ManifoldGroupUtils.jl

algebra(G)

The tangent space at identity of the group G.

compose_lie_matrix_op(
    G, # group
    op, # operator Alg(G) -> Alg(G)
    mat, # matrix in the basis
    B # basis of Alg(G)
    )

Compute the matrix in the basis B of the composition of op a linear endomorphism of Alg(G) and a matrix mat, also expressed in the basis B.

compose_matrix_op(
  G, # group
  M,p, # Manifold, point
  op, # operator Alg(G) -> T_pM
  mat, # matrix in basis BG
  BG, # basis of Alg(G)
  BM, # basis of T_pM
  )

Compose a matrix mat of a linear endomorphism of Alg(G) in some basis BG with an operator op : $Alg(G) → T_p M$ itself equipped with a basis BM.

get_id_matrix_lie(G)

The identity matrix on the Lie algebra of the group G.

get_op_matrix(G, # group
  M, p, # manifold + point
  op, # operator from Alg(G) -> T_pM
  BG, # Lie algebra basis
  BM, # basis at T_pM
  )

Matrix of an operator op : $Alg(G) → T_p M$ computed in the basis BG and BM.

get_proj_matrix(A::GroupAction, x, BG, BM)

From a group action $G ⊂ Diff(M)$, and a point $x ∈ M$, compute the projection matrix in the basis BG of Alg(G) and BM of $T_x M$ of the operator $ξ ↦ ξ ⋅x$, where $⋅$ denotes the infinitesimal group action above.

matrix_from_lin_endomorphism(
    G, # group
    op, # Alg(G) -> Alg(G)
    B # basis of Alg(G)
    )

Compute the matrix in the basis B of an operator op on a Lie algebra Alg(G).

rand_lie(rng::AbstractRNG, G)

Random element in the Lie algebra of the group G.

translate_from_id(G, χ, ξ, ::GroupActionSide)

The left translation $T_L(g,ξ) = gξ$ for the Left side, or right translation $T_R(g,ξ) = ξg$ for the Right side.

translate_to_id(G, χ, v, ::GroupActionSide)

Compute $η = v χ⁻¹$, or $χ⁻¹ v$ depending on whether the group action side is Right or Left respectively.