gh-35630: Implement the octonions

    
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The octonions are a nonassociative normed division algebra, one of only
4 over the reals, and used to create the 7 dimensional cross product. We
implement a slight generalization with 3 parameters over an arbitrary
commutative ring of characteristic not equal to 2, which allows us to
also define the split-octonions as another special case.

Fixes #32940.

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URL: #35630
Reported by: Travis Scrimshaw
Reviewer(s): Frédéric Chapoton, Matthias Köppe

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

@@ -110,6 +110,7 @@ Non-associative algebras
 .. toctree::
    :maxdepth: 1
 
+   sage/algebras/octonion_algebra
    lie_algebras
    lie_conformal_algebras
    sage/algebras/jordan_algebra

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

@@ -1900,9 +1900,9 @@ REFERENCES:
 .. [CW2005] \J. E. Cremona and M. Watkins. Computing isogenies of elliptic
             curves. preprint, 2005.
 
-.. [CHW2015] Cui, Shawn X.; Hong, Seung-Moon; Wang, Zhenghan Universal quantum computation 
+.. [CHW2015] Cui, Shawn X.; Hong, Seung-Moon; Wang, Zhenghan Universal quantum computation
              with weakly integral anyons. Quantum Inf. Process. 14 (2015),
-             no. 8, 2687-2727. 
+             no. 8, 2687-2727.
 
 .. [CW2015] Cui, S. X. and Wang, Z. (2015). Universal quantum computation with
             metaplectic anyons. Journal of Mathematical Physics, 56(3), 032202.
@@ -2287,7 +2287,7 @@ REFERENCES:
 .. [Dy1993] \M. J. Dyer. *Hecke algebras and shellings of Bruhat
             intervals*. Compositio Mathematica, 1993, 89(1): 91-115.
 
-.. [Dy1994] \M. J. Dyer. *Bruhat intervals, polyhedral cones and 
+.. [Dy1994] \M. J. Dyer. *Bruhat intervals, polyhedral cones and
             Kazhdan-Lusztig-Stanley polynomials*. Math.Z., 215(2):223-236, 1994.
 
 .. _ref-E:
@@ -3526,7 +3526,7 @@ REFERENCES:
             :doi:`10.1016/j.bbr.2011.03.031`, :arxiv:`0909.2442`.
 
 .. [JS2021] \D. Jahn, C. Stump.
-            *Bruhat intervals, subword complexes and brick polyhedra for 
+            *Bruhat intervals, subword complexes and brick polyhedra for
             finite Coxeter groups*, 2021, :arxiv:`2103.03715`.
 
 .. [JV2000] \J. Justin, L. Vuillon, *Return words in Sturmian and
@@ -3946,7 +3946,7 @@ REFERENCES:
              set as the intersection of super greedy linear extensions. Order 4,
              293-311 (1987).
              :doi:`10.1007/BF00337892`
-             
+
 .. [Kuh1987] \W. Kühnel, "Minimal triangulations of Kummer varieties",
              Abh. Math. Sem. Univ. Hamburg 57 (1987), 7-20.
 
@@ -4210,7 +4210,7 @@ REFERENCES:
              889-929. :doi:`10.1090/mcom/3358`.
 
 .. [Lon2013] \S. London,
-            *Constructing New Turyn Type Sequences, T-Sequences and Hadamard Matrices*. 
+            *Constructing New Turyn Type Sequences, T-Sequences and Hadamard Matrices*.
             PhD Thesis, University of Illinois at Chicago, 2013.
             https://hdl.handle.net/10027/9916
 
@@ -5448,6 +5448,9 @@ REFERENCES:
 .. [Sch1996] \E. Schaefer. A simplified data encryption
              algorithm. Cryptologia, 20(1):77--84, 1996.
 
+.. [Scha1996] Richard D. Schaefer. *An Introduction to Nonassociative Algebras*.
+              Dover, New York, 1996.
+
 .. [Sch1999] Gilles Schaeffer, *Random Sampling of Large Planar Maps and Convex
              Polyhedra*, Annual ACM Symposium on Theory of Computing
              (Atlanta, GA, 1999). :doi:`10.1145/301250.301448`.
@@ -5621,7 +5624,7 @@ REFERENCES:
              Journal of Cryptology. 12. 193-196. 1999.
              :doi:`10.1007/s001459900052`.
 
-.. [Smi2023] \D. Smith, J. S. Myers, C. S. Kaplan and C. Goodman-Strauss, 
+.. [Smi2023] \D. Smith, J. S. Myers, C. S. Kaplan and C. Goodman-Strauss,
              *An aperiodic monotile*,
              :arxiv:`2303.10798`
 
@@ -6501,7 +6504,7 @@ REFERENCES:
             Combinatorics, 19(1), P53, 2012. :doi:`10.37236/2087`
 
 .. [Zie1959] \N. Zierler. *Linear Recurring Sequences*.
-             Journal of the Society for Industrial and Applied Mathematics 7(1) 
+             Journal of the Society for Industrial and Applied Mathematics 7(1)
              (1959): 31-48. :doi:`10.1137/0107003`
 
 .. [Zie1998] \G. M. Ziegler. *Shelling polyhedral 3-balls and

src/sage/algebras/all.py

@@ -48,6 +48,8 @@
 
 lazy_import('sage.algebras.jordan_algebra', 'JordanAlgebra')
 
+lazy_import('sage.algebras.octonion_algebra', 'OctonionAlgebra')
+
 lazy_import('sage.algebras.shuffle_algebra', 'ShuffleAlgebra')
 
 from .clifford_algebra import CliffordAlgebra, ExteriorAlgebra

src/sage/algebras/catalog.py

@@ -48,6 +48,7 @@
 - :class:`algebras.MalvenutoReutenauer <sage.combinat.fqsym.FreeQuasisymmetricFunctions>`
 - :class:`algebras.NilCoxeter
   <sage.algebras.nil_coxeter_algebra.NilCoxeterAlgebra>`
+- :class:`algebras.Octonion <sage.algebras.octonion_algebra.OctonionAlgebra>`
 - :class:`algebras.OrlikTerao
   <sage.algebras.orlik_terao.OrlikTeraoAlgebra>`
 - :class:`algebras.OrlikSolomon
@@ -137,5 +138,6 @@
            'ACEQuantumOnsagerAlgebra', 'AlternatingCentralExtensionQuantumOnsager')
 lazy_import('sage.algebras.down_up_algebra', 'DownUpAlgebra', 'DownUp')
 lazy_import('sage.algebras.yangian', 'Yangian')
+lazy_import('sage.algebras.octonion_algebra', 'OctonionAlgebra', 'Octonion')
 
 del lazy_import  # We remove the object from here so it doesn't appear under tab completion

src/sage/algebras/octonion_algebra.pxd

@@ -0,0 +1,19 @@
+"""
+Octonions
+"""
+
+from sage.structure.element cimport AlgebraElement
+from sage.modules.free_module_element cimport FreeModuleElement
+
+cdef class Octonion_generic(AlgebraElement):
+    cdef FreeModuleElement vec
+
+    cpdef Octonion_generic conjugate(self)
+    cpdef quadratic_form(self)
+    cpdef norm(self)
+    cpdef abs(self)
+    cpdef real_part(self)
+    cpdef Octonion_generic imag_part(self)
+
+cdef class Octonion(Octonion_generic):
+    pass