X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=a0ba1c68bb30bf54383807c52961ad82e4a69f03;hb=cc75c4e093a920c44f1eaaef4a1baa525b5c5727;hp=8f21db7406a2d336a43f6fec3c8bc4438d0536a8;hpb=81bdc4bf9f5ad16f25b5854d271ed8762e4c3ee5;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index 8f21db7..a0ba1c6 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -21,8 +21,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): assume_associative=False, category=None, rank=None, - natural_basis=None, - inner_product=None): + natural_basis=None): n = len(mult_table) mult_table = [b.base_extend(field) for b in mult_table] for b in mult_table: @@ -46,18 +45,17 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): names=names, category=cat, rank=rank, - natural_basis=natural_basis, - inner_product=inner_product) + natural_basis=natural_basis) - def __init__(self, field, + def __init__(self, + field, mult_table, names='e', assume_associative=False, category=None, rank=None, - natural_basis=None, - inner_product=None): + natural_basis=None): """ EXAMPLES: @@ -73,7 +71,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ self._rank = rank self._natural_basis = natural_basis - self._inner_product = inner_product + self._multiplication_table = mult_table fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, mult_table, @@ -89,11 +87,109 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return fmt.format(self.degree(), self.base_ring()) + + @cached_method + def _charpoly_coeff(self, i): + """ + Return the coefficient polynomial "a_{i}" of this algebra's + general characteristic polynomial. + + Having this be a separate cached method lets us compute and + store the trace/determinant (a_{r-1} and a_{0} respectively) + separate from the entire characteristic polynomial. + """ + (A_of_x, x) = self._charpoly_matrix() + R = A_of_x.base_ring() + A_cols = A_of_x.columns() + A_cols[i] = (x**self.rank()).vector() + numerator = column_matrix(A_of_x.base_ring(), A_cols).det() + denominator = A_of_x.det() + + # We're relying on the theory here to ensure that each a_i is + # indeed back in R, and the added negative signs are to make + # the whole charpoly expression sum to zero. + return R(-numerator/denominator) + + + @cached_method + def _charpoly_matrix(self): + """ + Compute the matrix whose entries A_ij are polynomials in + X1,...,XN. This same matrix is used in more than one method and + it's not so fast to construct. + """ + r = self.rank() + n = self.dimension() + + # Construct a new algebra over a multivariate polynomial ring... + names = ['X' + str(i) for i in range(1,n+1)] + R = PolynomialRing(self.base_ring(), names) + J = FiniteDimensionalEuclideanJordanAlgebra(R, + self._multiplication_table, + rank=r) + + idmat = identity_matrix(J.base_ring(), n) + + x = J(vector(R, R.gens())) + l1 = [column_matrix((x**k).vector()) for k in range(r)] + l2 = [idmat.column(k-1).column() for k in range(r+1, n+1)] + A_of_x = block_matrix(R, 1, n, (l1 + l2)) + return (A_of_x, x) + + + @cached_method + def characteristic_polynomial(self): + """ + EXAMPLES: + + The characteristic polynomial in the spin algebra is given in + Alizadeh, Example 11.11:: + + sage: J = JordanSpinEJA(3) + sage: p = J.characteristic_polynomial(); p + X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2 + sage: xvec = J.one().vector() + sage: p(*xvec) + t^2 - 2*t + 1 + + """ + r = self.rank() + n = self.dimension() + + # The list of coefficient polynomials a_1, a_2, ..., a_n. + a = [ self._charpoly_coeff(i) for i in range(n) ] + + # We go to a bit of trouble here to reorder the + # indeterminates, so that it's easier to evaluate the + # characteristic polynomial at x's coordinates and get back + # something in terms of t, which is what we want. + R = a[0].parent() + S = PolynomialRing(self.base_ring(),'t') + t = S.gen(0) + S = PolynomialRing(S, R.variable_names()) + t = S(t) + + # Note: all entries past the rth should be zero. The + # coefficient of the highest power (x^r) is 1, but it doesn't + # appear in the solution vector which contains coefficients + # for the other powers (to make them sum to x^r). + if (r < n): + a[r] = 1 # corresponds to x^r + else: + # When the rank is equal to the dimension, trying to + # assign a[r] goes out-of-bounds. + a.append(1) # corresponds to x^r + + return sum( a[k]*(t**k) for k in range(len(a)) ) + + 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. + Defaults to the trace inner product, but can be overridden by + subclasses if they are sure that the necessary properties are + satisfied. EXAMPLES: @@ -111,10 +207,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ 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) + return x.trace_inner_product(y) def natural_basis(self): @@ -132,7 +225,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = RealSymmetricSimpleEJA(2) + sage: J = RealSymmetricEJA(2) sage: J.basis() Family (e0, e1, e2) sage: J.natural_basis() @@ -143,7 +236,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): :: - sage: J = JordanSpinSimpleEJA(2) + sage: J = JordanSpinEJA(2) sage: J.basis() Family (e0, e1) sage: J.natural_basis() @@ -180,14 +273,14 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): The identity in `S^n` is converted to the identity in the EJA:: - sage: J = RealSymmetricSimpleEJA(3) + 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 = RealSymmetricSimpleEJA(3) + sage: J = RealSymmetricEJA(3) sage: A = matrix(QQ,3, lambda i,j: i-j) sage: J(A) Traceback (most recent call last): @@ -270,19 +363,82 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return A( (self.operator_matrix()**(n-1))*self.vector() ) + def apply_univariate_polynomial(self, p): + """ + Apply the univariate polynomial ``p`` to this element. + + A priori, SageMath won't allow us to apply a univariate + polynomial to an element of an EJA, because we don't know + that EJAs are rings (they are usually not associative). Of + course, we know that EJAs are power-associative, so the + operation is ultimately kosher. This function sidesteps + the CAS to get the answer we want and expect. + + EXAMPLES:: + + sage: R = PolynomialRing(QQ, 't') + sage: t = R.gen(0) + sage: p = t^4 - t^3 + 5*t - 2 + sage: J = RealCartesianProductEJA(5) + sage: J.one().apply_univariate_polynomial(p) == 3*J.one() + True + + TESTS: + + We should always get back an element of the algebra:: + + sage: set_random_seed() + sage: p = PolynomialRing(QQ, 't').random_element() + sage: J = random_eja() + sage: x = J.random_element() + sage: x.apply_univariate_polynomial(p) in J + True + + """ + if len(p.variables()) > 1: + raise ValueError("not a univariate polynomial") + P = self.parent() + R = P.base_ring() + # Convert the coeficcients to the parent's base ring, + # because a priori they might live in an (unnecessarily) + # larger ring for which P.sum() would fail below. + cs = [ R(c) for c in p.coefficients(sparse=False) ] + return P.sum( cs[k]*(self**k) for k in range(len(cs)) ) + + def characteristic_polynomial(self): """ - Return my characteristic polynomial (if I'm a regular - element). + Return the characteristic polynomial of this element. + + EXAMPLES: + + The rank of `R^3` is three, and the minimal polynomial of + the identity element is `(t-1)` from which it follows that + the characteristic polynomial should be `(t-1)^3`:: + + sage: J = RealCartesianProductEJA(3) + sage: J.one().characteristic_polynomial() + t^3 - 3*t^2 + 3*t - 1 + + Likewise, the characteristic of the zero element in the + rank-three algebra `R^{n}` should be `t^{3}`:: + + sage: J = RealCartesianProductEJA(3) + sage: J.zero().characteristic_polynomial() + t^3 + + The characteristic polynomial of an element should evaluate + to zero on that element:: + + sage: set_random_seed() + sage: x = RealCartesianProductEJA(3).random_element() + sage: p = x.characteristic_polynomial() + sage: x.apply_univariate_polynomial(p) + 0 - Eventually this should be implemented in terms of the parent - algebra's characteristic polynomial that works for ALL - elements. """ - if self.is_regular(): - return self.minimal_polynomial() - else: - raise NotImplementedError('irregular element') + p = self.parent().characteristic_polynomial() + return p(*self.vector()) def inner_product(self, other): @@ -295,7 +451,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): 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 = JordanSpinSimpleEJA(3) + sage: J = JordanSpinEJA(3) sage: x = vector(QQ,[1,2,3]) sage: y = vector(QQ,[4,5,6]) sage: x.inner_product(y) @@ -308,7 +464,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): so the inner product of the identity matrix with itself should be the `n`:: - sage: J = RealSymmetricSimpleEJA(3) + sage: J = RealSymmetricEJA(3) sage: J.one().inner_product(J.one()) 3 @@ -317,7 +473,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): part because the product of Hermitian matrices may not be Hermitian:: - sage: J = ComplexHermitianSimpleEJA(3) + 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 @@ -384,12 +546,12 @@ 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() @@ -418,7 +580,7 @@ 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() @@ -490,8 +652,36 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): We can't use the superclass method because it relies on the algebra being associative. + + ALGORITHM: + + The usual way to do this is to check if the determinant is + zero, but we need the characteristic polynomial for the + determinant. The minimal polynomial is a lot easier to get, + so we use Corollary 2 in Chapter V of Koecher to check + whether or not the paren't algebra's zero element is a root + of this element's minimal polynomial. + + TESTS: + + The identity element is always invertible:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.one().is_invertible() + True + + The zero element is never invertible:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.zero().is_invertible() + False + """ - return not self.det().is_zero() + zero = self.parent().zero() + p = self.minimal_polynomial() + return not (p(zero) == zero) def is_nilpotent(self): @@ -548,7 +738,7 @@ 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 @@ -573,7 +763,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(4) + sage: J = JordanSpinEJA(4) sage: J.one().degree() 1 sage: e0,e1,e2,e3 = J.gens() @@ -585,7 +775,7 @@ 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 @@ -596,14 +786,30 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): def minimal_polynomial(self): """ - EXAMPLES:: + Return the minimal polynomial of this element, + as a function of the variable `t`. + + ALGORITHM: + + We restrict ourselves to the associative subalgebra + generated by this element, and then return the minimal + polynomial of this element's operator matrix (in that + subalgebra). This works by Baes Proposition 2.3.16. + + TESTS: + + The minimal polynomial of the identity and zero elements are + always the same:: sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.degree() == x.minimal_polynomial().degree() - True + sage: J = random_eja() + sage: J.one().minimal_polynomial() + t - 1 + sage: J.zero().minimal_polynomial() + t - :: + The degree of an element is (by one definition) the degree + of its minimal polynomial:: sage: set_random_seed() sage: x = random_eja().random_element() @@ -617,38 +823,38 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: n = ZZ.random_element(2,10) - sage: J = JordanSpinSimpleEJA(n) + 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: t = PolynomialRing(J.base_ring(),'t').gen(0) + sage: expected = t^2 - 2*y0*t + (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 + The minimal polynomial should always kill its element:: - 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())) + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: p = x.minimal_polynomial() + sage: x.apply_univariate_polynomial(p) + 0 - # Recursive call, but should work since elt lives in an - # associative algebra. - return elt.minimal_polynomial() + """ + 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())) + + # We get back a symbolic polynomial in 'x' but want a real + # polynomial in 't'. + p_of_x = elt.operator_matrix().minimal_polynomial() + return p_of_x.change_variable_name('t') def natural_representation(self): @@ -663,7 +869,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = ComplexHermitianSimpleEJA(3) + sage: J = ComplexHermitianEJA(3) sage: J.one() e0 + e5 + e8 sage: J.one().natural_representation() @@ -674,6 +880,25 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): [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() @@ -758,7 +983,7 @@ 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] @@ -832,7 +1057,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): # The dimension of the subalgebra can't be greater than # the big algebra, so just put everything into a list # and let span() get rid of the excess. - V = self.vector().parent() + # + # We do the extra ambient_vector_space() in case we're messing + # with polynomials and the direct parent is a module. + V = self.vector().parent().ambient_vector_space() return V.span( (self**d).vector() for d in xrange(V.dimension()) ) @@ -901,11 +1129,11 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): TESTS:: sage: set_random_seed() - sage: J = eja_rn(5) + sage: J = RealCartesianProductEJA(5) 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 @@ -961,18 +1189,35 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(3) - sage: e0,e1,e2 = J.gens() - sage: x = e0 + e1 + e2 + sage: J = JordanSpinEJA(3) + sage: x = sum(J.gens()) sage: x.trace() 2 + :: + + sage: J = RealCartesianProductEJA(5) + sage: J.one().trace() + 5 + + TESTS: + + The trace of an element is a real number:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.random_element().trace() in J.base_ring() + True + """ - cs = self.characteristic_polynomial().coefficients(sparse=False) - if len(cs) >= 2: - return -1*cs[-2] - else: - raise ValueError('charpoly had fewer than 2 coefficients') + P = self.parent() + r = P.rank() + p = P._charpoly_coeff(r-1) + # The _charpoly_coeff function already adds the factor of + # -1 to ensure that _charpoly_coeff(r-1) is really what + # appears in front of t^{r-1} in the charpoly. However, + # we want the negative of THAT for the trace. + return -p(*self.vector()) def trace_inner_product(self, other): @@ -985,16 +1230,20 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return (self*other).trace() -def eja_rn(dimension, field=QQ): +class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): """ Return the Euclidean Jordan Algebra corresponding to the set `R^n` under the Hadamard product. + Note: this is nothing more than the Cartesian product of ``n`` + copies of the spin algebra. Once Cartesian product algebras + are implemented, this can go. + EXAMPLES: This multiplication table can be verified by hand:: - sage: J = eja_rn(3) + sage: J = RealCartesianProductEJA(3) sage: e0,e1,e2 = J.gens() sage: e0*e0 e0 @@ -1010,19 +1259,21 @@ def eja_rn(dimension, field=QQ): e2 """ - # The FiniteDimensionalAlgebra constructor takes a list of - # matrices, the ith representing right multiplication by the ith - # basis element in the vector space. So if e_1 = (1,0,0), then - # right (Hadamard) multiplication of x by e_1 picks out the first - # component of x; and likewise for the ith basis element e_i. - Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i)) - for i in xrange(dimension) ] - - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=dimension, - inner_product=_usual_ip) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + # The FiniteDimensionalAlgebra constructor takes a list of + # matrices, the ith representing right multiplication by the ith + # basis element in the vector space. So if e_1 = (1,0,0), then + # right (Hadamard) multiplication of x by e_1 picks out the first + # component of x; and likewise for the ith basis element e_i. + Qs = [ matrix(field, n, n, lambda k,j: 1*(k == j == i)) + for i in xrange(n) ] + + fdeja = super(RealCartesianProductEJA, cls) + return fdeja.__classcall_private__(cls, field, Qs, rank=n) + def inner_product(self, x, y): + return _usual_ip(x,y) def random_eja(): @@ -1042,6 +1293,12 @@ 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. @@ -1051,11 +1308,17 @@ def random_eja(): Euclidean Jordan algebra of degree... """ - n = ZZ.random_element(1,5) - constructor = choice([eja_rn, - JordanSpinSimpleEJA, - RealSymmetricSimpleEJA, - ComplexHermitianSimpleEJA]) + + # The max_n component lets us choose different upper bounds on the + # value "n" that gets passed to the constructor. This is needed + # because e.g. R^{10} is reasonable to test, while the Hermitian + # 10-by-10 quaternion matrices are not. + (constructor, max_n) = choice([(RealCartesianProductEJA, 6), + (JordanSpinEJA, 6), + (RealSymmetricEJA, 5), + (ComplexHermitianEJA, 4), + (QuaternionHermitianEJA, 3)]) + n = ZZ.random_element(1, max_n) return constructor(n, field=QQ) @@ -1116,6 +1379,48 @@ def _complex_hermitian_basis(n, field=QQ): 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()) @@ -1190,6 +1495,20 @@ def _embed_complex_matrix(M): [ 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: @@ -1216,8 +1535,19 @@ 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: @@ -1235,14 +1565,123 @@ 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 ValueError('bad real submatrix') + raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0]: - raise ValueError('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 _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()) @@ -1256,7 +1695,7 @@ def _matrix_ip(X,Y): return (X_mat*Y_mat).trace() -def RealSymmetricSimpleEJA(n, field=QQ): +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 @@ -1264,7 +1703,7 @@ def RealSymmetricSimpleEJA(n, field=QQ): EXAMPLES:: - sage: J = RealSymmetricSimpleEJA(2) + sage: J = RealSymmetricEJA(2) sage: e0, e1, e2 = J.gens() sage: e0*e0 e0 @@ -1279,22 +1718,44 @@ 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, T) = _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) - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=n, - natural_basis=T, - inner_product=_matrix_ip) + fdeja = super(RealSymmetricEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) + + 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, @@ -1307,47 +1768,110 @@ 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, T) = _multiplication_table_from_matrix_basis(S) + @staticmethod + def __classcall_private__(cls, n, field=QQ): + S = _complex_hermitian_basis(n) + (Qs, T) = _multiplication_table_from_matrix_basis(S) - # 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. - ip = lambda X,Y: _matrix_ip(X,Y)/2 + fdeja = super(ComplexHermitianEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=n, - natural_basis=T, - inner_product=ip) + 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) -def JordanSpinSimpleEJA(n, field=QQ): + 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 + + +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 = @@ -1358,7 +1882,7 @@ 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 @@ -1375,31 +1899,27 @@ def JordanSpinSimpleEJA(n, field=QQ): sage: e2*e3 0 - In one dimension, this is the reals under multiplication:: - - sage: J1 = JordanSpinSimpleEJA(1) - sage: J2 = eja_rn(1) - sage: J1 == J2 - True - """ - 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), - inner_product=_usual_ip) + @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) + + # 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). + fdeja = super(JordanSpinEJA, cls) + return fdeja.__classcall_private__(cls, field, Qs, rank=min(n,2)) + + def inner_product(self, x, y): + return _usual_ip(x,y)