOneCocycles return lists of vectors which form a basis for the images of the representations of Z^1(L,V) and B^1(L,V), respectively.
The functions LieOneCohomology returns a linear map onto the factor of the results of LieOneCocycles modulo LieOneCobounds. Thus the image of this map is a faithful representation of H^1(L,V).
2.2 Second cohomology
2.2-1 LieTwoCohomology
‣ LieTwoCohomology ( L, B, M ) | ( function ) |
‣ LieTwoCobounds ( L, B, M ) | ( function ) |
‣ LieTwoCocycles ( L, B, M ) | ( function ) |
These functions take as input a Lie algebra L, a basis B of L and a list of matrices M such that the map mapping the elements of B to the elements of M induces a representation of L.
The functions LieTwoCobounds and LieTwoCocycles return lists of vectors which form a basis for the images of the representations of Z^2(L,V) and B^2(L,V), respectively.
The functions LieTwoCohomology returns a linear map onto the factor of the results of LieTwoCocycles modulo LieTwoCobounds. Thus the image of this map is a faithful representation of H^2(L,V).
2.3 Extensions by cocycles
2.3-1 LieExtensionByCocycle
‣ LieExtensionByCocycle ( L, B, M, c ) | ( function ) |
This function takes as input a Lie algebra L, a basis B of L and a list of matrices M such that the map mapping the elements of B to the elements of M induces a representation of L and a 2-cocycle c in the representation of Z^2(L,V).
The function returns a Lie algebra which is an extension of L by the natural L-module V defined by the representation via v.