Improving the documentation.

src/sage/algebras/down_up_algebra.py

@@ -57,9 +57,10 @@ class DownUpAlgebra(CombinatorialFreeModule):
  d(y) = \sum_x x \qquad\qquad u(y) = \sum_z z,
 
  where `y` covers `x` and `z` covers `y`. For `r`-differential posets
- we have `du - ud = r 1` and afford a representation of a
+ we have `du - ud = r 1`, and thus it affords a representation of a
  :class:`Weyl algebra <sage.algebras.weyl_algebra.DifferentialWeylAlgebra>`.
- This is obtained from `DU(0, 1, 2r)`. For a `(q,r)`-differential poset,
+ This Weyl algebra is obtained as the quotient of `DU(0, 1, 2r)` by the
+ ideal generated by `du - ud - r`. For a `(q, r)`-differential poset,
  we have the `d` and `u` operators satisfying
 
  .. MATH::
@@ -72,6 +73,13 @@ class DownUpAlgebra(CombinatorialFreeModule):
  or `\alpha = q(q+1)`, `\beta = -q^3`, and `\gamma = r`. Specializing
  `q = -1` recovers the `r`-differential poset relation.
 
+ Two other noteworthy quotients are:
+
+ - the `q`-Weyl algebra from `DU(0, q^2, q+1)` by the ideal generated by
+ `du - qud - 1`, and
+ - the quantum plane `R_q[d, u]`, where `du = qud`, from `DU(2q, -q^2, 0)`
+ by the ideal generated by `du - qud`.
+
  EXAMPLES:
 
  We begin by constructing the down-up algebra and perform some
@@ -104,11 +112,12 @@ class DownUpAlgebra(CombinatorialFreeModule):
  satisfying `z = [x, y]`, `[x, z] = \gamma x`, and `[z, y] = \gamma y`::
 
  sage: R.<g> = QQ[]
- sage: L = LieAlgebra(R, {('x','y'): {'z': 1}, ('x','z'): {'x': g}, ('z','y'): {'y': g}}, names='x,y,z')
+ sage: L = LieAlgebra(R, {('x','y'): {'z': 1}, ('x','z'): {'x': g}, ('z','y'): {'y': g}},
+ ....: names='x,y,z')
  sage: x, y, z = L.basis()
  sage: (L[x, y], L[x, z], L[z, y])
  (z, g*x, g*y)
- sage: x,y,z = L.pbw_basis().gens()
+ sage: x, y, z = L.pbw_basis().gens()
  sage: x^2*y - 2*x*y*x + y*x^2 == g*x
  True
  sage: x*y^2 - 2*y*x*y + y^2*x == g*y