Classical Lie Algebras

These are the Lie algebras corresponding to types \(A_n\), \(B_n\), \(C_n\), and \(D_n\). We also include support for the exceptional types \(E_{6,7,8}\), \(F_4\), and \(G_2\) in the Chevalley basis, and we give the matrix representation given in [HRT2000].

AUTHORS:

  • Travis Scrimshaw (2013-05-03): Initial version
  • Sebastian Oehms (2018-03-18): matrix method of the element class of ClassicalMatrixLieAlgebra added
sage.algebras.lie_algebras.classical_lie_algebra.ClassicalMatrixLieAlgebra

A classical Lie algebra represented using matrices.

INPUT:

  • R – the base ring
  • ct – the finite Cartan type

EXAMPLES:

sage: lie_algebras.ClassicalMatrix(QQ, ['A', 4])
Special linear Lie algebra of rank 5 over Rational Field
sage: lie_algebras.ClassicalMatrix(QQ, CartanType(['B',4]))
Special orthogonal Lie algebra of rank 9 over Rational Field
sage: lie_algebras.ClassicalMatrix(QQ, 'C4')
Symplectic Lie algebra of rank 8 over Rational Field
sage: lie_algebras.ClassicalMatrix(QQ, cartan_type=['D',4])
Special orthogonal Lie algebra of rank 8 over Rational Field
sage.algebras.lie_algebras.classical_lie_algebra.ExceptionalMatrixLieAlgebra

A matrix Lie algebra of exceptional type.

sage.algebras.lie_algebras.classical_lie_algebra.LieAlgebraChevalleyBasis

A simple finite dimensional Lie algebra in the Chevalley basis.

Let \(L\) be a simple (complex) Lie algebra with roots \(\Phi\), then the Chevalley basis is given by \(e_{\alpha}\) for all \(\alpha \in \Phi\) and \(h_{\alpha_i} := h_i\) where \(\alpha_i\) is a simple root subject. These generators are subject to the relations:

\[\begin{split}\begin{aligned} [h_i, h_j] & = 0 \\ [h_i, e_{\beta}] & = A_{\alpha_i, \beta} e_{\beta} \\ [e_{\beta}, e_{-\beta}] & = \sum_i A_{\beta, \alpha_i} h_i \\ [e_{\beta}, e_{\gamma}] & = \begin{cases} N_{\beta,\gamma} e_{\beta + \gamma} & \beta + \gamma \in \Phi \\ 0 & \text{otherwise.} \end{cases} \end{aligned}\end{split}\]

where \(A_{\alpha, \beta} = \frac{2 (\alpha, \beta)}{(\alpha, \alpha)}\) and \(N_{\alpha, \beta}\) is the maximum such that \(\alpha - N_{\alpha, \beta} \beta \in \Phi\).

For computing the signs of the coefficients, see Section 3 of [CMT2003].

sage.algebras.lie_algebras.classical_lie_algebra.e6

The matrix Lie algebra \(\mathfrak{e}_6\).

The simple Lie algebra \(\mathfrak{e}_6\) of type \(E_6\). The matrix representation is given following [HRT2000].

sage.algebras.lie_algebras.classical_lie_algebra.f4

The matrix Lie algebra \(\mathfrak{f}_4\).

The simple Lie algebra \(\mathfrak{f}_f\) of type \(F_4\). The matrix representation is given following [HRT2000] but indexed in the reversed order (i.e., interchange 1 with 4 and 2 with 3).

sage.algebras.lie_algebras.classical_lie_algebra.g2

The matrix Lie algebra \(\mathfrak{g}_2\).

The simple Lie algebra \(\mathfrak{g}_2\) of type \(G_2\). The matrix representation is given following [HRT2000].

sage.algebras.lie_algebras.classical_lie_algebra.gl

The matrix Lie algebra \(\mathfrak{gl}_n\).

The Lie algebra \(\mathfrak{gl}_n\) which consists of all \(n \times n\) matrices.

INPUT:

  • R – the base ring
  • n – the size of the matrix
sage.algebras.lie_algebras.classical_lie_algebra.sl

The matrix Lie algebra \(\mathfrak{sl}_n\).

The Lie algebra \(\mathfrak{sl}_n\), which consists of all \(n \times n\) matrices with trace 0. This is the Lie algebra of type \(A_{n-1}\).

sage.algebras.lie_algebras.classical_lie_algebra.so

The matrix Lie algebra \(\mathfrak{so}_n\).

The Lie algebra \(\mathfrak{so}_n\), which consists of all real anti-symmetric \(n \times n\) matrices. This is the Lie algebra of type \(B_{(n-1)/2}\) or \(D_{n/2}\) if \(n\) is odd or even respectively.

sage.algebras.lie_algebras.classical_lie_algebra.sp

The matrix Lie algebra \(\mathfrak{sp}_n\).

The Lie algebra \(\mathfrak{sp}_{2k}\), which consists of all \(2k \times 2k\) matrices \(X\) that satisfy the equation:

\[X^T M - M X = 0\]

where

\[\begin{split}M = \begin{pmatrix} 0 & I_k \\ -I_k & 0 \end{pmatrix}.\end{split}\]

This is the Lie algebra of type \(C_k\).