X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=30d04f4a352b7d553470bd456d18d224e2ec5428;hb=31baec0eee0c53b0cfe379c744cdf174aa57ebd9;hp=59e1537854a09f7058498cdd0959bbc82ccbde8d;hpb=3a5ca72aad3917ad670f5b2e918275a50a00e67c;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index 59e1537..30d04f4 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -5,12 +5,405 @@ are used in optimization, and have some additional nice methods beyond what can be supported in a general Jordan Algebra. """ -from sage.categories.magmatic_algebras import MagmaticAlgebras +from sage.categories.finite_dimensional_algebras_with_basis import FiniteDimensionalAlgebrasWithBasis +from sage.categories.morphism import SetMorphism from sage.structure.element import is_Matrix from sage.structure.category_object import normalize_names from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement +from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_morphism import FiniteDimensionalAlgebraMorphism, FiniteDimensionalAlgebraHomset + + +class FiniteDimensionalEuclideanJordanAlgebraHomset(FiniteDimensionalAlgebraHomset): + + def has_coerce_map_from(self, S): + """ + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: H = J.Hom(J) + sage: H.has_coerce_map_from(QQ) + True + + """ + try: + # The Homset classes override has_coerce_map_from() with + # something that crashes when it's given e.g. QQ. + if S.is_subring(self.codomain().base_ring()): + return True + except: + pclass = super(FiniteDimensionalEuclideanJordanAlgebraHomset, self) + return pclass.has_coerce_map_from(S) + + + def _coerce_map_from_(self, S): + """ + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: H = J.Hom(J) + sage: H.coerce(2) + Morphism from Euclidean Jordan algebra of degree 3 over Rational + Field to Euclidean Jordan algebra of degree 3 over Rational Field + given by matrix + [2 0 0] + [0 2 0] + [0 0 2] + + """ + C = self.codomain() + R = C.base_ring() + if S.is_subring(R): + h = S.hom(self.codomain()) + return SetMorphism(Hom(S,C), lambda x: h(x).operator()) + + + def __call__(self, x): + """ + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: H = J.Hom(J) + sage: H(2) + Morphism from Euclidean Jordan algebra of degree 3 over Rational + Field to Euclidean Jordan algebra of degree 3 over Rational Field + given by matrix + [2 0 0] + [0 2 0] + [0 0 2] + + """ + if x in self.base_ring(): + cols = self.domain().dimension() + rows = self.codomain().dimension() + x = x*identity_matrix(self.codomain().base_ring(), rows, cols) + return FiniteDimensionalEuclideanJordanAlgebraMorphism(self, x) + + + def one(self): + """ + Return the identity morphism, but as a member of the right + space (so that we can add it, multiply it, etc.) + """ + cols = self.domain().dimension() + rows = self.codomain().dimension() + mat = identity_matrix(self.base_ring(), rows, cols) + return FiniteDimensionalEuclideanJordanAlgebraMorphism(self, mat) + + + +class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMorphism): + """ + A linear map between two finite-dimensional EJAs. + + This is a very thin wrapper around FiniteDimensionalAlgebraMorphism + that does only a few things: + + 1. Avoids the ``unitary`` and ``check`` arguments to the constructor + that will always be ``False``. This is necessary because these + are homomorphisms with respect to ADDITION, but the SageMath + machinery wants to check that they're homomorphisms with respect + to (Jordan) MULTIPLICATION. That obviously doesn't work. + + 2. Inputs and outputs the underlying matrix with respect to COLUMN + vectors, unlike the parent class. + + 3. Allows us to add, subtract, negate, multiply (compose), and + invert morphisms in the obvious way. + + If this seems a bit heavyweight, it is. I would have been happy to + use a the ring morphism that underlies the finite-dimensional + algebra morphism, but they don't seem to be callable on elements of + our EJA, and you can't add/multiply/etc. them. + """ + def _add_(self, other): + """ + Add two EJA morphisms in the obvious way. + + EXAMPLES:: + + sage: J = RealSymmetricEJA(3) + sage: x = J.zero() + sage: y = J.one() + sage: x.operator() + y.operator() + Morphism from Euclidean Jordan algebra of degree 6 over Rational + Field to Euclidean Jordan algebra of degree 6 over Rational Field + given by matrix + [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] + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: (x.operator() + y.operator()) in J.Hom(J) + True + + """ + P = self.parent() + if not other in P: + raise ValueError("summands must live in the same space") + + return FiniteDimensionalEuclideanJordanAlgebraMorphism( + P, + self.matrix() + other.matrix() ) + + + def __init__(self, parent, f): + FiniteDimensionalAlgebraMorphism.__init__(self, + parent, + f.transpose(), + unitary=False, + check=False) + + + def __invert__(self): + """ + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens())) + sage: x.is_invertible() + True + sage: ~x.operator() + Morphism from Euclidean Jordan algebra of degree 3 over Rational + Field to Euclidean Jordan algebra of degree 3 over Rational Field + given by matrix + [-3/2 2 -1/2] + [ 1 0 0] + [-1/2 0 1/2] + sage: x.operator_matrix().inverse() + [-3/2 2 -1/2] + [ 1 0 0] + [-1/2 0 1/2] + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: not x.is_invertible() or ( + ....: (~x.operator()).matrix() == x.operator_matrix().inverse() ) + True + + """ + A = self.matrix() + if not A.is_invertible(): + raise ValueError("morphism is not invertible") + + P = self.parent() + return FiniteDimensionalEuclideanJordanAlgebraMorphism(self.parent(), + A.inverse()) + + def _lmul_(self, right): + """ + Compose two EJA morphisms using multiplicative notation. + + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: x = J.zero() + sage: y = J.one() + sage: x.operator() * y.operator() + Morphism from Euclidean Jordan algebra of degree 3 over Rational + Field to Euclidean Jordan algebra of degree 3 over Rational Field + given by matrix + [0 0 0] + [0 0 0] + [0 0 0] + + :: + + sage: J = RealSymmetricEJA(2) + sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens())) + sage: x.operator() + Morphism from Euclidean Jordan algebra of degree 3 over Rational + Field to Euclidean Jordan algebra of degree 3 over Rational Field + given by matrix + [ 0 1 0] + [1/2 1 1/2] + [ 0 1 2] + sage: 2*x.operator() + Morphism from Euclidean Jordan algebra of degree 3 over Rational + Field to Euclidean Jordan algebra of degree 3 over Rational Field + given by matrix + [0 2 0] + [1 2 1] + [0 2 4] + sage: x.operator()*2 + Morphism from Euclidean Jordan algebra of degree 3 over Rational + Field to Euclidean Jordan algebra of degree 3 over Rational Field + given by matrix + [0 2 0] + [1 2 1] + [0 2 4] + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: (x.operator() * y.operator()) in J.Hom(J) + True + + """ + try: + # I think the morphism classes break the coercion framework + # somewhere along the way, so we have to do this ourselves. + right = self.parent().coerce(right) + except: + pass + + if not right.codomain() is self.domain(): + raise ValueError("(co)domains must agree for composition") + + return FiniteDimensionalEuclideanJordanAlgebraMorphism( + self.parent(), + self.matrix()*right.matrix() ) + + __mul__ = _lmul_ + + + def __pow__(self, n): + """ + + TESTS:: + + sage: J = JordanSpinEJA(4) + sage: e0,e1,e2,e3 = J.gens() + sage: x = -5/2*e0 + 1/2*e2 + 20*e3 + sage: Qx = x.quadratic_representation() + sage: Qx^0 + Morphism from Euclidean Jordan algebra of degree 4 over Rational + Field to Euclidean Jordan algebra of degree 4 over Rational Field + given by matrix + [1 0 0 0] + [0 1 0 0] + [0 0 1 0] + [0 0 0 1] + sage: (x^0).quadratic_representation() == Qx^0 + True + + """ + if n == 0: + # We get back the stupid identity morphism which doesn't + # live in the right space. + return self.parent().one() + elif n == 1: + return self + else: + return FiniteDimensionalAlgebraMorphism.__pow__(self,n) + + + def _neg_(self): + """ + Negate this morphism. + + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: x = J.one() + sage: -x.operator() + Morphism from Euclidean Jordan algebra of degree 3 over Rational + Field to Euclidean Jordan algebra of degree 3 over Rational Field + given by matrix + [-1 0 0] + [ 0 -1 0] + [ 0 0 -1] + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: -x.operator() in J.Hom(J) + True + + """ + return FiniteDimensionalEuclideanJordanAlgebraMorphism( + self.parent(), + -self.matrix() ) + + + def _repr_(self): + """ + We override only the representation that is shown to the user, + because we want the matrix to be with respect to COLUMN vectors. + + TESTS: + + Ensure that we see the transpose of the underlying matrix object: + + sage: J = RealSymmetricEJA(3) + sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens())) + sage: L = x.operator() + sage: L + Morphism from Euclidean Jordan algebra of degree 6 over Rational + Field to Euclidean Jordan algebra of degree 6 over Rational Field + given by matrix + [ 0 1 2 0 0 0] + [1/2 3/2 2 1/2 1 0] + [ 1 2 5/2 0 1/2 1] + [ 0 1 0 3 4 0] + [ 0 1 1/2 2 4 2] + [ 0 0 2 0 4 5] + sage: L._matrix + [ 0 1/2 1 0 0 0] + [ 1 3/2 2 1 1 0] + [ 2 2 5/2 0 1/2 2] + [ 0 1/2 0 3 2 0] + [ 0 1 1/2 4 4 4] + [ 0 0 1 0 2 5] + + """ + return "Morphism from {} to {} given by matrix\n{}".format( + self.domain(), self.codomain(), self.matrix()) + + + def __sub__(self, other): + """ + Subtract one morphism from another using addition and negation. + + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: L1 = J.one().operator() + sage: L1 - L1 + Morphism from Euclidean Jordan algebra of degree 3 over Rational + Field to Euclidean Jordan algebra of degree 3 over Rational + Field given by matrix + [0 0 0] + [0 0 0] + [0 0 0] + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: x.operator() - y.operator() in J.Hom(J) + True + + """ + return self + (-other) + + + def matrix(self): + """ + Return the matrix of this morphism with respect to a left-action + on column vectors. + """ + return FiniteDimensionalAlgebraMorphism.matrix(self).transpose() + class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): @staticmethod @@ -30,7 +423,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): raise ValueError("input is not a multiplication table") mult_table = tuple(mult_table) - cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis() + cat = FiniteDimensionalAlgebrasWithBasis(field) cat.or_subcategory(category) if assume_associative: cat = cat.Associative() @@ -48,7 +441,17 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): natural_basis=natural_basis) - def __init__(self, field, + def _Hom_(self, B, cat): + """ + Construct a homset of ``self`` and ``B``. + """ + return FiniteDimensionalEuclideanJordanAlgebraHomset(self, + B, + category=cat) + + + def __init__(self, + field, mult_table, names='e', assume_associative=False, @@ -70,6 +473,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): """ self._rank = rank self._natural_basis = natural_basis + self._multiplication_table = mult_table fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, mult_table, @@ -85,6 +489,202 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return fmt.format(self.degree(), self.base_ring()) + def _a_regular_element(self): + """ + Guess a regular element. Needed to compute the basis for our + characteristic polynomial coefficients. + """ + gs = self.gens() + z = self.sum( (i+1)*gs[i] for i in range(len(gs)) ) + if not z.is_regular(): + raise ValueError("don't know a regular element") + return z + + + @cached_method + def _charpoly_basis_space(self): + """ + Return the vector space spanned by the basis used in our + characteristic polynomial coefficients. This is used not only to + compute those coefficients, but also any time we need to + evaluate the coefficients (like when we compute the trace or + determinant). + """ + z = self._a_regular_element() + V = z.vector().parent().ambient_vector_space() + V1 = V.span_of_basis( (z**k).vector() for k in range(self.rank()) ) + b = (V1.basis() + V1.complement().basis()) + return V.span_of_basis(b) + + + @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, xr, detA) = self._charpoly_matrix_system() + R = A_of_x.base_ring() + if i >= self.rank(): + # Guaranteed by theory + return R.zero() + + # Danger: the in-place modification is done for performance + # reasons (reconstructing a matrix with huge polynomial + # entries is slow), but I don't know how cached_method works, + # so it's highly possible that we're modifying some global + # list variable by reference, here. In other words, you + # probably shouldn't call this method twice on the same + # algebra, at the same time, in two threads + Ai_orig = A_of_x.column(i) + A_of_x.set_column(i,xr) + numerator = A_of_x.det() + A_of_x.set_column(i,Ai_orig) + + # 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/detA) + + + @cached_method + def _charpoly_matrix_system(self): + """ + Compute the matrix whose entries A_ij are polynomials in + X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector + corresponding to `x^r` and the determinent of the matrix A = + [A_ij]. In other words, all of the fixed (cachable) data needed + to compute the coefficients of the characteristic polynomial. + """ + 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) + + W = self._charpoly_basis_space() + W = W.change_ring(R.fraction_field()) + + # Starting with the standard coordinates x = (X1,X2,...,Xn) + # and then converting the entries to W-coordinates allows us + # to pass in the standard coordinates to the charpoly and get + # back the right answer. Specifically, with x = (X1,X2,...,Xn), + # we have + # + # W.coordinates(x^2) eval'd at (standard z-coords) + # = + # W-coords of (z^2) + # = + # W-coords of (standard coords of x^2 eval'd at std-coords of z) + # + # We want the middle equivalent thing in our matrix, but use + # the first equivalent thing instead so that we can pass in + # standard coordinates. + x = J(vector(R, R.gens())) + l1 = [column_matrix(W.coordinates((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)) + xr = W.coordinates((x**r).vector()) + return (A_of_x, x, xr, A_of_x.det()) + + + @cached_method + def characteristic_polynomial(self): + """ + + .. WARNING:: + + This implementation doesn't guarantee that the polynomial + denominator in the coefficients is not identically zero, so + theoretically it could crash. The way that this is handled + in e.g. Faraut and Koranyi is to use a basis that guarantees + the denominator is non-zero. But, doing so requires knowledge + of at least one regular element, and we don't even know how + to do that. The trade-off is that, if we use the standard basis, + the resulting polynomial will accept the "usual" coordinates. In + other words, we don't have to do a change of basis before e.g. + computing the trace or determinant. + + 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. + + Defaults to the trace inner product, but can be overridden by + subclasses if they are sure that the necessary properties are + satisfied. + + 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") + return x.trace_inner_product(y) + + def natural_basis(self): """ Return a more-natural representation of this algebra's basis. @@ -100,7 +700,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = RealSymmetricSimpleEJA(2) + sage: J = RealSymmetricEJA(2) sage: J.basis() Family (e0, e1, e2) sage: J.natural_basis() @@ -111,7 +711,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): :: - sage: J = JordanSpinSimpleEJA(2) + sage: J = JordanSpinEJA(2) sage: J.basis() Family (e0, e1) sage: J.natural_basis() @@ -142,6 +742,61 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): An element of a Euclidean Jordan algebra. """ + def __dir__(self): + """ + Oh man, I should not be doing this. This hides the "disabled" + methods ``left_matrix`` and ``matrix`` from introspection; + in particular it removes them from tab-completion. + """ + return filter(lambda s: s not in ['left_matrix', 'matrix'], + dir(self.__class__) ) + + + 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``. @@ -193,19 +848,144 @@ 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 + + """ + p = self.parent().characteristic_polynomial() + return p(*self.vector()) + - Eventually this should be implemented in terms of the parent - algebra's characteristic polynomial that works for ALL - elements. + def inner_product(self, other): """ - if self.is_regular(): - return self.minimal_polynomial() - else: - raise NotImplementedError('irregular element') + 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): @@ -238,7 +1018,7 @@ 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() @@ -251,32 +1031,49 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(2) + sage: J = JordanSpinEJA(2) sage: e0,e1 = J.gens() - sage: x = e0 + e1 + sage: x = sum( J.gens() ) 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 = sum( J.gens() ) sage: x.det() -1 + TESTS: + + An element is invertible if and only if its determinant is + non-zero:: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: x.is_invertible() == (x.det() != 0) + True + """ - cs = self.characteristic_polynomial().coefficients(sparse=False) - r = len(cs) - 1 - if r >= 0: - return cs[0] * (-1)**r - else: - raise ValueError('charpoly had no coefficients') + P = self.parent() + r = P.rank() + p = P._charpoly_coeff(0) + # The _charpoly_coeff function already adds the factor of + # -1 to ensure that _charpoly_coeff(0) is really what + # appears in front of t^{0} in the charpoly. However, + # we want (-1)^r times THAT for the determinant. + return ((-1)**r)*p(*self.vector()) def inverse(self): """ Return the Jordan-multiplicative inverse of this element. - We can't use the superclass method because it relies on the - algebra being associative. + ALGORITHM: + + We appeal to the quadratic representation as in Koecher's + Theorem 12 in Chapter III, Section 5. EXAMPLES: @@ -285,14 +1082,14 @@ 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(): + sage: while not x.is_invertible(): ....: x = J.random_element() sage: x_vec = x.vector() sage: x0 = x_vec[0] sage: x_bar = x_vec[1:] - sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar)) + sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar)) sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list()) sage: x_inverse = coeff*inv_vec sage: x.inverse() == J(x_inverse) @@ -307,48 +1104,35 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: J.one().inverse() == J.one() True - If an element has an inverse, it acts like one. TODO: this - can be a lot less ugly once ``is_invertible`` doesn't crash - on irregular elements:: + If an element has an inverse, it acts like one:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() - sage: try: - ....: x.inverse()*x == J.one() - ....: except: - ....: True + sage: (not x.is_invertible()) or (x.inverse()*x == J.one()) True - """ - if self.parent().is_associative(): - elt = FiniteDimensionalAlgebraElement(self.parent(), self) - return elt.inverse() + The inverse of the inverse is what we started with:: - # 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') + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: (not x.is_invertible()) or (x.inverse().inverse() == x) + True - # We do this a little different than the usual recursive - # call to a finite-dimensional algebra element, because we - # wind up with an inverse that lives in the subalgebra and - # we need information about the parent to convert it back. - 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())) + The zero element is never invertible:: - # This will be in the subalgebra's coordinates... - fda_elt = FiniteDimensionalAlgebraElement(assoc_subalg, elt) - subalg_inverse = fda_elt.inverse() + sage: set_random_seed() + sage: J = random_eja().zero().inverse() + Traceback (most recent call last): + ... + ValueError: element is not invertible + + """ + if not self.is_invertible(): + raise ValueError("element is not invertible") - # So we have to convert back... - basis = [ self.parent(v) for v in V.basis() ] - pairs = zip(subalg_inverse.vector(), basis) - return self.parent().linear_combination(pairs) + return (~self.quadratic_representation())(self) def is_invertible(self): @@ -357,8 +1141,36 @@ 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): @@ -415,7 +1227,7 @@ 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 @@ -440,7 +1252,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = JordanSpinSimpleEJA(4) + sage: J = JordanSpinEJA(4) sage: J.one().degree() 1 sage: e0,e1,e2,e3 = J.gens() @@ -452,7 +1264,7 @@ 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 @@ -461,72 +1273,87 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return self.span_of_powers().dimension() + def left_matrix(self): + """ + Our parent class defines ``left_matrix`` and ``matrix`` + methods whose names are misleading. We don't want them. + """ + raise NotImplementedError("use operator_matrix() instead") - def operator_matrix(self): + matrix = left_matrix + + + def minimal_polynomial(self): """ - Return the matrix that represents left- (or right-) - multiplication by this element in the parent algebra. + Return the minimal polynomial of this element, + as a function of the variable `t`. - We have to override this because the superclass method - returns a matrix that acts on row vectors (that is, on - the right). + ALGORITHM: - EXAMPLES: + 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. - Test the first polarization identity from my notes, Koecher Chapter - III, or from Baes (2.3):: + TESTS: + + The minimal polynomial of the identity and zero elements are + always the same:: sage: set_random_seed() sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: Lx = x.operator_matrix() - sage: Ly = y.operator_matrix() - sage: Lxx = (x*x).operator_matrix() - sage: Lxy = (x*y).operator_matrix() - sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly) - True + sage: J.one().minimal_polynomial() + t - 1 + sage: J.zero().minimal_polynomial() + t - Test the second polarization identity from my notes or from - Baes (2.4):: + The degree of an element is (by one definition) the degree + of its minimal polynomial:: sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() - sage: Lx = x.operator_matrix() - sage: Ly = y.operator_matrix() - sage: Lz = z.operator_matrix() - sage: Lzy = (z*y).operator_matrix() - sage: Lxy = (x*y).operator_matrix() - sage: Lxz = (x*z).operator_matrix() - sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly) + sage: x = random_eja().random_element() + sage: x.degree() == x.minimal_polynomial().degree() True - Test the third polarization identity from my notes or from - Baes (2.5):: + 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: J = random_eja() - sage: u = J.random_element() + sage: n = ZZ.random_element(2,10) + sage: J = JordanSpinEJA(n) sage: y = J.random_element() - sage: z = J.random_element() - sage: Lu = u.operator_matrix() - sage: Ly = y.operator_matrix() - sage: Lz = z.operator_matrix() - sage: Lzy = (z*y).operator_matrix() - sage: Luy = (u*y).operator_matrix() - sage: Luz = (u*z).operator_matrix() - sage: Luyz = (u*(y*z)).operator_matrix() - sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz - sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly - sage: bool(lhs == rhs) + 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: 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 minimal polynomial should always kill its element:: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: p = x.minimal_polynomial() + sage: x.apply_univariate_polynomial(p) + 0 + """ - fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self) - return fda_elt.matrix().transpose() + 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): @@ -541,7 +1368,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): EXAMPLES:: - sage: J = ComplexHermitianSimpleEJA(3) + sage: J = ComplexHermitianEJA(3) sage: J.one() e0 + e5 + e8 sage: J.one().natural_representation() @@ -552,67 +1379,119 @@ 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() return W.linear_combination(zip(self.vector(), B)) - def minimal_polynomial(self): + def operator(self): """ - EXAMPLES:: + Return the left-multiplication-by-this-element + operator on the ambient algebra. + + TESTS:: sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.degree() == x.minimal_polynomial().degree() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: x.operator()(y) == x*y + True + sage: y.operator()(x) == x*y True - :: + """ + P = self.parent() + return FiniteDimensionalEuclideanJordanAlgebraMorphism( + Hom(P,P), + self.operator_matrix() ) + + + + def operator_matrix(self): + """ + Return the matrix that represents left- (or right-) + multiplication by this element in the parent algebra. + + We implement this ourselves to work around the fact that + our parent class represents everything with row vectors. + + EXAMPLES: + + Test the first polarization identity from my notes, Koecher Chapter + III, or from Baes (2.3):: sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.degree() == x.minimal_polynomial().degree() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: Lx = x.operator_matrix() + sage: Ly = y.operator_matrix() + sage: Lxx = (x*x).operator_matrix() + sage: Lxy = (x*y).operator_matrix() + sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly) + True + + Test the second polarization identity from my notes or from + Baes (2.4):: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: Lx = x.operator_matrix() + sage: Ly = y.operator_matrix() + sage: Lz = z.operator_matrix() + sage: Lzy = (z*y).operator_matrix() + sage: Lxy = (x*y).operator_matrix() + sage: Lxz = (x*z).operator_matrix() + sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly) 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:: + Test the third polarization identity from my notes or from + Baes (2.5):: sage: set_random_seed() - sage: n = ZZ.random_element(2,10) - sage: J = JordanSpinSimpleEJA(n) + sage: J = random_eja() + sage: u = J.random_element() 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) + sage: z = J.random_element() + sage: Lu = u.operator_matrix() + sage: Ly = y.operator_matrix() + sage: Lz = z.operator_matrix() + sage: Lzy = (z*y).operator_matrix() + sage: Luy = (u*y).operator_matrix() + sage: Luz = (u*z).operator_matrix() + sage: Luyz = (u*(y*z)).operator_matrix() + sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz + sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly + sage: bool(lhs == rhs) 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() + fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self) + return fda_elt.matrix().transpose() def quadratic_representation(self, other=None): @@ -626,7 +1505,7 @@ 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] @@ -638,7 +1517,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: D = (x0^2 - x_bar.inner_product(x_bar))*D sage: D = D + 2*x_bar.tensor_product(x_bar) sage: Q = block_matrix(2,2,[A,B,C,D]) - sage: Q == x.quadratic_representation() + sage: Q == x.quadratic_representation().matrix() True Test all of the properties from Theorem 11.2 in Alizadeh:: @@ -647,49 +1526,87 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() + sage: Lx = x.operator() + sage: Lxx = (x*x).operator() + sage: Qx = x.quadratic_representation() + sage: Qy = y.quadratic_representation() + sage: Qxy = x.quadratic_representation(y) + sage: Qex = J.one().quadratic_representation(x) + sage: n = ZZ.random_element(10) + sage: Qxn = (x^n).quadratic_representation() Property 1: - sage: actual = x.quadratic_representation(y) - sage: expected = ( (x+y).quadratic_representation() - ....: -x.quadratic_representation() - ....: -y.quadratic_representation() ) / 2 - sage: actual == expected + sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy True Property 2: sage: alpha = QQ.random_element() - sage: actual = (alpha*x).quadratic_representation() - sage: expected = (alpha^2)*x.quadratic_representation() - sage: actual == expected + sage: (alpha*x).quadratic_representation() == (alpha^2)*Qx + True + + Property 3: + + sage: not x.is_invertible() or ( Qx(x.inverse()) == x ) + True + + sage: not x.is_invertible() or ( + ....: ~Qx + ....: == + ....: x.inverse().quadratic_representation() ) + True + + sage: Qxy(J.one()) == x*y + True + + Property 4: + + sage: not x.is_invertible() or ( + ....: x.quadratic_representation(x.inverse())*Qx + ....: == Qx*x.quadratic_representation(x.inverse()) ) + True + + sage: not x.is_invertible() or ( + ....: x.quadratic_representation(x.inverse())*Qx + ....: == + ....: 2*x.operator()*Qex - Qx ) + True + + sage: 2*x.operator()*Qex - Qx == Lxx True Property 5: - sage: Qy = y.quadratic_representation() - sage: actual = J(Qy*x.vector()).quadratic_representation() - sage: expected = Qy*x.quadratic_representation()*Qy - sage: actual == expected + sage: Qy(x).quadratic_representation() == Qy*Qx*Qy True Property 6: - sage: k = ZZ.random_element(1,10) - sage: actual = (x^k).quadratic_representation() - sage: expected = (x.quadratic_representation())^k - sage: actual == expected + sage: Qxn == (Qx)^n + True + + Property 7: + + sage: not x.is_invertible() or ( + ....: Qx*x.inverse().operator() == Lx ) + True + + Property 8: + + sage: not x.operator_commutes_with(y) or ( + ....: Qx(y)^n == Qxn(y^n) ) True """ 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() - return ( L*M + M*L - (self*other).operator_matrix() ) + L = self.operator() + M = other.operator() + return ( L*M + M*L - (self*other).operator() ) def span_of_powers(self): @@ -700,7 +1617,10 @@ 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()) ) @@ -769,12 +1689,11 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): TESTS:: sage: set_random_seed() - sage: J = eja_rn(5) - sage: c = J.random_element().subalgebra_idempotent() - sage: c^2 == c - True - sage: J = JordanSpinSimpleEJA(5) - sage: c = J.random_element().subalgebra_idempotent() + sage: J = random_eja() + sage: x = J.random_element() + sage: while x.is_nilpotent(): + ....: x = J.random_element() + sage: c = x.subalgebra_idempotent() sage: c^2 == c True @@ -829,40 +1748,102 @@ 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): """ Return the trace inner product of myself and ``other``. + + TESTS: + + The trace inner product is commutative:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element(); y = J.random_element() + sage: x.trace_inner_product(y) == y.trace_inner_product(x) + True + + The trace inner product is bilinear:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: a = QQ.random_element(); + sage: actual = (a*(x+z)).trace_inner_product(y) + sage: expected = ( a*x.trace_inner_product(y) + + ....: a*z.trace_inner_product(y) ) + sage: actual == expected + True + sage: actual = x.trace_inner_product(a*(y+z)) + sage: expected = ( a*x.trace_inner_product(y) + + ....: a*x.trace_inner_product(z) ) + sage: actual == expected + True + + The trace inner product satisfies the compatibility + condition in the definition of 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).trace_inner_product(z) == y.trace_inner_product(x*z) + True + """ 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() -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 @@ -878,16 +1859,21 @@ 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) ] + @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) ] - return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension) + 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(): @@ -907,6 +1893,12 @@ 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. @@ -916,11 +1908,17 @@ 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) @@ -981,6 +1979,54 @@ 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()) + +def _vec2mat(v): + return matrix(v.base_ring(), sqrt(v.degree()), v.list()) + def _multiplication_table_from_matrix_basis(basis): """ At least three of the five simple Euclidean Jordan algebras have the @@ -1001,19 +2047,13 @@ 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 = tuple( vec2mat(b) for b in W.basis() ) + S = tuple( _vec2mat(b) for b in W.basis() ) Qs = [] for s in S: @@ -1026,7 +2066,7 @@ 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) @@ -1047,24 +2087,38 @@ 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) @@ -1081,14 +2135,25 @@ 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() @@ -1100,16 +2165,137 @@ 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 @@ -1117,7 +2303,7 @@ def RealSymmetricSimpleEJA(n, field=QQ): EXAMPLES:: - sage: J = RealSymmetricSimpleEJA(2) + sage: J = RealSymmetricEJA(2) sage: e0, e1, e2 = J.gens() sage: e0*e0 e0 @@ -1132,21 +2318,44 @@ 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) + + fdeja = super(RealSymmetricEJA, cls) + return fdeja.__classcall_private__(cls, + field, + Qs, + rank=n, + natural_basis=T) - return FiniteDimensionalEuclideanJordanAlgebra(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, @@ -1159,36 +2368,110 @@ 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 - """ - S = _complex_hermitian_basis(n) - (Qs, T) = _multiplication_table_from_matrix_basis(S) - return FiniteDimensionalEuclideanJordanAlgebra(field, - Qs, - rank=n, - natural_basis=T) + 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 -def QuaternionHermitianSimpleEJA(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 + + +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. - """ - n = 3 - pass + 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 -def JordanSpinSimpleEJA(n, field=QQ): + """ + @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 + + +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 = @@ -1199,7 +2482,7 @@ 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 @@ -1216,28 +2499,27 @@ 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)) + @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)