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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 if check_axioms:
180 # The tables are square if we're verifying that they
181 # are commutative.
182 mult_table = [[W.zero() for j in range(n)] for i in range(n)]
183 ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j])
184 for j in range(n) ]
185 for i in range(n) ]
186 else:
187 mult_table = [[W.zero() for j in range(i+1)] for i in range(n)]
188 ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j])
189 for j in range(i+1) ]
190 for i in range(n) ]
191
192 for i in range(n):
193 for j in range(i+1):
194 product = basis[i]*basis[j]
195 # product.to_vector() might live in a vector subspace
196 # if our parent algebra is already a subalgebra. We
197 # use V.from_vector() to make it "the right size" in
198 # that case.
199 product_vector = V.from_vector(product.to_vector())
200 mult_table[i][j] = W.coordinate_vector(product_vector)
201 if check_axioms:
202 mult_table[j][i] = mult_table[i][j]
203
204 matrix_basis = tuple( b.to_matrix() for b in basis )
205
206
207 self._vector_space = W
208
209 fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
210 fdeja.__init__(field,
211 mult_table,
212 ip_table,
213 prefix=prefix,
214 category=category,
215 matrix_basis=matrix_basis,
216 check_field=False,
217 check_axioms=check_axioms)
218
219
220
221 def _element_constructor_(self, elt):
222 """
223 Construct an element of this subalgebra from the given one.
224 The only valid arguments are elements of the parent algebra
225 that happen to live in this subalgebra.
226
227 SETUP::
228
229 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
230 sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
231
232 EXAMPLES::
233
234 sage: J = RealSymmetricEJA(3)
235 sage: X = matrix(AA, [ [0,0,1],
236 ....: [0,1,0],
237 ....: [1,0,0] ])
238 sage: x = J(X)
239 sage: basis = ( x, x^2 ) # x^2 is the identity matrix
240 sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J, basis)
241 sage: K(J.one())
242 f1
243 sage: K(J.one() + x)
244 f0 + f1
245
246 ::
247
248 """
249 if elt not in self.superalgebra():
250 raise ValueError("not an element of this subalgebra")
251
252 # The extra hackery is because foo.to_vector() might not live
253 # in foo.parent().vector_space()! Subspaces of subspaces still
254 # have user bases in the ambient space, though, so only one
255 # level of coordinate_vector() is needed. In other words, if V
256 # is itself a subspace, the basis elements for W will be of
257 # the same length as the basis elements for V -- namely
258 # whatever the dimension of the ambient (parent of V?) space is.
259 V = self.superalgebra().vector_space()
260 W = self.vector_space()
261
262 # Multiply on the left because basis_matrix() is row-wise.
263 ambient_coords = elt.to_vector()*V.basis_matrix()
264 W_coords = W.coordinate_vector(ambient_coords)
265 return self.from_vector(W_coords)
266
267
268
269 def matrix_space(self):
270 """
271 Return the matrix space of this algebra, which is identical to
272 that of its superalgebra.
273
274 This is correct "by definition," and avoids a mismatch when
275 the subalgebra is trivial (with no matrix basis elements to
276 infer anything from) and the parent is not.
277 """
278 return self.superalgebra().matrix_space()
279
280
281 def superalgebra(self):
282 """
283 Return the superalgebra that this algebra was generated from.
284 """
285 return self._superalgebra
286
287
288 def vector_space(self):
289 """
290 SETUP::
291
292 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
293 sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
294
295 EXAMPLES::
296
297 sage: J = RealSymmetricEJA(3)
298 sage: E11 = matrix(ZZ, [ [1,0,0],
299 ....: [0,0,0],
300 ....: [0,0,0] ])
301 sage: E22 = matrix(ZZ, [ [0,0,0],
302 ....: [0,1,0],
303 ....: [0,0,0] ])
304 sage: b1 = J(E11)
305 sage: b2 = J(E22)
306 sage: basis = (b1, b2)
307 sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis)
308 sage: K.vector_space()
309 Vector space of degree 6 and dimension 2 over...
310 User basis matrix:
311 [1 0 0 0 0 0]
312 [0 0 1 0 0 0]
313 sage: b1.to_vector()
314 (1, 0, 0, 0, 0, 0)
315 sage: b2.to_vector()
316 (0, 0, 1, 0, 0, 0)
317
318 """
319 return self._vector_space
320
321
322 Element = FiniteDimensionalEuclideanJordanSubalgebraElement