Lie algebra modules

Lie algebra modules in OSCAR are always finite dimensional and represented by the type LieAlgebraModule{C}. Similar to other types in OSCAR, there is the corresponding element type LieAlgebraModuleElem{C}. The type parameter C is the element type of the coefficient ring.

base_lie_algebra — Method

base_lie_algebra(V::LieAlgebraModule{C}) -> LieAlgebra{C}

Return the Lie algebra V is a module over.

zero — Method

zero(V::LieAlgebraModule{C}) -> LieAlgebraModuleElem{C}

Return the zero element of the Lie algebra module V.

iszero — Method

iszero(v::LieAlgebraModuleElem{C}) -> Bool

Check whether the Lie algebra module element v is zero.

dim — Method

dim(V::LieAlgebraModule{C}) -> Int

Return the dimension of the Lie algebra module V.

basis — Method

basis(V::LieAlgebraModule{C}) -> Vector{LieAlgebraModuleElem{C}}

Return a basis of the Lie algebra module V.

basis — Method

basis(V::LieAlgebraModule{C}, i::Int) -> LieAlgebraModuleElem{C}

Return the i-th basis element of the Lie algebra module V.

coefficients — Method

coefficients(v::LieAlgebraModuleElem{C}) -> Vector{C}

Return the coefficients of v with respect to basis(::LieAlgebraModule).

coeff — Method

coeff(v::LieAlgebraModuleElem{C}, i::Int) -> C

Return the i-th coefficient of v with respect to basis(::LieAlgebraModule).

getindex — Method

getindex(v::LieAlgebraModuleElem{C}, i::Int) -> C

Return the i-th coefficient of v with respect to basis(::LieAlgebraModule).

symbols — Method

symbols(V::LieAlgebraModule{C}) -> Vector{Symbol}

Return the symbols used for printing basis elements of the Lie algebra module V.

Element constructors

(V::LieAlgebraModule{C})() returns the zero element of the Lie algebra module V.

(V::LieAlgebraModule{C})(v::LieAlgebraModuleElem{C}) returns v if v is an element of L. If V is the dual module of the parent of v, it returns the dual of v. In all other cases, it fails.

(V::LieAlgebraModule{C})(v) constructs the element of V with coefficient vector v. v can be of type Vector{C}, Vector{Int}, SRow{C}, or MatElem{C} (of size $1 \times \dim(L)$).

(V::LieAlgebraModule{C})(a::Vector{T}) where {T<:LieAlgebraModuleElem{C}}): If V is a direct sum, return its element, where the $i$-th component is equal to a[i]. If V is a tensor product, return the tensor product of the a[i]. If V is a exterior (symmetric, tensor) power, return the wedge product (product, tensor product) of the a[i]. Requires that a has a suitable length, and that the a[i] are elements of the correct modules, where correct depends on the case above.

Arithmetics

The usual arithmetics, e.g. +, -, and * (scalar multiplication), are defined for LieAlgebraModuleElems.

The module action is defined as *.

* — Method

action(x::LieAlgebraElem{C}, v::LieAlgebraModuleElem{C}) -> LieAlgebraModuleElem{C}
*(x::LieAlgebraElem{C}, v::LieAlgebraModuleElem{C}) -> LieAlgebraModuleElem{C}

Apply the action of x on v.

Module constructors

trivial_module — Function

trivial_module(L::LieAlgebra{C}, d=1) -> LieAlgebraModule{C}

Construct the d-dimensional module of the Lie algebra L with trivial action.

Examples

julia> L = special_linear_lie_algebra(QQ, 3);

julia> trivial_module(L)
Abstract Lie algebra module
  of dimension 1
over special linear Lie algebra of degree 3 over QQ

standard_module — Method

standard_module(L::LinearLieAlgebra{C}) -> LieAlgebraModule{C}

Construct the standard module of the linear Lie algebra L. If L is a Lie subalgebra of $\mathfrak{gl}_n(R)$, then the standard module is $R^n$ with the action of $L$ given by left multiplication.

Examples

julia> L = special_linear_lie_algebra(QQ, 3);

julia> standard_module(L)
Standard module
  of dimension 3
over special linear Lie algebra of degree 3 over QQ

dual — Method

dual(V::LieAlgebraModule{C}) -> LieAlgebraModule{C}

Construct the dual module of V.

Examples

julia> L = special_linear_lie_algebra(QQ, 3);

julia> V = exterior_power(standard_module(L), 2); # some module

julia> dual(V)
Dual module
  of dimension 3
  dual of
    2nd exterior power of
      standard module
over special linear Lie algebra of degree 3 over QQ

direct_sum — Method

direct_sum(V::LieAlgebraModule{C}...) -> LieAlgebraModule{C}
⊕(V::LieAlgebraModule{C}...) -> LieAlgebraModule{C}

Construct the direct sum of the modules V....

Examples

julia> L = special_linear_lie_algebra(QQ, 3);

julia> V1 = exterior_power(standard_module(L), 2); # some module

julia> V2 = symmetric_power(standard_module(L), 3); # some module

julia> direct_sum(V1, V2)
Direct sum module
  of dimension 13
  direct sum with direct summands
    2nd exterior power of
      standard module
    3rd symmetric power of
      standard module
over special linear Lie algebra of degree 3 over QQ

tensor_product — Method

tensor_product(V::LieAlgebraModule{C}...) -> LieAlgebraModule{C} ⊗(V::LieAlgebraModule{C}...) -> LieAlgebraModule{C}

Construct the tensor product of the modules V....

Examples

julia> L = special_linear_lie_algebra(QQ, 3);

julia> V1 = exterior_power(standard_module(L), 2); # some module

julia> V2 = symmetric_power(standard_module(L), 3); # some module

julia> tensor_product(V1, V2)
Tensor product module
  of dimension 30
  tensor product with tensor factors
    2nd exterior power of
      standard module
    3rd symmetric power of
      standard module
over special linear Lie algebra of degree 3 over QQ

exterior_power — Method

exterior_power(V::LieAlgebraModule{C}, k::Int) -> LieAlgebraModule{C}

Construct the k-th exterior power $\bigwedge^k (V)$ of the module V.

Examples

julia> L = special_linear_lie_algebra(QQ, 3);

julia> V = symmetric_power(standard_module(L), 3); # some module

julia> exterior_power(V, 2)
Exterior power module
  of dimension 45
  2nd exterior power of
    3rd symmetric power of
      standard module
over special linear Lie algebra of degree 3 over QQ

symmetric_power — Method

symmetric_power(V::LieAlgebraModule{C}, k::Int) -> LieAlgebraModule{C}

Construct the k-th symmetric power $S^k (V)$ of the module V.

Examples

julia> L = special_linear_lie_algebra(QQ, 3);

julia> V = exterior_power(standard_module(L), 2); # some module

julia> symmetric_power(V, 3)
Symmetric power module
  of dimension 10
  3rd symmetric power of
    2nd exterior power of
      standard module
over special linear Lie algebra of degree 3 over QQ

tensor_power — Method

tensor_power(V::LieAlgebraModule{C}, k::Int) -> LieAlgebraModule{C}

Construct the k-th tensor power $T^k (V)$ of the module V.

Examples

julia> L = special_linear_lie_algebra(QQ, 3);

julia> V = exterior_power(standard_module(L), 2); # some module

julia> tensor_power(V, 3)
Tensor power module
  of dimension 27
  3rd tensor power of
    2nd exterior power of
      standard module
over special linear Lie algebra of degree 3 over QQ

abstract_module — Method

abstract_module(L::LieAlgebra{C}, dimV::Int, transformation_matrices::Vector{<:MatElem{C}}, s::Vector{<:VarName}; check::Bool) -> LieAlgebraModule{C}

Construct the the Lie algebra module over L of dimension dimV given by transformation_matrices and with basis element names s.

transformation_matrices: The action of the $i$-th basis element of L on some element $v$ of the constructed module is given by left multiplication of the matrix transformation_matrices[i] to the coefficient vector of $v$.
s: A vector of basis element names. This is [Symbol("v_$i") for i in 1:dimV] by default.
check: If true, check that the structure constants are anti-symmetric and satisfy the Jacobi identity. This is true by default.

abstract_module — Method

abstract_module(L::LieAlgebra{C}, dimV::Int, struct_consts::Matrix{SRow{C}}, s::Vector{<:VarName}; check::Bool) -> LieAlgebraModule{C}

Construct the the Lie algebra module over L of dimension dimV given by structure constants struct_consts and with basis element names s.

The action on the newly constructed Lie algebra module V is determined by the structure constants in struct_consts as follows: let $x_i$ denote the $i$-th standard basis vector of L, and $v_i$ the $i$-th standard basis vector of V. Then the entry struct_consts[i,j][k] is a scalar $a_{i,j,k}$ such that $x_i * v_j = \sum_k a_{i,j,k} v_k$.

s: A vector of basis element names. This is [Symbol("v_$i") for i in 1:dimV] by default.
check: If true, check that the structure constants are anti-symmetric and satisfy the Jacobi identity. This is true by default.

Type-dependent getters

is_standard_module — Method

is_standard_module(V::LieAlgebraModule{C}) -> Bool

Check whether V has been constructed as a standard module.

is_dual — Method

is_dual(V::LieAlgebraModule{C}) -> Bool

Check whether V has been constructed as a dual module.

is_direct_sum — Method

is_direct_sum(V::LieAlgebraModule{C}) -> Bool

Check whether V has been constructed as a direct sum of modules.

is_tensor_product — Method

is_tensor_product(V::LieAlgebraModule{C}) -> Bool

Check whether V has been constructed as a tensor product of modules.

is_exterior_power — Method

is_exterior_power(V::LieAlgebraModule{C}) -> Bool

Check whether V has been constructed as an exterior power of a module.

is_symmetric_power — Method

is_symmetric_power(V::LieAlgebraModule{C}) -> Bool

Check whether V has been constructed as a symmetric power of a module.

is_tensor_power — Method

is_tensor_power(V::LieAlgebraModule{C}) -> Bool

Check whether V has been constructed as a tensor power of a module.

base_module — Method

base_module(V::LieAlgebraModule{C}) -> LieAlgebraModule{C}

Returns the base module of V, if V has been constructed as a power module.

base_modules — Method

base_modules(V::LieAlgebraModule{C}) -> Vector{LieAlgebraModule{C}}

Returns the summands or tensor factors of V, if V has been constructed as a direct sum or tensor product of modules.