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eja: use the associativity of one-generator subalgebras.
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1 """
2 Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
3 specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
4 are used in optimization, and have some additional nice methods beyond
5 what can be supported in a general Jordan Algebra.
6 """
7
8 from sage.categories.magmatic_algebras import MagmaticAlgebras
9 from sage.structure.element import is_Matrix
10 from sage.structure.category_object import normalize_names
11
12 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra
13 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement
14
15 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
16 @staticmethod
17 def __classcall_private__(cls,
18 field,
19 mult_table,
20 names='e',
21 assume_associative=False,
22 category=None,
23 rank=None):
24 n = len(mult_table)
25 mult_table = [b.base_extend(field) for b in mult_table]
26 for b in mult_table:
27 b.set_immutable()
28 if not (is_Matrix(b) and b.dimensions() == (n, n)):
29 raise ValueError("input is not a multiplication table")
30 mult_table = tuple(mult_table)
31
32 cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis()
33 cat.or_subcategory(category)
34 if assume_associative:
35 cat = cat.Associative()
36
37 names = normalize_names(n, names)
38
39 fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls)
40 return fda.__classcall__(cls,
41 field,
42 mult_table,
43 assume_associative=assume_associative,
44 names=names,
45 category=cat,
46 rank=rank)
47
48
49 def __init__(self, field,
50 mult_table,
51 names='e',
52 assume_associative=False,
53 category=None,
54 rank=None):
55 self._rank = rank
56 fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
57 fda.__init__(field,
58 mult_table,
59 names=names,
60 category=category)
61
62
63 def _repr_(self):
64 """
65 Return a string representation of ``self``.
66 """
67 fmt = "Euclidean Jordan algebra of degree {} over {}"
68 return fmt.format(self.degree(), self.base_ring())
69
70 def rank(self):
71 """
72 Return the rank of this EJA.
73 """
74 if self._rank is None:
75 raise ValueError("no rank specified at genesis")
76 else:
77 return self._rank
78
79
80 class Element(FiniteDimensionalAlgebraElement):
81 """
82 An element of a Euclidean Jordan algebra.
83
84 Since EJAs are commutative, the "right multiplication" matrix is
85 also the left multiplication matrix and must be symmetric::
86
87 sage: set_random_seed()
88 sage: n = ZZ.random_element(1,10).abs()
89 sage: J = eja_rn(5)
90 sage: J.random_element().matrix().is_symmetric()
91 True
92 sage: J = eja_ln(5)
93 sage: J.random_element().matrix().is_symmetric()
94 True
95
96 """
97
98 def __pow__(self, n):
99 """
100 Return ``self`` raised to the power ``n``.
101
102 Jordan algebras are always power-associative; see for
103 example Faraut and Koranyi, Proposition II.1.2 (ii).
104 """
105 A = self.parent()
106 if n == 0:
107 return A.one()
108 elif n == 1:
109 return self
110 else:
111 return A.element_class(A, self.vector()*(self.matrix()**(n-1)))
112
113
114 def span_of_powers(self):
115 """
116 Return the vector space spanned by successive powers of
117 this element.
118 """
119 # The dimension of the subalgebra can't be greater than
120 # the big algebra, so just put everything into a list
121 # and let span() get rid of the excess.
122 V = self.vector().parent()
123 return V.span( (self**d).vector() for d in xrange(V.dimension()) )
124
125
126 def degree(self):
127 """
128 Compute the degree of this element the straightforward way
129 according to the definition; by appending powers to a list
130 and figuring out its dimension (that is, whether or not
131 they're linearly dependent).
132
133 EXAMPLES::
134
135 sage: J = eja_ln(4)
136 sage: J.one().degree()
137 1
138 sage: e0,e1,e2,e3 = J.gens()
139 sage: (e0 - e1).degree()
140 2
141
142 In the spin factor algebra (of rank two), all elements that
143 aren't multiples of the identity are regular::
144
145 sage: set_random_seed()
146 sage: n = ZZ.random_element(1,10).abs()
147 sage: J = eja_ln(n)
148 sage: x = J.random_element()
149 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
150 True
151
152 """
153 return self.span_of_powers().dimension()
154
155
156 def subalgebra_generated_by(self):
157 """
158 Return the associative subalgebra of the parent EJA generated
159 by this element.
160
161 TESTS::
162
163 sage: set_random_seed()
164 sage: n = ZZ.random_element(1,10).abs()
165 sage: J = eja_rn(n)
166 sage: x = J.random_element()
167 sage: x.subalgebra_generated_by().is_associative()
168 True
169 sage: J = eja_ln(n)
170 sage: x = J.random_element()
171 sage: x.subalgebra_generated_by().is_associative()
172 True
173
174 """
175 # First get the subspace spanned by the powers of myself...
176 V = self.span_of_powers()
177 F = self.base_ring()
178
179 # Now figure out the entries of the right-multiplication
180 # matrix for the successive basis elements b0, b1,... of
181 # that subspace.
182 mats = []
183 for b_right in V.basis():
184 eja_b_right = self.parent()(b_right)
185 b_right_rows = []
186 # The first row of the right-multiplication matrix by
187 # b1 is what we get if we apply that matrix to b1. The
188 # second row of the right multiplication matrix by b1
189 # is what we get when we apply that matrix to b2...
190 for b_left in V.basis():
191 eja_b_left = self.parent()(b_left)
192 # Multiply in the original EJA, but then get the
193 # coordinates from the subalgebra in terms of its
194 # basis.
195 this_row = V.coordinates((eja_b_left*eja_b_right).vector())
196 b_right_rows.append(this_row)
197 b_right_matrix = matrix(F, b_right_rows)
198 mats.append(b_right_matrix)
199
200 # It's an algebra of polynomials in one element, and EJAs
201 # are power-associative.
202 return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True)
203
204
205 def minimal_polynomial(self):
206 """
207 EXAMPLES::
208
209 sage: set_random_seed()
210 sage: n = ZZ.random_element(1,10).abs()
211 sage: J = eja_rn(n)
212 sage: x = J.random_element()
213 sage: x.degree() == x.minimal_polynomial().degree()
214 True
215
216 ::
217
218 sage: set_random_seed()
219 sage: n = ZZ.random_element(1,10).abs()
220 sage: J = eja_ln(n)
221 sage: x = J.random_element()
222 sage: x.degree() == x.minimal_polynomial().degree()
223 True
224
225 The minimal polynomial and the characteristic polynomial coincide
226 and are known (see Alizadeh, Example 11.11) for all elements of
227 the spin factor algebra that aren't scalar multiples of the
228 identity::
229
230 sage: set_random_seed()
231 sage: n = ZZ.random_element(2,10).abs()
232 sage: J = eja_ln(n)
233 sage: y = J.random_element()
234 sage: while y == y.coefficient(0)*J.one():
235 ....: y = J.random_element()
236 sage: y0 = y.vector()[0]
237 sage: y_bar = y.vector()[1:]
238 sage: actual = y.minimal_polynomial()
239 sage: x = SR.symbol('x', domain='real')
240 sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
241 sage: bool(actual == expected)
242 True
243
244 """
245 if self.parent().is_associative():
246 return self.matrix().minimal_polynomial()
247
248 V = self.span_of_powers()
249 assoc_subalg = self.subalgebra_generated_by()
250 # Mis-design warning: the basis used for span_of_powers()
251 # and subalgebra_generated_by() must be the same, and in
252 # the same order!
253 subalg_self = assoc_subalg(V.coordinates(self.vector()))
254 # Recursive call, but should work since the subalgebra is
255 # associative.
256 return subalg_self.minimal_polynomial()
257
258
259 def characteristic_polynomial(self):
260 return self.matrix().characteristic_polynomial()
261
262
263 def eja_rn(dimension, field=QQ):
264 """
265 Return the Euclidean Jordan Algebra corresponding to the set
266 `R^n` under the Hadamard product.
267
268 EXAMPLES:
269
270 This multiplication table can be verified by hand::
271
272 sage: J = eja_rn(3)
273 sage: e0,e1,e2 = J.gens()
274 sage: e0*e0
275 e0
276 sage: e0*e1
277 0
278 sage: e0*e2
279 0
280 sage: e1*e1
281 e1
282 sage: e1*e2
283 0
284 sage: e2*e2
285 e2
286
287 """
288 # The FiniteDimensionalAlgebra constructor takes a list of
289 # matrices, the ith representing right multiplication by the ith
290 # basis element in the vector space. So if e_1 = (1,0,0), then
291 # right (Hadamard) multiplication of x by e_1 picks out the first
292 # component of x; and likewise for the ith basis element e_i.
293 Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
294 for i in xrange(dimension) ]
295
296 return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension)
297
298
299 def eja_ln(dimension, field=QQ):
300 """
301 Return the Jordan algebra corresponding to the Lorentz "ice cream"
302 cone of the given ``dimension``.
303
304 EXAMPLES:
305
306 This multiplication table can be verified by hand::
307
308 sage: J = eja_ln(4)
309 sage: e0,e1,e2,e3 = J.gens()
310 sage: e0*e0
311 e0
312 sage: e0*e1
313 e1
314 sage: e0*e2
315 e2
316 sage: e0*e3
317 e3
318 sage: e1*e2
319 0
320 sage: e1*e3
321 0
322 sage: e2*e3
323 0
324
325 In one dimension, this is the reals under multiplication::
326
327 sage: J1 = eja_ln(1)
328 sage: J2 = eja_rn(1)
329 sage: J1 == J2
330 True
331
332 """
333 Qs = []
334 id_matrix = identity_matrix(field,dimension)
335 for i in xrange(dimension):
336 ei = id_matrix.column(i)
337 Qi = zero_matrix(field,dimension)
338 Qi.set_row(0, ei)
339 Qi.set_column(0, ei)
340 Qi += diagonal_matrix(dimension, [ei[0]]*dimension)
341 # The addition of the diagonal matrix adds an extra ei[0] in the
342 # upper-left corner of the matrix.
343 Qi[0,0] = Qi[0,0] * ~field(2)
344 Qs.append(Qi)
345
346 # The rank of the spin factor algebra is two, UNLESS we're in a
347 # one-dimensional ambient space (the rank is bounded by the
348 # ambient dimension).
349 rank = min(dimension,2)
350 return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=rank)