gh-35484: Implement down-up algebras and their Verma modules

    
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Down-up algebras arose from certain operators on posets. In particular,
this is a generalization of the algebra of those operators for
$r$-differential posets. They have a PBW-type basis and corresponding
Verma modules with a triangular-type decomposition. We provide an
implementation of these.

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URL: #35484
Reported by: Travis Scrimshaw
Reviewer(s): Frédéric Chapoton, Martin Rubey, Travis Scrimshaw

src/doc/en/reference/algebras/index.rst

@@ -48,6 +48,7 @@ Named associative algebras
  sage/algebras/clifford_algebra
  sage/algebras/cluster_algebra
  sage/combinat/descent_algebra
+ sage/algebras/down_up_algebra
  fusion_rings
  sage/algebras/hall_algebra
  sage/combinat/posets/incidence_algebras

src/doc/en/reference/references/index.rst

@@ -1127,6 +1127,10 @@ REFERENCES:
  Achtzelle (Oktatope)", Verhand. Konik. Akad. Wetenschap,
  Erste Sectie, 10 (1910)
 
+.. [BR1998] Georgia Benkart and Tom Roby. *Down-up algebras*.
+ J. Algebra, **209** no. 1 (1999), pp. 305-335.
+ :doi:`10.1006/jabr.1998.7511`.
+
 .. [Br2000] Kenneth S. Brown, *Semigroups, rings, and Markov chains*,
  :arxiv:`math/0006145v1`.
 
@@ -1619,6 +1623,10 @@ REFERENCES:
  \J. Algebr. Comb. **39** (2014) pp. 17-51.
  :doi:`10.1007/s10801-013-0437-x`, :arxiv:`1108.1776`.
 
+.. [CM2000] Paula A.A.B. Carvalho and Ian M. Musson. *Down-up algebras and
+ their representation theory*. J. Algebra. **228** no. 1, (2000),
+ pp. 286-310. :doi:`10.1006/jabr.1999.8263`
+
 .. [CM2012] \M. Cabanes, I. Marin, *On ternary quotients of cubic Hecke
  algebras*, Comm. Math. Phys. (2012), Volume 314, Issue 1,
  pp 57-92. :doi:`10.1007/s00220-012-1519-7`, :arxiv:`1010.1465`.

src/sage/algebras/catalog.py

@@ -23,6 +23,7 @@
 - :class:`algebras.Descent <sage.combinat.descent_algebra.DescentAlgebra>`
 - :class:`algebras.DifferentialWeyl
  <sage.algebras.weyl_algebra.DifferentialWeylAlgebra>`
+- :class:`algebras.DownUp <sage.algebras.down_up_algebra.DownUpAlgebra>`
 - :class:`algebras.Exterior <sage.algebras.clifford_algebra.ExteriorAlgebra>`
 - :class:`algebras.FiniteDimensional
  <sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra.FiniteDimensionalAlgebra>`
@@ -134,6 +135,7 @@
 lazy_import('sage.algebras.quantum_groups.quantum_group_gap', 'QuantumGroup')
 lazy_import('sage.algebras.quantum_groups.ace_quantum_onsager',
  'ACEQuantumOnsagerAlgebra', 'AlternatingCentralExtensionQuantumOnsager')
+lazy_import('sage.algebras.down_up_algebra', 'DownUpAlgebra', 'DownUp')
 lazy_import('sage.algebras.yangian', 'Yangian')
 
 del lazy_import # We remove the object from here so it doesn't appear under tab completion