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1 from sage.matrix.constructor import matrix
2
3 from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
4 from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
5
6 class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
7 """
8 SETUP::
9
10 sage: from mjo.eja.eja_algebra import random_eja
11
12 TESTS::
13
14 The matrix representation of an element in the subalgebra is
15 the same as its matrix representation in the superalgebra::
16
17 sage: set_random_seed()
18 sage: A = random_eja().random_element().subalgebra_generated_by()
19 sage: y = A.random_element()
20 sage: actual = y.to_matrix()
21 sage: expected = y.superalgebra_element().to_matrix()
22 sage: actual == expected
23 True
24
25 The left-multiplication-by operator for elements in the subalgebra
26 works like it does in the superalgebra, even if we orthonormalize
27 our basis::
28
29 sage: set_random_seed()
30 sage: x = random_eja(AA).random_element()
31 sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
32 sage: y = A.random_element()
33 sage: y.operator()(A.one()) == y
34 True
35
36 """
37
38 def superalgebra_element(self):
39 """
40 Return the object in our algebra's superalgebra that corresponds
41 to myself.
42
43 SETUP::
44
45 sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
46 ....: random_eja)
47
48 EXAMPLES::
49
50 sage: J = RealSymmetricEJA(3)
51 sage: x = sum(J.gens())
52 sage: x
53 e0 + e1 + e2 + e3 + e4 + e5
54 sage: A = x.subalgebra_generated_by()
55 sage: A(x)
56 f1
57 sage: A(x).superalgebra_element()
58 e0 + e1 + e2 + e3 + e4 + e5
59 sage: y = sum(A.gens())
60 sage: y
61 f0 + f1
62 sage: B = y.subalgebra_generated_by()
63 sage: B(y)
64 g1
65 sage: B(y).superalgebra_element()
66 f0 + f1
67
68 TESTS:
69
70 We can convert back and forth faithfully::
71
72 sage: set_random_seed()
73 sage: J = random_eja()
74 sage: x = J.random_element()
75 sage: A = x.subalgebra_generated_by()
76 sage: A(x).superalgebra_element() == x
77 True
78 sage: y = A.random_element()
79 sage: A(y.superalgebra_element()) == y
80 True
81 sage: B = y.subalgebra_generated_by()
82 sage: B(y).superalgebra_element() == y
83 True
84
85 """
86 # As with the _element_constructor_() method on the
87 # algebra... even in a subspace of a subspace, the basis
88 # elements belong to the ambient space. As a result, only one
89 # level of coordinate_vector() is needed, regardless of how
90 # deeply we're nested.
91 W = self.parent().vector_space()
92 V = self.parent().superalgebra().vector_space()
93
94 # Multiply on the left because basis_matrix() is row-wise.
95 ambient_coords = self.to_vector()*W.basis_matrix()
96 V_coords = V.coordinate_vector(ambient_coords)
97 return self.parent().superalgebra().from_vector(V_coords)
98
99
100
101
102 class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
103 """
104 A subalgebra of an EJA with a given basis.
105
106 SETUP::
107
108 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
109 ....: JordanSpinEJA,
110 ....: RealSymmetricEJA)
111 sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
112
113 EXAMPLES:
114
115 The following Peirce subalgebras of the 2-by-2 real symmetric
116 matrices do not contain the superalgebra's identity element::
117
118 sage: J = RealSymmetricEJA(2)
119 sage: E11 = matrix(AA, [ [1,0],
120 ....: [0,0] ])
121 sage: E22 = matrix(AA, [ [0,0],
122 ....: [0,1] ])
123 sage: K1 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E11),))
124 sage: K1.one().to_matrix()
125 [1 0]
126 [0 0]
127 sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),))
128 sage: K2.one().to_matrix()
129 [0 0]
130 [0 1]
131
132 TESTS:
133
134 Ensure that our generator names don't conflict with the superalgebra::
135
136 sage: J = JordanSpinEJA(3)
137 sage: J.one().subalgebra_generated_by().gens()
138 (f0,)
139 sage: J = JordanSpinEJA(3, prefix='f')
140 sage: J.one().subalgebra_generated_by().gens()
141 (g0,)
142 sage: J = JordanSpinEJA(3, prefix='b')
143 sage: J.one().subalgebra_generated_by().gens()
144 (c0,)
145
146 Ensure that we can find subalgebras of subalgebras::
147
148 sage: A = ComplexHermitianEJA(3).one().subalgebra_generated_by()
149 sage: B = A.one().subalgebra_generated_by()
150 sage: B.dimension()
151 1
152
153 """
154 def __init__(self, superalgebra, basis, category=None, check_axioms=True):
155 self._superalgebra = superalgebra
156 V = self._superalgebra.vector_space()
157 field = self._superalgebra.base_ring()
158 if category is None:
159 category = self._superalgebra.category()
160
161 # A half-assed attempt to ensure that we don't collide with
162 # the superalgebra's prefix (ignoring the fact that there
163 # could be super-superelgrbas in scope). If possible, we
164 # try to "increment" the parent algebra's prefix, although
165 # this idea goes out the window fast because some prefixen
166 # are off-limits.
167 prefixen = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ]
168 try:
169 prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1]
170 except ValueError:
171 prefix = prefixen[0]
172
173 # If our superalgebra is a subalgebra of something else, then
174 # these vectors won't have the right coordinates for
175 # V.span_of_basis() unless we use V.from_vector() on them.
176 W = V.span_of_basis( V.from_vector(b.to_vector()) for b in basis )
177
178 n = len(basis)
179 mult_table = [[W.zero() for i in range(n)] for j in range(n)]
180 ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j])
181 for i in range(n) ]
182 for j in range(n) ]
183
184 for i in range(n):
185 for j in range(n):
186 product = basis[i]*basis[j]
187 # product.to_vector() might live in a vector subspace
188 # if our parent algebra is already a subalgebra. We
189 # use V.from_vector() to make it "the right size" in
190 # that case.
191 product_vector = V.from_vector(product.to_vector())
192 mult_table[i][j] = W.coordinate_vector(product_vector)
193
194 self._inner_product_matrix = matrix(field, ip_table)
195 matrix_basis = tuple( b.to_matrix() for b in basis )
196
197
198 self._vector_space = W
199
200 fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
201 fdeja.__init__(field,
202 mult_table,
203 prefix=prefix,
204 category=category,
205 matrix_basis=matrix_basis,
206 check_field=False,
207 check_axioms=check_axioms)
208
209
210
211 def _element_constructor_(self, elt):
212 """
213 Construct an element of this subalgebra from the given one.
214 The only valid arguments are elements of the parent algebra
215 that happen to live in this subalgebra.
216
217 SETUP::
218
219 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
220 sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
221
222 EXAMPLES::
223
224 sage: J = RealSymmetricEJA(3)
225 sage: X = matrix(AA, [ [0,0,1],
226 ....: [0,1,0],
227 ....: [1,0,0] ])
228 sage: x = J(X)
229 sage: basis = ( x, x^2 ) # x^2 is the identity matrix
230 sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J, basis)
231 sage: K(J.one())
232 f1
233 sage: K(J.one() + x)
234 f0 + f1
235
236 ::
237
238 """
239 if elt not in self.superalgebra():
240 raise ValueError("not an element of this subalgebra")
241
242 # The extra hackery is because foo.to_vector() might not live
243 # in foo.parent().vector_space()! Subspaces of subspaces still
244 # have user bases in the ambient space, though, so only one
245 # level of coordinate_vector() is needed. In other words, if V
246 # is itself a subspace, the basis elements for W will be of
247 # the same length as the basis elements for V -- namely
248 # whatever the dimension of the ambient (parent of V?) space is.
249 V = self.superalgebra().vector_space()
250 W = self.vector_space()
251
252 # Multiply on the left because basis_matrix() is row-wise.
253 ambient_coords = elt.to_vector()*V.basis_matrix()
254 W_coords = W.coordinate_vector(ambient_coords)
255 return self.from_vector(W_coords)
256
257
258
259 def matrix_space(self):
260 """
261 Return the matrix space of this algebra, which is identical to
262 that of its superalgebra.
263
264 This is correct "by definition," and avoids a mismatch when
265 the subalgebra is trivial (with no matrix basis elements to
266 infer anything from) and the parent is not.
267 """
268 return self.superalgebra().matrix_space()
269
270
271 def superalgebra(self):
272 """
273 Return the superalgebra that this algebra was generated from.
274 """
275 return self._superalgebra
276
277
278 def vector_space(self):
279 """
280 SETUP::
281
282 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
283 sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
284
285 EXAMPLES::
286
287 sage: J = RealSymmetricEJA(3)
288 sage: E11 = matrix(ZZ, [ [1,0,0],
289 ....: [0,0,0],
290 ....: [0,0,0] ])
291 sage: E22 = matrix(ZZ, [ [0,0,0],
292 ....: [0,1,0],
293 ....: [0,0,0] ])
294 sage: b1 = J(E11)
295 sage: b2 = J(E22)
296 sage: basis = (b1, b2)
297 sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis)
298 sage: K.vector_space()
299 Vector space of degree 6 and dimension 2 over...
300 User basis matrix:
301 [1 0 0 0 0 0]
302 [0 0 1 0 0 0]
303 sage: b1.to_vector()
304 (1, 0, 0, 0, 0, 0)
305 sage: b2.to_vector()
306 (0, 0, 1, 0, 0, 0)
307
308 """
309 return self._vector_space
310
311
312 Element = FiniteDimensionalEuclideanJordanSubalgebraElement