X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=32481621975ab3aabc217eae5ef72d9abf191980;hb=2cfb1e2864c14542d101334bac962000f85e017d;hp=c0b7787a535b7754c446fcb0046c300b21695579;hpb=6ca4fb6c40e9e8f65eec8cafd0044992fae79f02;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index c0b7787..3248162 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -20,7 +20,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): names='e', assume_associative=False, category=None, - rank=None): + rank=None, + natural_basis=None, + inner_product=None): n = len(mult_table) mult_table = [b.base_extend(field) for b in mult_table] for b in mult_table: @@ -43,15 +45,20 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): assume_associative=assume_associative, names=names, category=cat, - rank=rank) + rank=rank, + natural_basis=natural_basis, + inner_product=inner_product) - def __init__(self, field, + def __init__(self, + field, mult_table, names='e', assume_associative=False, category=None, - rank=None): + rank=None, + natural_basis=None, + inner_product=None): """ EXAMPLES: @@ -66,6 +73,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ self._rank = rank + self._natural_basis = natural_basis + self._inner_product = inner_product fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, mult_table, @@ -80,6 +89,77 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): fmt = "Euclidean Jordan algebra of degree {} over {}" return fmt.format(self.degree(), self.base_ring()) + + def inner_product(self, x, y): + """ + The inner product associated with this Euclidean Jordan algebra. + + Will default to the trace inner product if nothing else. + + EXAMPLES: + + The inner product must satisfy its axiom for this algebra to truly + be a Euclidean Jordan Algebra:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: (x*y).inner_product(z) == y.inner_product(x*z) + True + + """ + if (not x in self) or (not y in self): + raise TypeError("arguments must live in this algebra") + if self._inner_product is None: + return x.trace_inner_product(y) + else: + return self._inner_product(x,y) + + + def natural_basis(self): + """ + Return a more-natural representation of this algebra's basis. + + Every finite-dimensional Euclidean Jordan Algebra is a direct + sum of five simple algebras, four of which comprise Hermitian + matrices. This method returns the original "natural" basis + for our underlying vector space. (Typically, the natural basis + is used to construct the multiplication table in the first place.) + + Note that this will always return a matrix. The standard basis + in `R^n` will be returned as `n`-by-`1` column matrices. + + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: J.basis() + Family (e0, e1, e2) + sage: J.natural_basis() + ( + [1 0] [0 1] [0 0] + [0 0], [1 0], [0 1] + ) + + :: + + sage: J = JordanSpinEJA(2) + sage: J.basis() + Family (e0, e1) + sage: J.natural_basis() + ( + [1] [0] + [0], [1] + ) + + """ + if self._natural_basis is None: + return tuple( b.vector().column() for b in self.basis() ) + else: + return self._natural_basis + + def rank(self): """ Return the rank of this EJA. @@ -95,6 +175,51 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): An element of a Euclidean Jordan algebra. """ + def __init__(self, A, elt=None): + """ + EXAMPLES: + + The identity in `S^n` is converted to the identity in the EJA:: + + sage: J = RealSymmetricEJA(3) + sage: I = identity_matrix(QQ,3) + sage: J(I) == J.one() + True + + This skew-symmetric matrix can't be represented in the EJA:: + + sage: J = RealSymmetricEJA(3) + sage: A = matrix(QQ,3, lambda i,j: i-j) + sage: J(A) + Traceback (most recent call last): + ... + ArithmeticError: vector is not in free module + + """ + # Goal: if we're given a matrix, and if it lives in our + # parent algebra's "natural ambient space," convert it + # into an algebra element. + # + # The catch is, we make a recursive call after converting + # the given matrix into a vector that lives in the algebra. + # This we need to try the parent class initializer first, + # to avoid recursing forever if we're given something that + # already fits into the algebra, but also happens to live + # in the parent's "natural ambient space" (this happens with + # vectors in R^n). + try: + FiniteDimensionalAlgebraElement.__init__(self, A, elt) + except ValueError: + natural_basis = A.natural_basis() + if elt in natural_basis[0].matrix_space(): + # Thanks for nothing! Matrix spaces aren't vector + # spaces in Sage, so we have to figure out its + # natural-basis coordinates ourselves. + V = VectorSpace(elt.base_ring(), elt.nrows()**2) + W = V.span( _mat2vec(s) for s in natural_basis ) + coords = W.coordinates(_mat2vec(elt)) + FiniteDimensionalAlgebraElement.__init__(self, A, coords) + def __pow__(self, n): """ Return ``self`` raised to the power ``n``. @@ -161,6 +286,68 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): raise NotImplementedError('irregular element') + def inner_product(self, other): + """ + Return the parent algebra's inner product of myself and ``other``. + + EXAMPLES: + + The inner product in the Jordan spin algebra is the usual + inner product on `R^n` (this example only works because the + basis for the Jordan algebra is the standard basis in `R^n`):: + + sage: J = JordanSpinEJA(3) + sage: x = vector(QQ,[1,2,3]) + sage: y = vector(QQ,[4,5,6]) + sage: x.inner_product(y) + 32 + sage: J(x).inner_product(J(y)) + 32 + + The inner product on `S^n` is ` = trace(X*Y)`, where + multiplication is the usual matrix multiplication in `S^n`, + so the inner product of the identity matrix with itself + should be the `n`:: + + sage: J = RealSymmetricEJA(3) + sage: J.one().inner_product(J.one()) + 3 + + Likewise, the inner product on `C^n` is ` = + Re(trace(X*Y))`, where we must necessarily take the real + part because the product of Hermitian matrices may not be + Hermitian:: + + sage: J = ComplexHermitianEJA(3) + sage: J.one().inner_product(J.one()) + 3 + + Ditto for the quaternions:: + + sage: J = QuaternionHermitianEJA(3) + sage: J.one().inner_product(J.one()) + 3 + + TESTS: + + Ensure that we can always compute an inner product, and that + it gives us back a real number:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: x.inner_product(y) in RR + True + + """ + P = self.parent() + if not other in P: + raise TypeError("'other' must live in the same algebra") + + return P.inner_product(self, other) + + def operator_commutes_with(self, other): """ Return whether or not this element operator-commutes @@ -191,7 +378,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ if not other in self.parent(): - raise ArgumentError("'other' must live in the same algebra") + raise TypeError("'other' must live in the same algebra") A = self.operator_matrix() B = other.operator_matrix() @@ -204,12 +391,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(2) + sage: J = JordanSpinEJA(2) sage: e0,e1 = J.gens() sage: x = e0 + e1 sage: x.det() 0 - sage: J = JordanSpinSimpleEJA(3) + sage: J = JordanSpinEJA(3) sage: e0,e1,e2 = J.gens() sage: x = e0 + e1 + e2 sage: x.det() @@ -238,7 +425,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: while x.is_zero(): ....: x = J.random_element() @@ -281,7 +468,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): # TODO: we can do better once the call to is_invertible() # doesn't crash on irregular elements. #if not self.is_invertible(): - # raise ArgumentError('element is not invertible') + # raise ValueError('element is not invertible') # We do this a little different than the usual recursive # call to a finite-dimensional algebra element, because we @@ -368,7 +555,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): The identity element always has degree one, but any element linearly-independent from it is regular:: - sage: J = JordanSpinSimpleEJA(5) + sage: J = JordanSpinEJA(5) sage: J.one().is_regular() False sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity @@ -393,7 +580,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(4) + sage: J = JordanSpinEJA(4) sage: J.one().degree() 1 sage: e0,e1,e2,e3 = J.gens() @@ -405,7 +592,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: x == x.coefficient(0)*J.one() or x.degree() == 2 True @@ -414,7 +601,112 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return self.span_of_powers().dimension() - def matrix(self): + def minimal_polynomial(self): + """ + EXAMPLES:: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: x.degree() == x.minimal_polynomial().degree() + True + + :: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: x.degree() == x.minimal_polynomial().degree() + True + + The minimal polynomial and the characteristic polynomial coincide + and are known (see Alizadeh, Example 11.11) for all elements of + the spin factor algebra that aren't scalar multiples of the + identity:: + + sage: set_random_seed() + sage: n = ZZ.random_element(2,10) + sage: J = JordanSpinEJA(n) + sage: y = J.random_element() + sage: while y == y.coefficient(0)*J.one(): + ....: y = J.random_element() + sage: y0 = y.vector()[0] + sage: y_bar = y.vector()[1:] + sage: actual = y.minimal_polynomial() + sage: x = SR.symbol('x', domain='real') + sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2) + sage: bool(actual == expected) + True + + """ + # The element we're going to call "minimal_polynomial()" on. + # Either myself, interpreted as an element of a finite- + # dimensional algebra, or an element of an associative + # subalgebra. + elt = None + + if self.parent().is_associative(): + elt = FiniteDimensionalAlgebraElement(self.parent(), self) + else: + V = self.span_of_powers() + assoc_subalg = self.subalgebra_generated_by() + # Mis-design warning: the basis used for span_of_powers() + # and subalgebra_generated_by() must be the same, and in + # the same order! + elt = assoc_subalg(V.coordinates(self.vector())) + + # Recursive call, but should work since elt lives in an + # associative algebra. + return elt.minimal_polynomial() + + + def natural_representation(self): + """ + Return a more-natural representation of this element. + + Every finite-dimensional Euclidean Jordan Algebra is a + direct sum of five simple algebras, four of which comprise + Hermitian matrices. This method returns the original + "natural" representation of this element as a Hermitian + matrix, if it has one. If not, you get the usual representation. + + EXAMPLES:: + + sage: J = ComplexHermitianEJA(3) + sage: J.one() + e0 + e5 + e8 + sage: J.one().natural_representation() + [1 0 0 0 0 0] + [0 1 0 0 0 0] + [0 0 1 0 0 0] + [0 0 0 1 0 0] + [0 0 0 0 1 0] + [0 0 0 0 0 1] + + :: + + sage: J = QuaternionHermitianEJA(3) + sage: J.one() + e0 + e9 + e14 + sage: J.one().natural_representation() + [1 0 0 0 0 0 0 0 0 0 0 0] + [0 1 0 0 0 0 0 0 0 0 0 0] + [0 0 1 0 0 0 0 0 0 0 0 0] + [0 0 0 1 0 0 0 0 0 0 0 0] + [0 0 0 0 1 0 0 0 0 0 0 0] + [0 0 0 0 0 1 0 0 0 0 0 0] + [0 0 0 0 0 0 1 0 0 0 0 0] + [0 0 0 0 0 0 0 1 0 0 0 0] + [0 0 0 0 0 0 0 0 1 0 0 0] + [0 0 0 0 0 0 0 0 0 1 0 0] + [0 0 0 0 0 0 0 0 0 0 1 0] + [0 0 0 0 0 0 0 0 0 0 0 1] + + """ + B = self.parent().natural_basis() + W = B[0].matrix_space() + return W.linear_combination(zip(self.vector(), B)) + + + def operator_matrix(self): """ Return the matrix that represents left- (or right-) multiplication by this element in the parent algebra. @@ -480,71 +772,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self) return fda_elt.matrix().transpose() - # - # The plan is to eventually phase out "matrix()", which sounds - # too much like "matrix_representation()", in favor of the more- - # accurate "operator_matrix()". But we need to override matrix() - # to keep parent class methods happy in the meantime. - # - operator_matrix = matrix - - - def minimal_polynomial(self): - """ - EXAMPLES:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.degree() == x.minimal_polynomial().degree() - True - - :: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.degree() == x.minimal_polynomial().degree() - True - - The minimal polynomial and the characteristic polynomial coincide - and are known (see Alizadeh, Example 11.11) for all elements of - the spin factor algebra that aren't scalar multiples of the - identity:: - - sage: set_random_seed() - sage: n = ZZ.random_element(2,10) - sage: J = JordanSpinSimpleEJA(n) - sage: y = J.random_element() - sage: while y == y.coefficient(0)*J.one(): - ....: y = J.random_element() - sage: y0 = y.vector()[0] - sage: y_bar = y.vector()[1:] - sage: actual = y.minimal_polynomial() - sage: x = SR.symbol('x', domain='real') - sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2) - sage: bool(actual == expected) - True - - """ - # The element we're going to call "minimal_polynomial()" on. - # Either myself, interpreted as an element of a finite- - # dimensional algebra, or an element of an associative - # subalgebra. - elt = None - - if self.parent().is_associative(): - elt = FiniteDimensionalAlgebraElement(self.parent(), self) - else: - V = self.span_of_powers() - assoc_subalg = self.subalgebra_generated_by() - # Mis-design warning: the basis used for span_of_powers() - # and subalgebra_generated_by() must be the same, and in - # the same order! - elt = assoc_subalg(V.coordinates(self.vector())) - - # Recursive call, but should work since elt lives in an - # associative algebra. - return elt.minimal_polynomial() - def quadratic_representation(self, other=None): """ @@ -557,7 +784,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = JordanSpinEJA(n) sage: x = J.random_element() sage: x_vec = x.vector() sage: x0 = x_vec[0] @@ -616,7 +843,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): if other is None: other=self elif not other in self.parent(): - raise ArgumentError("'other' must live in the same algebra") + raise TypeError("'other' must live in the same algebra") L = self.operator_matrix() M = other.operator_matrix() @@ -704,7 +931,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True - sage: J = JordanSpinSimpleEJA(5) + sage: J = JordanSpinEJA(5) sage: c = J.random_element().subalgebra_idempotent() sage: c^2 == c True @@ -760,7 +987,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(3) + sage: J = JordanSpinEJA(3) sage: e0,e1,e2 = J.gens() sage: x = e0 + e1 + e2 sage: x.trace() @@ -779,7 +1006,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): Return the trace inner product of myself and ``other``. """ if not other in self.parent(): - raise ArgumentError("'other' must live in the same algebra") + raise TypeError("'other' must live in the same algebra") return (self*other).trace() @@ -817,7 +1044,10 @@ def eja_rn(dimension, field=QQ): Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i)) for i in xrange(dimension) ] - return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension) + return FiniteDimensionalEuclideanJordanAlgebra(field, + Qs, + rank=dimension, + inner_product=_usual_ip) @@ -838,6 +1068,12 @@ def random_eja(): * The ``n``-by-``n`` rational symmetric matrices with the symmetric product. + * The ``n``-by-``n`` complex-rational Hermitian matrices embedded + in the space of ``2n``-by-``2n`` real symmetric matrices. + + * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded + in the space of ``4n``-by-``4n`` real symmetric matrices. + Later this might be extended to return Cartesian products of the EJAs above. @@ -849,9 +1085,10 @@ def random_eja(): """ n = ZZ.random_element(1,5) constructor = choice([eja_rn, - JordanSpinSimpleEJA, - RealSymmetricSimpleEJA, - ComplexHermitianSimpleEJA]) + JordanSpinEJA, + RealSymmetricEJA, + ComplexHermitianEJA, + QuaternionHermitianEJA]) return constructor(n, field=QQ) @@ -872,7 +1109,7 @@ def _real_symmetric_basis(n, field=QQ): # Beware, orthogonal but not normalized! Sij = Eij + Eij.transpose() S.append(Sij) - return S + return tuple(S) def _complex_hermitian_basis(n, field=QQ): @@ -909,8 +1146,56 @@ def _complex_hermitian_basis(n, field=QQ): S.append(Sij_real) Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose()) S.append(Sij_imag) - return S + return tuple(S) + + +def _quaternion_hermitian_basis(n, field=QQ): + """ + Returns a basis for the space of quaternion Hermitian n-by-n matrices. + + TESTS:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) ) + True + + """ + Q = QuaternionAlgebra(QQ,-1,-1) + I,J,K = Q.gens() + + # This is like the symmetric case, but we need to be careful: + # + # * We want conjugate-symmetry, not just symmetry. + # * The diagonal will (as a result) be real. + # + S = [] + for i in xrange(n): + for j in xrange(i+1): + Eij = matrix(Q, n, lambda k,l: k==i and l==j) + if i == j: + Sij = _embed_quaternion_matrix(Eij) + S.append(Sij) + else: + # Beware, orthogonal but not normalized! The second, + # third, and fourth ones have a minus because they're + # conjugated. + Sij_real = _embed_quaternion_matrix(Eij + Eij.transpose()) + S.append(Sij_real) + Sij_I = _embed_quaternion_matrix(I*Eij - I*Eij.transpose()) + S.append(Sij_I) + Sij_J = _embed_quaternion_matrix(J*Eij - J*Eij.transpose()) + S.append(Sij_J) + Sij_K = _embed_quaternion_matrix(K*Eij - K*Eij.transpose()) + S.append(Sij_K) + return tuple(S) + + +def _mat2vec(m): + return vector(m.base_ring(), m.list()) +def _vec2mat(v): + return matrix(v.base_ring(), sqrt(v.degree()), v.list()) def _multiplication_table_from_matrix_basis(basis): """ @@ -919,7 +1204,10 @@ def _multiplication_table_from_matrix_basis(basis): multiplication on the right is matrix multiplication. Given a basis for the underlying matrix space, this function returns a multiplication table (obtained by looping through the basis - elements) for an algebra of those matrices. + elements) for an algebra of those matrices. A reordered copy + of the basis is also returned to work around the fact that + the ``span()`` in this function will change the order of the basis + from what we think it is, to... something else. """ # In S^2, for example, we nominally have four coordinates even # though the space is of dimension three only. The vector space V @@ -929,19 +1217,13 @@ def _multiplication_table_from_matrix_basis(basis): field = basis[0].base_ring() dimension = basis[0].nrows() - def mat2vec(m): - return vector(field, m.list()) - - def vec2mat(v): - return matrix(field, dimension, v.list()) - V = VectorSpace(field, dimension**2) - W = V.span( mat2vec(s) for s in basis ) + W = V.span( _mat2vec(s) for s in basis ) # Taking the span above reorders our basis (thanks, jerk!) so we # need to put our "matrix basis" in the same order as the # (reordered) vector basis. - S = [ vec2mat(b) for b in W.basis() ] + S = tuple( _vec2mat(b) for b in W.basis() ) Qs = [] for s in S: @@ -954,12 +1236,12 @@ def _multiplication_table_from_matrix_basis(basis): # why we're computing rows here and not columns. Q_rows = [] for t in S: - this_row = mat2vec((s*t + t*s)/2) + this_row = _mat2vec((s*t + t*s)/2) Q_rows.append(W.coordinates(this_row)) Q = matrix(field, W.dimension(), Q_rows) Qs.append(Q) - return Qs + return (Qs, S) def _embed_complex_matrix(M): @@ -975,24 +1257,38 @@ def _embed_complex_matrix(M): sage: x2 = F(1 + 2*i) sage: x3 = F(-i) sage: x4 = F(6) - sage: M = matrix(F,2,[x1,x2,x3,x4]) + sage: M = matrix(F,2,[[x1,x2],[x3,x4]]) sage: _embed_complex_matrix(M) - [ 4 2| 1 -2] - [-2 4| 2 1] + [ 4 -2| 1 2] + [ 2 4|-2 1] [-----+-----] - [ 0 1| 6 0] - [-1 0| 0 6] + [ 0 -1| 6 0] + [ 1 0| 0 6] + + TESTS: + + Embedding is a homomorphism (isomorphism, in fact):: + + sage: set_random_seed() + sage: n = ZZ.random_element(5) + sage: F = QuadraticField(-1, 'i') + sage: X = random_matrix(F, n) + sage: Y = random_matrix(F, n) + sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y) + sage: expected = _embed_complex_matrix(X*Y) + sage: actual == expected + True """ n = M.nrows() if M.ncols() != n: - raise ArgumentError("the matrix 'M' must be square") + raise ValueError("the matrix 'M' must be square") field = M.base_ring() blocks = [] for z in M.list(): a = z.real() b = z.imag() - blocks.append(matrix(field, 2, [[a,-b],[b,a]])) + blocks.append(matrix(field, 2, [[a,b],[-b,a]])) # We can drop the imaginaries here. return block_matrix(field.base_ring(), n, blocks) @@ -1009,14 +1305,25 @@ def _unembed_complex_matrix(M): ....: [ 9, 10, 11, 12], ....: [-10, 9, -12, 11] ]) sage: _unembed_complex_matrix(A) - [ -2*i + 1 -4*i + 3] - [ -10*i + 9 -12*i + 11] + [ 2*i + 1 4*i + 3] + [ 10*i + 9 12*i + 11] + + TESTS: + + Unembedding is the inverse of embedding:: + + sage: set_random_seed() + sage: F = QuadraticField(-1, 'i') + sage: M = random_matrix(F, 3) + sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M + True + """ n = ZZ(M.nrows()) if M.ncols() != n: - raise ArgumentError("the matrix 'M' must be square") + raise ValueError("the matrix 'M' must be square") if not n.mod(2).is_zero(): - raise ArgumentError("the matrix 'M' must be a complex embedding") + raise ValueError("the matrix 'M' must be a complex embedding") F = QuadraticField(-1, 'i') i = F.gen() @@ -1028,16 +1335,137 @@ def _unembed_complex_matrix(M): for j in xrange(n/2): submat = M[2*k:2*k+2,2*j:2*j+2] if submat[0,0] != submat[1,1]: - raise ArgumentError('bad real submatrix') + raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0]: - raise ArgumentError('bad imag submatrix') - z = submat[0,0] + submat[1,0]*i + raise ValueError('bad off-diagonal submatrix') + z = submat[0,0] + submat[0,1]*i elements.append(z) return matrix(F, n/2, elements) -def RealSymmetricSimpleEJA(n, field=QQ): +def _embed_quaternion_matrix(M): + """ + Embed the n-by-n quaternion matrix ``M`` into the space of real + matrices of size 4n-by-4n by first sending each quaternion entry + `z = a + bi + cj + dk` to the block-complex matrix + ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into + a real matrix. + + EXAMPLES:: + + sage: Q = QuaternionAlgebra(QQ,-1,-1) + sage: i,j,k = Q.gens() + sage: x = 1 + 2*i + 3*j + 4*k + sage: M = matrix(Q, 1, [[x]]) + sage: _embed_quaternion_matrix(M) + [ 1 2 3 4] + [-2 1 -4 3] + [-3 4 1 -2] + [-4 -3 2 1] + + Embedding is a homomorphism (isomorphism, in fact):: + + sage: set_random_seed() + sage: n = ZZ.random_element(5) + sage: Q = QuaternionAlgebra(QQ,-1,-1) + sage: X = random_matrix(Q, n) + sage: Y = random_matrix(Q, n) + sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y) + sage: expected = _embed_quaternion_matrix(X*Y) + sage: actual == expected + True + + """ + quaternions = M.base_ring() + n = M.nrows() + if M.ncols() != n: + raise ValueError("the matrix 'M' must be square") + + F = QuadraticField(-1, 'i') + i = F.gen() + + blocks = [] + for z in M.list(): + t = z.coefficient_tuple() + a = t[0] + b = t[1] + c = t[2] + d = t[3] + cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i], + [-c + d*i, a - b*i]]) + blocks.append(_embed_complex_matrix(cplx_matrix)) + + # We should have real entries by now, so use the realest field + # we've got for the return value. + return block_matrix(quaternions.base_ring(), n, blocks) + + +def _unembed_quaternion_matrix(M): + """ + The inverse of _embed_quaternion_matrix(). + + EXAMPLES:: + + sage: M = matrix(QQ, [[ 1, 2, 3, 4], + ....: [-2, 1, -4, 3], + ....: [-3, 4, 1, -2], + ....: [-4, -3, 2, 1]]) + sage: _unembed_quaternion_matrix(M) + [1 + 2*i + 3*j + 4*k] + + TESTS: + + Unembedding is the inverse of embedding:: + + sage: set_random_seed() + sage: Q = QuaternionAlgebra(QQ, -1, -1) + sage: M = random_matrix(Q, 3) + sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M + True + + """ + n = ZZ(M.nrows()) + if M.ncols() != n: + raise ValueError("the matrix 'M' must be square") + if not n.mod(4).is_zero(): + raise ValueError("the matrix 'M' must be a complex embedding") + + Q = QuaternionAlgebra(QQ,-1,-1) + i,j,k = Q.gens() + + # Go top-left to bottom-right (reading order), converting every + # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1 + # quaternion block. + elements = [] + for l in xrange(n/4): + for m in xrange(n/4): + submat = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4]) + if submat[0,0] != submat[1,1].conjugate(): + raise ValueError('bad on-diagonal submatrix') + if submat[0,1] != -submat[1,0].conjugate(): + raise ValueError('bad off-diagonal submatrix') + z = submat[0,0].real() + submat[0,0].imag()*i + z += submat[0,1].real()*j + submat[0,1].imag()*k + elements.append(z) + + return matrix(Q, n/4, elements) + + +# The usual inner product on R^n. +def _usual_ip(x,y): + return x.vector().inner_product(y.vector()) + +# The inner product used for the real symmetric simple EJA. +# We keep it as a separate function because e.g. the complex +# algebra uses the same inner product, except divided by 2. +def _matrix_ip(X,Y): + X_mat = X.natural_representation() + Y_mat = Y.natural_representation() + return (X_mat*Y_mat).trace() + + +class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of real symmetric n-by-n matrices, the usual symmetric Jordan product, and the trace inner @@ -1045,7 +1473,7 @@ def RealSymmetricSimpleEJA(n, field=QQ): EXAMPLES:: - sage: J = RealSymmetricSimpleEJA(2) + sage: J = RealSymmetricEJA(2) sage: e0, e1, e2 = J.gens() sage: e0*e0 e0 @@ -1060,18 +1488,44 @@ def RealSymmetricSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricSimpleEJA(n) + sage: J = RealSymmetricEJA(n) sage: J.degree() == (n^2 + n)/2 True + The Jordan multiplication is what we think it is:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = RealSymmetricEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: actual = (x*y).natural_representation() + sage: X = x.natural_representation() + sage: Y = y.natural_representation() + sage: expected = (X*Y + Y*X)/2 + sage: actual == expected + True + sage: J(expected) == x*y + True + """ - S = _real_symmetric_basis(n, field=field) - Qs = _multiplication_table_from_matrix_basis(S) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _real_symmetric_basis(n, field=field) + (Qs, T) = _multiplication_table_from_matrix_basis(S) + + fdeja = super(RealSymmetricEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) - return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=n) + def inner_product(self, x, y): + return _matrix_ip(x,y) -def ComplexHermitianSimpleEJA(n, field=QQ): +class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of complex Hermitian n-by-n matrices over the real numbers, the usual symmetric Jordan product, @@ -1084,33 +1538,110 @@ def ComplexHermitianSimpleEJA(n, field=QQ): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianSimpleEJA(n) + sage: J = ComplexHermitianEJA(n) sage: J.degree() == n^2 True + The Jordan multiplication is what we think it is:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = ComplexHermitianEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: actual = (x*y).natural_representation() + sage: X = x.natural_representation() + sage: Y = y.natural_representation() + sage: expected = (X*Y + Y*X)/2 + sage: actual == expected + True + sage: J(expected) == x*y + True + """ - S = _complex_hermitian_basis(n) - Qs = _multiplication_table_from_matrix_basis(S) - return FiniteDimensionalEuclideanJordanAlgebra(field, Qs, rank=n) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _complex_hermitian_basis(n) + (Qs, T) = _multiplication_table_from_matrix_basis(S) + + fdeja = super(ComplexHermitianEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) + + def inner_product(self, x, y): + # Since a+bi on the diagonal is represented as + # + # a + bi = [ a b ] + # [ -b a ], + # + # we'll double-count the "a" entries if we take the trace of + # the embedding. + return _matrix_ip(x,y)/2 -def QuaternionHermitianSimpleEJA(n): +class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-n simple EJA consisting of self-adjoint n-by-n quaternion matrices, the usual symmetric Jordan product, and the real-part-of-trace inner product. It has dimension `2n^2 - n` over the reals. - """ - pass -def OctonionHermitianSimpleEJA(n): - """ - This shit be crazy. It has dimension 27 over the reals. + TESTS: + + The degree of this algebra is `n^2`:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = QuaternionHermitianEJA(n) + sage: J.degree() == 2*(n^2) - n + True + + The Jordan multiplication is what we think it is:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = QuaternionHermitianEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: actual = (x*y).natural_representation() + sage: X = x.natural_representation() + sage: Y = y.natural_representation() + sage: expected = (X*Y + Y*X)/2 + sage: actual == expected + True + sage: J(expected) == x*y + True + """ - n = 3 - pass + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _quaternion_hermitian_basis(n) + (Qs, T) = _multiplication_table_from_matrix_basis(S) + + fdeja = super(QuaternionHermitianEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) + + def inner_product(self, x, y): + # Since a+bi+cj+dk on the diagonal is represented as + # + # a + bi +cj + dk = [ a b c d] + # [ -b a -d c] + # [ -c d a -b] + # [ -d -c b a], + # + # we'll quadruple-count the "a" entries if we take the trace of + # the embedding. + return _matrix_ip(x,y)/4 -def JordanSpinSimpleEJA(n, field=QQ): + +class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): """ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` with the usual inner product and jordan product ``x*y = @@ -1121,7 +1652,7 @@ def JordanSpinSimpleEJA(n, field=QQ): This multiplication table can be verified by hand:: - sage: J = JordanSpinSimpleEJA(4) + sage: J = JordanSpinEJA(4) sage: e0,e1,e2,e3 = J.gens() sage: e0*e0 e0 @@ -1138,28 +1669,34 @@ def JordanSpinSimpleEJA(n, field=QQ): sage: e2*e3 0 - In one dimension, this is the reals under multiplication:: + """ + @staticmethod + def __classcall_private__(cls, n, field=QQ): + Qs = [] + id_matrix = identity_matrix(field, n) + for i in xrange(n): + ei = id_matrix.column(i) + Qi = zero_matrix(field, n) + Qi.set_row(0, ei) + Qi.set_column(0, ei) + Qi += diagonal_matrix(n, [ei[0]]*n) + # The addition of the diagonal matrix adds an extra ei[0] in the + # upper-left corner of the matrix. + Qi[0,0] = Qi[0,0] * ~field(2) + Qs.append(Qi) + + fdeja = super(JordanSpinEJA, cls) + return fdeja.__classcall_private__(cls, field, Qs) + + def rank(self): + """ + Return the rank of this Jordan Spin Algebra. - sage: J1 = JordanSpinSimpleEJA(1) - sage: J2 = eja_rn(1) - sage: J1 == J2 - True + The rank of the spin algebra is two, unless we're in a + one-dimensional ambient space (because the rank is bounded by + the ambient dimension). + """ + return min(self.dimension(),2) - """ - Qs = [] - id_matrix = identity_matrix(field, n) - for i in xrange(n): - ei = id_matrix.column(i) - Qi = zero_matrix(field, n) - Qi.set_row(0, ei) - Qi.set_column(0, ei) - Qi += diagonal_matrix(n, [ei[0]]*n) - # The addition of the diagonal matrix adds an extra ei[0] in the - # upper-left corner of the matrix. - Qi[0,0] = Qi[0,0] * ~field(2) - Qs.append(Qi) - - # The rank of the spin factor algebra is two, UNLESS we're in a - # one-dimensional ambient space (the rank is bounded by the - # ambient dimension). - return FiniteDimensionalEuclideanJordanAlgebra(field, Qs, rank=min(n,2)) + def inner_product(self, x, y): + return _usual_ip(x,y)