Fixing some details.

Author: tscrim

src/sage/algebras/lie_algebras/lie_algebra_element.pyx

@@ -1234,7 +1234,7 @@ cdef class UntwistedAffineLieAlgebraElement(Element):
 
     cpdef _acted_upon_(self, scalar, bint self_on_left) noexcept:
         """
-        Return ``self`` acted upon by ``x``.
+        Return ``self`` acted upon by ``scalar``.
 
         EXAMPLES::
 

src/sage/algebras/lie_algebras/representation.py

@@ -189,16 +189,25 @@ def __classcall_private__(cls, lie_algebra, f=None, index_set=None, on_basis=Fal
 
             sage: L.<x,y> = LieAlgebra(QQ, {('x','y'): {'y':1}})
             sage: f = {'x': Matrix([[1,0]]), 'y': Matrix([[0,1]])}
-            sage: R = L.representation(f)
+            sage: L.representation(f)
             Traceback (most recent call last):
             ...
             ValueError: all matrices must be square
 
             sage: f = {'x': Matrix([[1,0],[0,0]]), 'y': Matrix([[0]])}
-            sage: R = L.representation(f)
+            sage: L.representation(f)
             Traceback (most recent call last):
             ...
             ValueError: all matrices must be square of size 2
+
+            sage: L.representation(index_set=[1,2,3])
+            Traceback (most recent call last):
+            ...
+            ValueError: either 'f' or 'on_basis' must be specified
+            sage: L.representation(on_basis=lambda x: QQ.zero())
+            Traceback (most recent call last):
+            ...
+            ValueError: the index set needs to be specified
         """
         from sage.sets.finite_enumerated_set import FiniteEnumeratedSet
         base = lie_algebra.base_ring()

src/sage/categories/lie_algebras.py

@@ -711,14 +711,14 @@ def trivial_representation(self):
             from sage.algebras.lie_algebras.representation import TrivialRepresentation
             return TrivialRepresentation(self)
 
-        def representation(self, f=None, index_set=None, on_basis=None, **kwargs):
+        def representation(self, f=None, index_set=None, on_basis=False, **kwargs):
             """
             Return a representation of ``self``.
 
             If no arguments are given, then this returns the trivial
             representation.
 
-            Currently the only implementated method of constructing a
+            Currently the only implemented method of constructing a
             representation is by explicitly specifying the action of
 
             * the elements of ``self`` by matrices;
@@ -750,10 +750,8 @@ def representation(self, f=None, index_set=None, on_basis=None, **kwargs):
                 sage: L.representation()
                 Trivial representation of Lie algebra on 2 generators (x, y) over Rational Field
             """
-            if f is None and on_basis is None and index_set is None:
+            if f is None and on_basis is False and index_set is None:
                 return self.trivial_representation(**kwargs)
-            if on_basis is None:
-                on_basis = False
             from sage.algebras.lie_algebras.representation import RepresentationByMorphism
             return RepresentationByMorphism(self, f, index_set, on_basis, **kwargs)