Trac #25008: Additional method "matrix" / "_matrix_" for the element …

…class of ClassicalMatrixLieAlgebra

A method to obtain the underlying matrix as element of the corresponding
full matrix space analog to the similar method of
MatrixGroupElement_generic should be added. This may be a shortcut for
the following:

{{{
sage: LM = lie_algebras.sp(QQ, 4, representation='matrix')
sage: e1, e2, f1, f2, h1, h2 = LM.gens()
sage: mh1 = LM.associative_algebra()(dict(h1)); mh1
[ 1  0  0  0]
[ 0 -1  0  0]
[ 0  0 -1  0]
[ 0  0  0  1]
sage: mh1.parent()
Full MatrixSpace of 4 by 4 sparse matrices over Rational Field
}}}

URL: https://trac.sagemath.org/25008
Reported by: soehms
Ticket author(s): Sebastian Oehms
Reviewer(s): Travis Scrimshaw

src/sage/algebras/lie_algebras/classical_lie_algebra.py

@@ -9,6 +9,7 @@
 AUTHORS:
 
 - Travis Scrimshaw (2013-05-03): Initial version
+- Sebastian Oehms  (2018-03-18): matrix method of the element class of ClassicalMatrixLieAlgebra added
 """
 
 #*****************************************************************************
@@ -333,6 +334,42 @@ def affine(self, kac_moody=False):
         from sage.algebras.lie_algebras.affine_lie_algebra import AffineLieAlgebra
         return AffineLieAlgebra(self, kac_moody)
 
+    class Element(LieAlgebraFromAssociative.Element):
+        def matrix(self):
+            r"""
+            Return ``self`` as element of the underlying matrix algebra.
+
+            OUTPUT:
+
+            An instance of the element class of MatrixSpace.
+
+            EXAMPLES::
+
+                sage: sl3m = lie_algebras.sl(ZZ, 3, representation='matrix')
+                sage: e1,e2, f1, f2, h1, h2 = sl3m.gens()
+                sage: h1m = h1.matrix(); h1m
+                [ 1  0  0]
+                [ 0 -1  0]
+                [ 0  0  0]
+                sage: h1m.parent()
+                Full MatrixSpace of 3 by 3 sparse matrices over Integer Ring
+                sage: matrix(h2)
+                [ 0  0  0]
+                [ 0  1  0]
+                [ 0  0 -1]
+                sage: L = lie_algebras.so(QQ['z'], 5, representation='matrix')
+                sage: matrix(L.an_element())
+                [ 1  1  0  0  0]
+                [ 1  1  0  0  2]
+                [ 0  0 -1 -1  0]
+                [ 0  0 -1 -1 -1]
+                [ 0  1  0 -2  0]
+            """
+            return self.value
+
+        _matrix_ = matrix
+
+
 class gl(LieAlgebraFromAssociative):
     r"""
     The matrix Lie algebra `\mathfrak{gl}_n`.
@@ -460,7 +497,7 @@ def monomial(self, i):
             return self.basis()['E_{}_{}'.format(*i)]
         return self.basis()[i]
 
-    class Element(LieAlgebraFromAssociative.Element):
+    class Element(ClassicalMatrixLieAlgebra.Element):
         def monomial_coefficients(self, copy=True):
             r"""
             Return the monomial coefficients of ``self``.