+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA)
+ sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
+
+ EXAMPLES:
+
+ The following Peirce subalgebras of the 2-by-2 real symmetric
+ matrices do not contain the superalgebra's identity element::
+
+ sage: J = RealSymmetricEJA(2)
+ sage: E11 = matrix(AA, [ [1,0],
+ ....: [0,0] ])
+ sage: E22 = matrix(AA, [ [0,0],
+ ....: [0,1] ])
+ sage: K1 = FiniteDimensionalEJASubalgebra(J, (J(E11),), associative=True)
+ sage: K1.one().to_matrix()
+ [1 0]
+ [0 0]
+ sage: K2 = FiniteDimensionalEJASubalgebra(J, (J(E22),), associative=True)
+ sage: K2.one().to_matrix()
+ [0 0]
+ [0 1]
+
+ TESTS:
+
+ Ensure that our generator names don't conflict with the
+ superalgebra::
+
+ sage: J = JordanSpinEJA(3)
+ sage: J.one().subalgebra_generated_by().gens()
+ (c0,)
+ sage: J = JordanSpinEJA(3, prefix='f')
+ sage: J.one().subalgebra_generated_by().gens()
+ (g0,)
+ sage: J = JordanSpinEJA(3, prefix='a')
+ sage: J.one().subalgebra_generated_by().gens()
+ (b0,)
+
+ Ensure that we can find subalgebras of subalgebras::
+
+ sage: A = ComplexHermitianEJA(3).one().subalgebra_generated_by()
+ sage: B = A.one().subalgebra_generated_by()
+ sage: B.dimension()
+ 1
+ """
+ def __init__(self, superalgebra, basis, **kwargs):
+ self._superalgebra = superalgebra
+ V = self._superalgebra.vector_space()
+ field = self._superalgebra.base_ring()
+
+ # A half-assed attempt to ensure that we don't collide with
+ # the superalgebra's prefix (ignoring the fact that there
+ # could be super-superelgrbas in scope). If possible, we
+ # try to "increment" the parent algebra's prefix, although
+ # this idea goes out the window fast because some prefixen
+ # are off-limits.
+ prefixen = ["b","c","d","e","f","g","h","l","m"]
+ try:
+ prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1]
+ except ValueError:
+ prefix = prefixen[0]
+
+ # The superalgebra constructor expects these to be in original matrix
+ # form, not algebra-element form.
+ matrix_basis = tuple( b.to_matrix() for b in basis )
+ def jordan_product(x,y):
+ return (self._superalgebra(x)*self._superalgebra(y)).to_matrix()
+
+ def inner_product(x,y):
+ return self._superalgebra(x).inner_product(self._superalgebra(y))
+
+ super().__init__(matrix_basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ matrix_space=superalgebra.matrix_space(),
+ prefix=prefix,
+ **kwargs)