Lie algebras

zero(L::LieAlgebra{C}) -> LieAlgebraElem{C}

Return the zero element of the Lie algebra L.

iszero(x::LieAlgebraElem{C}) -> Bool

Check whether the Lie algebra element x is zero.

dim(L::LieAlgebra) -> Int

Return the dimension of the Lie algebra L.

basis(L::LieAlgebra{C}) -> Vector{LieAlgebraElem{C}}

Return a basis of the Lie algebra L.

basis(L::LieAlgebra{C}, i::Int) -> LieAlgebraElem{C}

Return the i-th basis element of the Lie algebra L.

coefficients(x::LieAlgebraElem{C}) -> Vector{C}

Return the coefficients of x with respect to basis(::LieAlgebra).

coeff(x::LieAlgebraElem{C}, i::Int) -> C

Return the i-th coefficient of x with respect to basis(::LieAlgebra).

getindex(x::LieAlgebraElem{C}, i::Int) -> C

Return the i-th coefficient of x with respect to basis(::LieAlgebra).

symbols(L::LieAlgebra{C}) -> Vector{Symbol}

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

characteristic(L::LieAlgebra) -> Int

Return the characteristic of the coefficient ring of the Lie algebra L.

coerce_to_lie_algebra_elem(L::LinearLieAlgebra{C}, x::MatElem{C}) -> LinearLieAlgebraElem{C}

Return the element of L whose matrix representation corresponds to x. If no such element exists, an error is thrown.

matrix_repr_basis(L::LinearLieAlgebra{C}) -> Vector{MatElem{C}}

Return the basis basis(L) of the Lie algebra L in the underlying matrix representation.

matrix_repr_basis(L::LinearLieAlgebra{C}, i::Int) -> MatElem{C}

Return the i-th element of the basis basis(L) of the Lie algebra L in the underlying matrix representation.

matrix_repr(a::Perm)

Return the permutation matrix as a sparse matrix representing a via natural embedding of the permutation group into the general linear group over $\mathbb{Z}$.

Examples

julia> p = Perm([2,3,1])
(1,2,3)

julia> matrix_repr(p)
3×3 SparseArrays.SparseMatrixCSC{Int64, Int64} with 3 stored entries:
 ⋅  1  ⋅
 ⋅  ⋅  1
 1  ⋅  ⋅

julia> Array(ans)
3×3 Matrix{Int64}:
 0  1  0
 0  0  1
 1  0  0

matrix_repr(Y::YoungTableau)

Construct sparse integer matrix representing the tableau.

Examples

julia> y = YoungTableau([4,3,1]);


julia> matrix_repr(y)
3×4 SparseArrays.SparseMatrixCSC{Int64, Int64} with 8 stored entries:
 1  2  3  4
 5  6  7  ⋅
 8  ⋅  ⋅  ⋅

is_abelian(L::LieAlgebra) -> Bool

Return true if L is abelian, i.e. $[L, L] = 0$.

is_nilpotent(L::LieAlgebra) -> Bool

Return true if L is nilpotent, i.e. the lower central series of L terminates in $0$.

is_perfect(L::LieAlgebra) -> Bool

Return true if L is perfect, i.e. $[L, L] = L$.

is_semisimple(L::LieAlgebra) -> Bool

Return true if L is semisimple, i.e. the solvable radical of L is zero.

is_simple(L::LieAlgebra) -> Bool

Return true if L is simple, i.e. L is not abelian and has no non-trivial ideals.

is_solvable(L::LieAlgebra) -> Bool

Return true if L is solvable, i.e. the derived series of L terminates in $0$.

center(L::LieAlgebra) -> LieAlgebraIdeal

Return the center of L, i.e. $\{x \in L \mid [x, L] = 0\}$

centralizer(L::LieAlgebra, xs::AbstractVector{<:LieAlgebraElem}) -> LieSubalgebra

Return the centralizer of xs in L, i.e. $\{y \in L \mid [x, y] = 0 \forall x \in xs\}$.

centralizer(L::LieAlgebra, x::LieAlgebraElem) -> LieSubalgebra

Return the centralizer of x in L, i.e. $\{y \in L \mid [x, y] = 0\}$.

derived_algebra(L::LieAlgebra) -> LieAlgebraIdeal

Return the derived algebra of L, i.e. $[L, L]$.

derived_series(L::LieAlgebra) -> Vector{LieAlgebraIdeal}

Return the derived series of L, i.e. the sequence of ideals $L^{(0)} = L$, $L^{(i + 1)} = [L^{(i)}, L^{(i)}]$.

lower_central_series(L::LieAlgebra) -> Vector{LieAlgebraIdeal}

Return the lower central series of L, i.e. the sequence of ideals $L^{(0)} = L$, $L^{(i + 1)} = [L, L^{(i)}]$.

lie_algebra(gapL::GapObj, s::Vector{<:VarName}) -> LieAlgebra{elem_type(R)}

Construct a Lie algebra isomorphic to the GAP Lie algebra gapL. Its basis element are named by s, or by x_i by default. We require gapL to be a finite-dimensional GAP Lie algebra. The return type is dependent on properties of gapL, in particular, whether GAP knows about a matrix representation.

lie_algebra(R::Field, struct_consts::Matrix{sparse_row_type(R)}, s::Vector{<:VarName}; check::Bool) -> AbstractLieAlgebra{elem_type(R)}

Construct the Lie algebra over the field R with structure constants struct_consts and with basis element names s.

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

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

lie_algebra(R::Field, struct_consts::Array{elem_type(R),3}, s::Vector{<:VarName}; check::Bool) -> AbstractLieAlgebra{elem_type(R)}

Construct the Lie algebra over the field R with structure constants struct_consts and with basis element names s.

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

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

Examples

julia> struct_consts = zeros(QQ, 3, 3, 3);

julia> struct_consts[1, 2, 3] = QQ(1);

julia> struct_consts[2, 1, 3] = QQ(-1);

julia> struct_consts[3, 1, 1] = QQ(2);

julia> struct_consts[1, 3, 1] = QQ(-2);

julia> struct_consts[3, 2, 2] = QQ(-2);

julia> struct_consts[2, 3, 2] = QQ(2);

julia> sl2 = lie_algebra(QQ, struct_consts, ["e", "f", "h"])
Abstract Lie algebra
  of dimension 3
over rational field

julia> e, f, h = basis(sl2)
3-element Vector{AbstractLieAlgebraElem{QQFieldElem}}:
 e
 f
 h

julia> e * f
h

julia> h * e
2*e

julia> h * f
-2*f

lie_algebra(R::Field, rs::RootSystem) -> AbstractLieAlgebra{elem_type(R)}

Construct a simple Lie algebra over the field R with the root system rs. The internally used basis of this Lie algebra is the Chevalley basis.

The experienced user may supply a boolean vector of length n_positive_roots(rs) - n_simple_roots(rs) via the kwarg extraspecial_pair_signs::Vector{Bool} to specify the concrete Lie algebra to be constructed. If $(\alpha,\beta)$ is the extraspecial pair for the non-simple root root(rs, i), then $\varepsilon_{\alpha,\beta} = 1$ iff extraspecial_pair_signs[i - n_simple_roots(rs)] = true. For the used notation and the definition of extraspecial pairs, see [CMT04].

Examples

julia> L = lie_algebra(QQ, root_system(:B, 4))
Abstract Lie algebra
  of type B4
  of dimension 36
over rational field

lie_algebra(R::Field, fam::Symbol, rk::Int) -> AbstractLieAlgebra{elem_type(R)}

Construct a simple Lie algebra over the field R with Dynkin type given by fam and rk. See cartan_matrix(fam::Symbol, rk::Int) for allowed combinations. The internally used basis of this Lie algebra is the Chevalley basis.

Examples

julia> L = lie_algebra(QQ, :C, 4)
Abstract Lie algebra
  of type C4
  of dimension 36
over rational field

lie_algebra(R::Field, n::Int, basis::Vector{<:MatElem{elem_type(R)}}, s::Vector{<:VarName}; check::Bool=true) -> LinearLieAlgebra{elem_type(R)}

Construct the Lie algebra over the field R with basis basis and basis element names given by s. The basis elements must be square matrices of size n.

We require basis to be linearly independent, and to contain the Lie bracket of any two basis elements in its span (this is currently not checked). Setting check=false disables these checks (once they are in place).

Examples

julia> e = matrix(QQ, [0 0 1; 0 0 0; 0 0 0]);

julia> f = matrix(QQ, [0 0 0; 0 0 0; 1 0 0]);

julia> h = matrix(QQ, [1 0 0; 0 0 0; 0 0 -1]);

julia> L = lie_algebra(QQ, 3, [e, f, h], [:e, :f, :h])
Linear Lie algebra with 3x3 matrices
  of dimension 3
over rational field

julia> root_system(L);

julia> L
Linear Lie algebra with 3x3 matrices
  of type A1
  of dimension 3
over rational field

julia> basis(L)
3-element Vector{LinearLieAlgebraElem{QQFieldElem}}:
 e
 f
 h

julia> matrix_repr_basis(L)
3-element Vector{QQMatrix}:
 [0 0 1; 0 0 0; 0 0 0]
 [0 0 0; 0 0 0; 1 0 0]
 [1 0 0; 0 0 0; 0 0 -1]

lie_algebra(S::LieSubalgebra) -> LieAlgebra, LieAlgebraHom

Return S as a Lie algebra LS, together with an embedding LS -> L, where L is the Lie algebra where S lives in.

lie_algebra(I::LieAlgebraIdeal) -> LieAlgebra, LieAlgebraHom

Return I as a Lie algebra LI, together with an embedding LI -> L, where L is the Lie algebra where I lives in.

abelian_lie_algebra(R::Field, n::Int) -> LinearLieAlgebra{elem_type(R)}
abelian_lie_algebra(::Type{LinearLieAlgebra}, R::Field, n::Int) -> LinearLieAlgebra{elem_type(R)}
abelian_lie_algebra(::Type{AbstractLieAlgebra}, R::Field, n::Int) -> AbstractLieAlgebra{elem_type(R)}

Return the abelian Lie algebra of dimension n over the field R. The first argument can be optionally provided to specify the type of the returned Lie algebra.

Examples

julia> abelian_lie_algebra(LinearLieAlgebra, QQ, 3)
Linear Lie algebra with 3x3 matrices
  of dimension 3
over rational field

julia> abelian_lie_algebra(AbstractLieAlgebra, QQ, 3)
Abstract Lie algebra
  of dimension 3
over rational field

general_linear_lie_algebra(R::Field, n::Int) -> LinearLieAlgebra{elem_type(R)}

Return the general linear Lie algebra $\mathfrak{gl}_n(R)$, i.e., the Lie algebra of all $n \times n$ matrices over the field R.

Examples

julia> L = general_linear_lie_algebra(QQ, 2)
General linear Lie algebra of degree 2
  of dimension 4
over rational field

julia> basis(L)
4-element Vector{LinearLieAlgebraElem{QQFieldElem}}:
 x_1_1
 x_1_2
 x_2_1
 x_2_2

julia> matrix_repr_basis(L)
4-element Vector{QQMatrix}:
 [1 0; 0 0]
 [0 1; 0 0]
 [0 0; 1 0]
 [0 0; 0 1]

special_linear_lie_algebra(R::Field, n::Int) -> LinearLieAlgebra{elem_type(R)}

Return the special linear Lie algebra $\mathfrak{sl}_n(R)$, i.e., the Lie algebra of all $n \times n$ matrices over the field R with trace zero.

Examples

julia> L = special_linear_lie_algebra(QQ, 2)
Special linear Lie algebra of degree 2
  of dimension 3
over rational field

julia> basis(L)
3-element Vector{LinearLieAlgebraElem{QQFieldElem}}:
 e_1_2
 f_1_2
 h_1

julia> matrix_repr_basis(L)
3-element Vector{QQMatrix}:
 [0 1; 0 0]
 [0 0; 1 0]
 [1 0; 0 -1]

special_orthogonal_lie_algebra(R::Field, n::Int) -> LinearLieAlgebra{elem_type(R)}
special_orthogonal_lie_algebra(R::Field, n::Int, gram::MatElem) -> LinearLieAlgebra{elem_type(R)}
special_orthogonal_lie_algebra(R::Field, n::Int, gram::Matrix) -> LinearLieAlgebra{elem_type(R)}

Return the special orthogonal Lie algebra $\mathfrak{so}_n(R)$.

Given a non-degenerate symmetric bilinear form $f$ via its Gram matrix gram, $\mathfrak{so}_n(R)$ is the Lie algebra of all $n \times n$ matrices $x$ over the field R such that $f(xv, w) = -f(v, xw)$ for all $v, w \in R^n$.

If gram is not provided, for $n = 2k$ the form defined by $\begin{matrix} 0 & I_k \\ -I_k & 0 \end{matrix}$ is used, and for $n = 2k + 1$ the form defined by $\begin{matrix} 1 & 0 & 0 \\ 0 & 0 I_k \\ 0 & I_k & 0 \end{matrix}$.

Examples

julia> L1 = special_orthogonal_lie_algebra(QQ, 4)
Special orthogonal Lie algebra of degree 4
  of dimension 6
over rational field

julia> basis(L1)
6-element Vector{LinearLieAlgebraElem{QQFieldElem}}:
 x_1
 x_2
 x_3
 x_4
 x_5
 x_6

julia> matrix_repr_basis(L1)
6-element Vector{QQMatrix}:
 [1 0 0 0; 0 0 0 0; 0 0 -1 0; 0 0 0 0]
 [0 1 0 0; 0 0 0 0; 0 0 0 0; 0 0 -1 0]
 [0 0 0 1; 0 0 -1 0; 0 0 0 0; 0 0 0 0]
 [0 0 0 0; 1 0 0 0; 0 0 0 -1; 0 0 0 0]
 [0 0 0 0; 0 1 0 0; 0 0 0 0; 0 0 0 -1]
 [0 0 0 0; 0 0 0 0; 0 1 0 0; -1 0 0 0]

julia> L2 = special_orthogonal_lie_algebra(QQ, 3, identity_matrix(QQ, 3))
Special orthogonal Lie algebra of degree 3
  of dimension 3
over rational field

julia> basis(L2)
3-element Vector{LinearLieAlgebraElem{QQFieldElem}}:
 x_1
 x_2
 x_3

julia> matrix_repr_basis(L2)
3-element Vector{QQMatrix}:
 [0 1 0; -1 0 0; 0 0 0]
 [0 0 1; 0 0 0; -1 0 0]
 [0 0 0; 0 0 1; 0 -1 0]

symplectic_lie_algebra(R::Field, n::Int) -> LinearLieAlgebra{elem_type(R)}
symplectic_lie_algebra(R::Field, n::Int, gram::MatElem) -> LinearLieAlgebra{elem_type(R)}
symplectic_lie_algebra(R::Field, n::Int, gram::Matrix) -> LinearLieAlgebra{elem_type(R)}

Return the symplectic Lie algebra $\mathfrak{sp}_n(R)$.

Given a non-degenerate skew-symmetric bilinear form $f$ via its Gram matrix gram, $\mathfrak{sp}_n(R)$ is the Lie algebra of all $n \times n$ matrices $x$ over the field R such that $f(xv, w) = -f(v, xw)$ for all $v, w \in R^n$.

If gram is not provided, for $n = 2k$ the form defined by $\begin{matrix} 0 & I_k \\ -I_k & 0 \end{matrix}$ is used. For odd $n$ there is no non-degenerate skew-symmetric bilinear form on $R^n$.

Examples

julia> L = symplectic_lie_algebra(QQ, 4)
Symplectic Lie algebra of degree 4
  of dimension 10
over rational field

julia> basis(L)
10-element Vector{LinearLieAlgebraElem{QQFieldElem}}:
 x_1
 x_2
 x_3
 x_4
 x_5
 x_6
 x_7
 x_8
 x_9
 x_10

julia> matrix_repr_basis(L)
10-element Vector{QQMatrix}:
 [1 0 0 0; 0 0 0 0; 0 0 -1 0; 0 0 0 0]
 [0 1 0 0; 0 0 0 0; 0 0 0 0; 0 0 -1 0]
 [0 0 1 0; 0 0 0 0; 0 0 0 0; 0 0 0 0]
 [0 0 0 1; 0 0 1 0; 0 0 0 0; 0 0 0 0]
 [0 0 0 0; 1 0 0 0; 0 0 0 -1; 0 0 0 0]
 [0 0 0 0; 0 1 0 0; 0 0 0 0; 0 0 0 -1]
 [0 0 0 0; 0 0 0 1; 0 0 0 0; 0 0 0 0]
 [0 0 0 0; 0 0 0 0; 1 0 0 0; 0 0 0 0]
 [0 0 0 0; 0 0 0 0; 0 1 0 0; 1 0 0 0]
 [0 0 0 0; 0 0 0 0; 0 0 0 0; 0 1 0 0]