Trac #34030: move supercommutator to superalgebras

which is the proper place for it, as suggested by a TODO

URL: https://trac.sagemath.org/34030
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/algebras/clifford_algebra.py

@@ -318,64 +318,6 @@ def conjugate(self):
 
     clifford_conjugate = conjugate
 
-    # TODO: This is a general function which should be moved to a
-    #   superalgebras category when one is implemented.
-    def supercommutator(self, x):
-        r"""
-        Return the supercommutator of ``self`` and ``x``.
-
-        Let `A` be a superalgebra. The *supercommutator* of homogeneous
-        elements `x, y \in A` is defined by
-
-        .. MATH::
-
-            [x, y\} = x y - (-1)^{|x| |y|} y x
-
-        and extended to all elements by linearity.
-
-        EXAMPLES::
-
-            sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
-            sage: Cl.<x,y,z> = CliffordAlgebra(Q)
-            sage: a = x*y - z
-            sage: b = x - y + y*z
-            sage: a.supercommutator(b)
-            -5*x*y + 8*x*z - 2*y*z - 6*x + 12*y - 5*z
-            sage: a.supercommutator(Cl.one())
-            0
-            sage: Cl.one().supercommutator(a)
-            0
-            sage: Cl.zero().supercommutator(a)
-            0
-            sage: a.supercommutator(Cl.zero())
-            0
-
-            sage: Q = QuadraticForm(ZZ, 2, [-1,1,-3])
-            sage: Cl.<x,y> = CliffordAlgebra(Q)
-            sage: [a.supercommutator(b) for a in Cl.basis() for b in Cl.basis()]
-            [0, 0, 0, 0, 0, -2, 1, -x - 2*y, 0, 1,
-             -6, 6*x + y, 0, x + 2*y, -6*x - y, 0]
-            sage: [a*b-b*a for a in Cl.basis() for b in Cl.basis()]
-            [0, 0, 0, 0, 0, 0, 2*x*y - 1, -x - 2*y, 0,
-             -2*x*y + 1, 0, 6*x + y, 0, x + 2*y, -6*x - y, 0]
-
-        Exterior algebras inherit from Clifford algebras, so
-        supercommutators work as well. We verify the exterior algebra
-        is supercommutative::
-
-            sage: E.<x,y,z,w> = ExteriorAlgebra(QQ)
-            sage: all(b1.supercommutator(b2) == 0
-            ....:     for b1 in E.basis() for b2 in E.basis())
-            True
-        """
-        P = self.parent()
-        ret = P.zero()
-        for ms,cs in self:
-            for mx,cx in x:
-                ret += P.term(ms, cs) * P.term(mx, cx)
-                s = (-1)**(P.degree_on_basis(ms) * P.degree_on_basis(mx))
-                ret -= s * P.term(mx, cx) * P.term(ms, cs)
-        return ret
 
 class CliffordAlgebra(CombinatorialFreeModule):
     r"""

src/sage/categories/super_algebras.py

@@ -1,19 +1,20 @@
 r"""
 Super Algebras
 """
-#*****************************************************************************
+# ****************************************************************************
 #  Copyright (C) 2015 Travis Scrimshaw <tscrim at ucdavis.edu>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
-#                  http://www.gnu.org/licenses/
-#******************************************************************************
+#                  https://www.gnu.org/licenses/
+# *****************************************************************************
 
 from sage.categories.super_modules import SuperModulesCategory
 from sage.categories.signed_tensor import SignedTensorProductsCategory, tensor_signed
 from sage.categories.tensor import tensor
 from sage.misc.lazy_import import LazyImport
 from sage.misc.cachefunc import cached_method
 
+
 class SuperAlgebras(SuperModulesCategory):
     r"""
     The category of super algebras.

src/sage/categories/super_algebras_with_basis.py

@@ -1,17 +1,18 @@
 r"""
 Super algebras with basis
 """
-#*****************************************************************************
+# ****************************************************************************
 #  Copyright (C) 2015,2019 Travis Scrimshaw <tcscrims at gmail.com>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
-#                  http://www.gnu.org/licenses/
-#******************************************************************************
+#                  https://www.gnu.org/licenses/
+# *****************************************************************************
 
 from sage.categories.super_modules import SuperModulesCategory
 from sage.categories.signed_tensor import SignedTensorProductsCategory
 from sage.misc.cachefunc import cached_method
 
+
 class SuperAlgebrasWithBasis(SuperModulesCategory):
     """
     The category of super algebras with a distinguished basis
@@ -59,6 +60,66 @@ def graded_algebra(self):
             from sage.algebras.associated_graded import AssociatedGradedAlgebra
             return AssociatedGradedAlgebra(self)
 
+    class ElementMethods:
+        def supercommutator(self, x):
+            r"""
+            Return the supercommutator of ``self`` and ``x``.
+
+            Let `A` be a superalgebra. The *supercommutator* of homogeneous
+            elements `x, y \in A` is defined by
+
+            .. MATH::
+
+                [x, y\} = x y - (-1)^{|x| |y|} y x
+
+            and extended to all elements by linearity.
+
+            EXAMPLES::
+
+                sage: Q = QuadraticForm(ZZ, 3, [1,2,3,4,5,6])
+                sage: Cl.<x,y,z> = CliffordAlgebra(Q)
+                sage: a = x*y - z
+                sage: b = x - y + y*z
+                sage: a.supercommutator(b)
+                -5*x*y + 8*x*z - 2*y*z - 6*x + 12*y - 5*z
+                sage: a.supercommutator(Cl.one())
+                0
+                sage: Cl.one().supercommutator(a)
+                0
+                sage: Cl.zero().supercommutator(a)
+                0
+                sage: a.supercommutator(Cl.zero())
+                0
+
+                sage: Q = QuadraticForm(ZZ, 2, [-1,1,-3])
+                sage: Cl.<x,y> = CliffordAlgebra(Q)
+                sage: [a.supercommutator(b) for a in Cl.basis() for b in Cl.basis()]
+                [0, 0, 0, 0, 0, -2, 1, -x - 2*y, 0, 1,
+                 -6, 6*x + y, 0, x + 2*y, -6*x - y, 0]
+                sage: [a*b-b*a for a in Cl.basis() for b in Cl.basis()]
+                [0, 0, 0, 0, 0, 0, 2*x*y - 1, -x - 2*y, 0,
+                 -2*x*y + 1, 0, 6*x + y, 0, x + 2*y, -6*x - y, 0]
+
+            Exterior algebras inherit from Clifford algebras, so
+            supercommutators work as well. We verify the exterior algebra
+            is supercommutative::
+
+                sage: E.<x,y,z,w> = ExteriorAlgebra(QQ)
+                sage: all(b1.supercommutator(b2) == 0
+                ....:     for b1 in E.basis() for b2 in E.basis())
+                True
+            """
+            P = self.parent()
+            ret = P.zero()
+            for ms, cs in self:
+                term_s = P.term(ms, cs)
+                sign_s = (-1)**P.degree_on_basis(ms)
+                for mx, cx in x:
+                    ret += term_s * P.term(mx, cx)
+                    s = sign_s**P.degree_on_basis(mx)
+                    ret -= s * P.term(mx, cx) * term_s
+            return ret
+
     class SignedTensorProducts(SignedTensorProductsCategory):
         """
         The category of super algebras with basis constructed by tensor